Systems Theory Research: Problemy Kibernetiki

SYSTEMS THEORY RESEARCH

IIPOBJIEMbl RßBEPHETßRß

PROBLEMY KIBERNETIKI

PROBLEMS OF CYBERNETICS

SYSTEMS THEOR Y RESEARCH

(Problemy Kibernetiki)

Edited by A. A. Lyapunov

Volume 23

Translated from Russian

® CONSULTANTS BUREAU • NEW YORK-LONDON • 1973

The original Russian text was published by Nauka Press in Moscow in 1970 under the general direction of the Scientific Council on Complex Problems of Cybernetics of the Academy of Sciences of the USSR, Academician A. I. Berg, Chairman. The present translation is published under an agreement with Mezhdunarodnaya Kniga,

the Soviet book export agency.

Library of Congress Catalog Card Number 68-15025

ISBN 978-1-4757-0081-7 ISBN 978-1-4757-0079-4 (eBook) DOl 10.1007/978-1-4757-0079-4

©1973 Consultants Bureau, New York A Division of Plenum Publishing Corporation

227 West 17th Street, New York, N. Y.10011

United Kingdom edition published by Consultants Bureau, London A Division of Plenum Publishing Company, Ltd.

Davis House (4th Floor), 8 Scrubs Lane, Harlesden, London, NWI0 6SE, England

All rights reserved

No part of this pUblication may be reproduced in any form without written permission from the publisher

CONTENTS

THEORY OF CONTROL SYSTEMS

On the Completeness of Functions Having Delays. . • . • . . • • . . • . . . . . • • • . • . • • . • . 3 L. A. Biryukova and V. B. Kudryavtsev

Asymptotically Stable Distributions of Charge on Vertices of an n- Dimensional C ube • . . . . . • . • • • • • • • • • • • • • . . • • • • . . • . • • • 25

V. K. Leont'ev On Networks Consisting of Functional Elements with Delays • • • • . • . • • . • • • • • • • • . 43

0. B. Lupanov Proof of Minimality of Circuits Consisting of Functional Elements • . • • • • • • • • • • • . • 85

N. P. Red'kin Full Test for Nonrepetitive Switching Circuits. • • • . • • . • . . • . . . • • • . . • . • • • • . • • 105

Kh. A. Madatyan On Finite Model Schemes Having Discrete Functioning • . • • . • . . . . . • • • . . . . • • • • . 121

Yu. A. Vinogradov On a Certain Generalization of Finite Automata, which Forms

a Hierarchy Analogaus to the Grzegorczyk Classification of Primitively Hecursive Functions. . . . . • . . . • • • • . . • • • . . • • • . . . . • 129

V. A. Kozmidiadi General Linear Automata. . . . • . • . • . • . • • • . . • . • . • • • • • . . • • • • • . • • • • • • • • • 179

A. A. Muchnik Distinguishability of Infinite Automata • . . . . . . • • • • • • • • • • • . • . • • • . • • • • • • . • • 219

Ch. Faisi

PROGRAMMING

On Algorithm Schemata WhichAre Defined on Situations . • . . . . • • • • • • • • . • . • • . • . 225 H. I. Podlovchenko

CONTROL PROCESSES IN LIVING ORGANISMS

On the Problem of Modeling for an Evolutionary Process with Regard to Methods of Selection • . • • • • • • . • . • • • • • • • • • • • • • • • • • • • • • • • . • 261

T. I. Bulgakova, 0. S. Kulagina, and A. A. Lyapunov On the Dynamics and Control of the Age Structure of a Population. • • • • . • • . • • • • • • . 273

L. R. Ginsburg On the C ontrol of C ardiac Rhythm. . • • . . • • • • • • • . • • • • • • • • • • • • • • • • • • • • • • • 287

Yu. A. Vlasov and A. T. Kolotov

V

vi CONTENTS

BRIEF COMMUNICATIONS

A Note on Deterministic Linear Languages • • • • • • • • • • • • • • • • . • . • • • • • • • • • • • . 295 A. Ya. Dikovskii

Nonrecurrent Codes with Minimal Decoding Complexity • • • • • • • • • • • • • • • • • • • • . • 301 A. A. Markov

Realization of Disjunctions and Conjunctions in Monotonic Bases • • • • • • • • • • • • • • • • 305 :E. I. Nechiporuk

C ircuits to R.aise ReHability. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 309 M. M. Rokhlina

THEOR Y OF CONTROL SYSTEMS

ON THE COMPLETENESS OF FUNCTIONS HA VING DELA YSt

L. A. Biryukova and V. B. Kudryavtsev

Moscow

This paper studies completeness conditions for a certain class of automata without feedbacks - so-called functions with delays [1]. The problems considered in the paper are related to investigations begun in [1] and to an obvious degree are a continuation of them.

Assurne 1F2 is the set of alllogic-algebra functions having delays not exceeding l and rol ~ 1 P 2 • The set rol ~ 1 P 2 is called l-c o m p 1 e t e if by means of "synchronous s uperposition" operations one may obtain any logic-algebra function having the delay l by starting from the elements of the set ~m . The paper inv.estigates the conditions which l-complete systems must satisfy. The functional system studied, as became clear, has numerous interesting properties.

In particular, it turned out that in the general case not every class in 'Pz can be expanded to an l-precomplete class, and therefore the criterion of l-completeness may not be formulated solely in terms of l-precomplete classes. This fact already holds for l == 1. However, in this case we were nevertheless able to show that a finite system is complete when and only when it does not belong to a certain finite number of 1-precomplete classes and three rigorously increasing chains of closed classes, none of which is contained in any of the 1-precomplete classes. From this, in particular, there derives the existence of an algorithm which establishes 1-completeness of any finite system of functions having delays. It is shown that in the general case a finite l-complete subsystem cannot be isolated from just any Zcomplete system, and that the power of the set of closed classes in 1 P2 is equal to the continuurn. The case of l = 1 is investigated in particular detail, although many results may easily be carried over to the case of arbitrary Z.

In the paper extensive use is made of the results of [1, 2, 3] and all notations which are not defined may be found there.

§ 1. Basic Notions

Assurne X= {x1, x2 , ••• , Xn, .•• } is the set of Boolean variables that take 0 or 1 as their values. Sometimes we shall use the notation x, y, z, ... to denote letters from X. Let us use P2 to denote the set of all logic-algebra functions which depend on variables from the set X. Hereafter logic-algebra functions shall be called simply functions for brevity. As-surne T is a parameter which takes one of the values 0, 1, ... , l. We shall call it a d e 1 ay. Let us consider the set 1 h of all pairs of the form (j (x1, ... , xn), T), where f (xt, ... , xn) E P 2

and -r E {0, 1, ... , l} . Sometimes the pair (j, T) shall be called a funct io n f ha ving

t Original article submitted November 15, 1968. 3

4 L. A. BffiYUKOVA AND V. B. KUDRYAVTSEV

the delay T. Weshall consider two pairs (j, r 1) and (cp, r 2) tobe equal and denote that fact by (j, T 1) = (cp, r 2) ü T 1 = r 2 and the functions f and cp düfer only, perhaps, by fictitious variables. Weshall assume ahead of timethat with the stipulation of the pair (f, r), all pairs equal to it are stipulated simultaneously. In the set 1P2 we shall inductively introduce synchronous-superposition operations as follows.

Definitions . 1. Assurne that we have the pair

(f (x1, X2, ... , Xi-b X; 1 Xi+h ... 1 Xn), <) (1)

andthevariablexi;thenthe pair (j(x 1 , ••• , xi-t• xi, xi+t• •.. , xn), T) is obtained from the pair (1) by means of synchronous-superposition operations (the rule for redesignating the variables).

2. Assurne we have the pairs

(f(x1, ... , Xn), <), (g;1 (x1, ... , Xm), 0), (g;2 (x1, ... , Xm), 0), ... , (gir (x1, ... , Xm), 0),

i<.ii<.n for 1-<.j<.r and i1 <i2 < ... <i,. (2)

Thenthe pair (f(G 1, G 2 , ••• , Gn), r), whereGkdenoteseitheravariableora functiongk,k=l, ••. ,n, is obtained from (2) by means of synchronoussuperposition operations (the rule of substituting functions with zero delays into a function with delay).

3. Assurne we have the pairs

(3)

then the p a i r ( f ( g 1 , g 2 , ••• , g n) , T ) , where T = T 1 + T 2 and T :::5 l , i s ob t a in e d from the pairs (3) by means of synchronous-superposition operations (the rule of substituting functions with delays into a function with delay).

Rem a r k . Let us note the following two important facts affecting the synchronoussuperposition operations. First, if the system m s 1P2 consists solely of functions having a zero delay, then in essence the operations 1-3 which we introduced for elements from ~J1 are completely analogous to the "conventional" Superpositionoperations in the class of functions P 2• Second, we wish to emphasize the fact that operation 3 together with the substitution öf some pair of the form (cp, r 2), r 2 > 0 in place of a certain variable of the function f in the pair (f, T 1) "compels" us to replace each of the remaining variables of the functions f by a pair having a delay which is likewise equal to T 2, the condition T 1 + T 2 :::::; l being valid under these conditions.

Assurne m s 1 P2• The set [lml is called the c 1 o s ur e of m ü it contains those and only those pairs which are obtained from the pairs of the set Wt by means of a finite nurober of applications of synchronous-superposition operations.

The set m is called closed ü lm=[lm].

Definition. Theset m iscalledl-complete if [lmJ containsallpairshaving delays equal to l.

Definition. The set 1m is called l-precomplete if

1) m- is not l-complete;

2) lmU{(g, •)} is l-complete for any pair (g, T) from 1P2 suchthat (g, •)U,R. It is obvious that an l-complete class is closed.

COMPLETENESS OF FUNCTIONS HAVING DELAYS 5

As is weil known, e ach function f (x1, ••• , x 11) roay be stipulated (and uniquely at that) by roeans of a Zhegalkin polynoroial; i.e., the equation

holds, where C;1 , ...• i., and d are equal to zero or unity, and roultiplication and addition is

carried out roodulo two. The d e g r e e o f t h e f u n c t i o n f is called the degree of its Zhegalkinpolynoroial. The degree of linearity of the function f is calledthe nurober of linear terros in its Zhegalkin polynoroial. The rank of the function f is called the nurober of terros in its Zhegalkin polynoroial.

Let us introduce the following notation for the sets which we shall require further on [2' 3]:

A all a-functions; B all ß -functions; r all y -functions; L:l all o -functions; C2 all functions f (x1, x2, ... , x11), such that f (0, ... , 0) = 0; C3 all functions j(x1, x2, ... , x11), suchthat /(1, ... , 1) = 1; A1 all roonotonic functions; D3 all self-dual functions; L1 alllinear functions; Y all even functions; A1 all functions which are negations of roonotonic functions; D1 allself-dual a-functions; F;' all a-functions which satisfy condition (A"");

Fä' all functions satisfying the condition (A"");

Fß all roonotonic a-functions satisfying the condition (A"");

F7 all roonotonic functions satisfying the condition (A"");

F~ all a-functions satisfying the condition (A~), 11 > 2; F~ all functions satisfying the conditions (A~), 11 > 2; Ff all roonotonic a-functions satisfying the condition (Alt); Fif all roonotonic functions satisfying the condition (A~); Fi all a-functions satisfying the condition (a"");

F~ all roonotonic a-functions satisfying the conditions (a~);

F;' all roonotonic a-functions satisfying the condition (a"");

Ff all a-functions satisfying the condition (a~); F 4' all functions satisfying the condition (a"");

F~ all functions satisfying the condition (a~); F3 all roonotonic functions satisfying the condition (a"");

F~ all roonotonic functions satisfying the condition (a~t); A2 all roonotonic a-functions and ß -functions; A3 all roonotonic a-functions and y -functions; A4 all roonotonic a-functions; D2 allself-dual roonotonic functions; 0 1 all functions equal to x, and all functions derived froro it by redesignating the

variables without identification; 0 2 all functions equal to the function 1; 0 3 all functions equal to the function O;

6

04

05

Os

07 Oa

09

st Sa s5 Sa p1 Pa p5 Pa L2 La L4 L5

L. A. BIRYUKOVA AND V. B. KUDRYAVTSEV

all functions equal to the functions x or x, and all functions derived from them by redesignating the variables without identüication; all functions equal to the functions 1 or x, and all functions derived from them by redesignating the variables without identüication; all functions equal to the functions 0 or x, and all functions derived from them by redesignating the variables without identification; all functions equal to the functions 0 or 1; all functions equal to the functions 0, 1, or x, and all functions derived from them by redesignating the variables without identification; all functions equal to the functions 1, 0, x, or x, and all functions derived from them by redesignating the variables without identüication; alllogical sums; all logical sums and all functions equal to 1; alllogical sums and all functions equal to O; alllogical sums and all functions equal to 0 or 1; all logical products; alllogical products and all functions equal to O; alllogical products and all functions equal to 1; alllogical products and all functions equal to 0 or 1; all linear a- and ß -functions; all linear a- and y-functions; alllinear a-functions; all linear self -dual functions.

From the basic Posttheorems [2] it follows that each of the sets enumerated above, with the exception ofthe sets B, r, d, Y, A1, is closed relative to Superposition operations, and no closed sets exist which are different from them.

Note that the notation for the sets, with the exception of the sets A, B, r, d, Y, Ä1, coincides with the notation in [2]. The sets A, B, r, d are denoted in the same way as they are in [1].

The notation which has been introduced shall be used in a certain modified form to denote the subsets from 1 P2. Thus, assume IR is any of the sets of functions just examined; then we use 'tln to denote the set of all pairs (f, T) such that I E ~.

We shall be interested in the following problem. Assurne there is a finite system of pairs m c 1 P2. It is required to clarüy the conditions under which this system has the property of l-completeness.

The paperwill carry out a detailed consideration of the case in which l = 1. Namely, a criterion of 1-completeness will be obtained which is formulated in terms of 1-precomplete classes and certain expanding chains of closed classes.

Assurne the set Wl s; 1 P2, and assume

(/1> 0), (!2, 0), ... ' (cpj, 1), (cp2, 1), ... ' ('!Jt, l), ('IJ2, l), ...

are all functions with 0, unitary, etc., delays, respectively (certain of these sets may turn out tobe empty), which are contained in m. In order to denote Wl it will sometimes be convenient for us to use the following notation:

{(f;,-.D), (cpj, 1), ... ' ('!Jh, l)},

where the parameters i, j, ... , k run the gamut of values from a natural series.

COMPLETENESS OF FUNCTIONS HAVING DELAYS

§ 2. Certain Necessary Conditions of 1-Completeness

of Finite Systems

Lemma 1. If ~ is a finite 1-complete system, then it necessarily contains the following pairs: (j, 0) and (cp, 1), where f and cp arenot constants.

7

Pro o f. In accordance with the conditions of the theorem, Iet us consider two cases.

a) Assurne condition 1) is not fulfilled. Then it is obvious that by using pairs from l8, one cannot obtain any pair of the form (j, T), where f depends essentially on a !arger number of variables than any of the functions appearing in the pairs of the system ~ , by means of synchronous-superposition operations.

b) Assurne that condition 2) has not been fulfilled. It is obvious that a pair of the form (cp, 1) must appear in ~. Assurne cp = const. Since functions having a unitary delay may not be substituted into one another, it follows that in order to obtain functions with unitary delays, one may substitute only constants with a unitary delay in place of allvariables in a function having a zero delay, or vice versa. It is obvious that as a result we again obtain a constant having a unitary delay; i.e., ~ is not a 1-complete system, which is what it has been required to prove.

Lemma 2. If l8 is a finite 1-complete system, then the pair(/, 0) ~ 0L1 i s c o n t a i ne d in i t.

Proof. Let us assume that \B, having zero delays, contains only linear functions.

Further, it is not difficult to see that the Substitution of a linear function having a zero delay into the pair (1/J, 1) does not raise the degree of 1/Ji furthermore, the Substitution of functions having a unitary delay into a linear function having a zero delay yields a function with a unitary delay whose degree does not exceed the maximum degree out of the degrees of the substituted functions. From this it follows that the degree of any function having delay from l8 has a degree no !arger than r, which it was required to prove.

Lemma 3. If [~] is a finite 1-complete system, then l8 contains the pairs (/, 0)~ 0P6 and (g, 0)~ 086 •

Pro o f. By virtue of the duality principle it is sufficient to prove, for example, that l8 contains the pair (/, 0) ~ 0P6 • Let us assume that the Iemma is invalid (i.e., a 1-complete finite system m, exists which contains only elements of the set 0P 6 as functions having zero delay). We shall show that in this case the system l8 cannot be 1-complete.

The set [l8] may be partitioned into two subsets which may, in general, cross. In the first subset [lS], we shall include all functions having delays from [l8]' which are obtained by the substitution of a function having zero delay into a function having unitary delay from m. Assurne the highest degree of linearity and the highest rank of functions appearing in pairs of the set m, are equal to m 2: 1 and r 2: 1 (r 2: m), respectively. It is obvious that each function contained in a pair from the set [l8]', has a degree of linearity no higher than r.

In the second subset [l8]" we shall include all functions having delays from [Q3], which are obtained by substitution of the functions having unitary delays from [Q3]' into functions having zero delay from m. It is obvious that [Q3]' U PE]"= [lEJ. We shall show that the set (lS]" cannot contain the pair (j, T), where j is an 0!-function whose degree of linearity is higher than r. Note that the set [lEJ contains only the pairs (0, 0), (1, 0), and pairs of the form (x1• x2 • ... • Xn, 0), whose degree of linearity is obviously no higher than r, as functions having zero delay. By virtue of this it is sufficient to consider functions having a unitary delay solely


from [18]". Note that each pair (cp, 1) E [~3]", where cp differs from a constant, may be represented in the form (j 1 • f 2 • ••• fs, 1), where {!;, 1) E [l8)', i = 1, ... , s. But proof of the fact that each a-function cp having delay 1 which belongs to [Q3]", has a degree of linearity no higher than r will be carried out by induction with respect to the nurober s. For s = 1 our statement is obvious. Assurne that it is true for all s :::; k, k::::: 1. We shall show that our statement is true for s = k + 1. Let us consider the pair (fd2· .•. ·IR.· fk+i> T) • We denote the product of the first k functions in this pair by g. Relative to the functions g and fk+t the following distribution of values of free terms c and d is possible in their Zhegalkin polynomials.

1) c = 0, d = 0. In this case, g·fk+1 is an a-function only when g and h+1 are themselves a-functions. The latter, according to the induction proposition, have a degree of linearity:::; r. From this it follows that the degree of linearity of g·f~<+t is no higher than r.

2) c = 0, d = 1. In this case, g. fk+t is an a-fWlCtion only when g is an a-function and fk+1 is a ß-function. Since, according to the induction proposition, the degree of linearity of g

is no higher than r, it follows that the degree of linearity of g · fn+t is no higher than r.

3) c = 1, d = 0. This case is completely analogous to case 2).

4) c = 1, d = 0. In this case g. fk+ 1 cannot be an a-function.

Thus, [l8] does not contain a singlepair (/, r) suchthat f is an a-function whose degree of linearity is higher than r. This contradicts the postulated 1-completeness of the system 18, which is what it was required to prove.

Let us consider the system N containing the following 19 sets of functions with delays (the symbol iQ denotes the set of all pairs of the form (j, i), where

911 = (0P2 U 1C), where C = {0, 1}, 9h=(0AU 1BU 1f), \Jh = (°Ca IJ 1Ca),

\R4 = (0 At U 1A1), ffi5=(0A4 U1A U1B U1f), \Ra= (OA4 ijlß IJ lß IJ lr),

\R1 = (0At U 1A1), ffis = (0Da U 1Da), \R9 = (0Da U 1Y),

\R10 = (0D1 IJ 1A IJ 1ß),

ffiu = (°F: IJ 1A IJ 1f U 1ß), \R12= (0A IJ 1ß IJ 1f}, \R1a = (0A2 IJ lß IJ 1f IJ (1, 1)),

\R 14 =(0A2 U1AU 1BU (0, 1)),

\R15 = (°C2 U 1C2),

iRta= (°F~ U 1A IJ 1B IJ 1ß),

\R11 = (0A IJ 1ß IJ 1B), ffits = (0Aa IJ 1A U 1 f U (1, 1)), iRt9=(0AaU 1ßU 1BU (0, 1)).

In §3 it will be proved that each of the sets of the system N is 1-precomplete.

§ 3. Proof of 1-Precompleteness of Classes

from the System N

The closure of each of the classes of the system N easily derives from the definition of these classes. Let us show that for any \R; and any pair (!, T) E 1F2 , suchthat {!, T) tf \R 1,

the set \R; U {(/, T)} is 1-complete. Let us consider each of the cases separately.

Lemma 4. The set \R1=(0P2U1C) is 1-precomplete.

Pro of. The lemmawill be proved if we show that the set im= \R 1 U {(!1 1)} , where f ~ c is 1-complete. Forthis purpose it is obviously sufficient to show that [?ffi] ~ 0P2 U {(x, 1)}. Since in the pair (j, 1) the function f (x1, ••• , xn) is different from a constant, it follows that one can find a variable (without loss of generality it may be assumed that this is x1) on which


the function depends essentially (i.e., for certain collections a :::: (0, 012, ... , an) and a' :::: (1, a 2, ... , a 11) the relation f(a)=l=f(a'). holds). Let us substitute the pair .(a;, O)E(0P2 U1C) into the pair (j, 1) in place of the variable xi, i ;t 1, while the pair (x, 0) is substituted in place of the variable x1 if f (a} = 0, and the pair (x, 0} if f (a} = 1 (obviously, the pairs (x, O} and (x, 0} belong to (0P2 U 1C)). It can easily be seen that as a result of this Substitution we obtain a pair equal to the pair (x, 1}, which is what it was required to prove.

Lemma 5. The set ffi2=( 0AU 1BU 1f) is 1-precomplete.

Pro of. Let us show that the set im= \R2 U {(/, T)} , where (/, T) 1 (0A U 1B U 1f) is 1-complete. First we show that in the case in which the pair (f, T) is equal to the pair (x, 1)

9

the system im will be 1-complete. The sorting of all remaining possible pairs (j, T} shall be reduced to this basic case. Thus, let us show that the system im'= (0A U 1B U ir) U {(x, 1)} is 1-complete. For the proof we note that for any function <p(x1, ... , x 11} one can find an a-function zp(y1, y2, x1, ... , X 11} of the form Y1 V qJ·Y2· Let us consider the pair ('ljJ, 0) E (0A U 1B U 1f). Let us substitute the pair (0, 1} into this pair instead of the variable y 1, the pair (1, 1} instead of the variable y2, and the pair (xi, 1) instead of the variable xi. Each of the enumerated pairs belongs to [il.J}'J. As a result we obviously obtain the pair (<p, 1) (i.e., the system W}' is 1-complete}.

Let us consider the logically possible cases for the pair (j, T).

1} (j, T) is an a-function having unitary delay. Since the pair (j (x, ... , x}, 1) is equal tc the pair (x, 1), then everything reduces to the basis case.

2} (j, T) is an o -function having unitary delay. Note that (0 AU 1B U 1 f) contains the pair (y1 V y2i, 0). Let us substitute the pair (0, 1} into this pair instead of the variable y1, the pair (1, 1} instead of the variable y2, and the pair (j (x, ... , x), 1), which is obviously equal to the pair (X, 1}, instead of the variable z. It can easily be seen that as a result we obtain the pair (x, 1). Thus, this case has been reduced to the basic one.

It remains for us to consider the case in which (j, T} is either a ß-function having zero delay, a y-function having zero delay, or a o-function having zero delay. The proof of 1-completeness in each of these cases derives from the following concepts. The set \R2 contains the pair (xV ii, 1). Let us substitute the pair (f (x, ... , x), 0) into this pair instead of the variable y, where the function f is either a ß- or a o-function; as a result we obtain the pair (x, 1}. If the function f is a y-function, then we Substitute the pair (j (x, ... , x), O) into the pair (x · ii, 1) E (0 A U 1 B U 1 f) in place of the variable y; as a result we obtain the pair (x, 1). Therefore, the consideration of these cases also can be reduced to the basic case, which is what it was required to prove.

Lemma 6. The set ffia=(°C3 U1C3) is 1-precomplete.

Proof. Let us show thatthe set im=\R3 U{(f, T)}, where (/, T)1~(°C3 , 1C3) is 1-complete. Since it is obvious that 1C3 U 1Cu where 1c3 denotes the set of all pairs (j, 1} such that (t, 1) ecs is a 1-complete set, it follows that for proof of 1-completeness of the system it is sufficient to show that either the pair (x, 1) E [im] or the pair (x, 0) E {im].


1} (j, T} is a ß-function with a zero delay.

The pair (f (x, ... , x}, 0} is obviously equal to the pair (1, 0}.

Note that the set ~ contains the pair (x • y, 1}. Let us replace the variables x by the pair (1, 0} in this pair, and the variable y by the pair (x, 0). It can easily be seen that as a result we obtain the pair (x, 1}.

10 L. A. BIRYUKOVA AND V. B. KUDRYAVTSEV

2) (f, T) is a o-function having a zero delay. It is obvious that the pair (f (x, ... , x), 0) coincides with the pair (i, 0).

3) (f, T) is a ß-function with a unitary delay. The pair (f (x, ••• , x), 0) is equal to the pair (1, 1). Note that the set in contains the pair (x • y, 0). In place of the variable x we substitute the pair (1, 1) in this pair, andin place of the variable y the pair (x, 1). As a result we obtain the pair (X, 1).

4) (f, T) is a o-f~ction having a unitary delay. It is obvious that the pair (j (x, ... , x), 0) coincides with the pair (x, 1), which is what it was required to prove.

Lemma 7. T he s et \Rd0A1 u 1A1) i s 1- pre complete.

Proof. Let us show that the set ?ffi=\R~U{(f, •)}, where (!, •)Ef(0A1 U1A1) is 1-complete. First we shall show that for the case in which the pair (/, T) is equal to the pair (x, 1), the system m will be 1-complete. But the sorting of all remaining possible pairs (f, T) shall be reduced to this basic case. Thus, let us show that the system ?m' = \R~ u (~ 1) is 1-complete. Since the pairs (x·y, 0), (xVy, 0), (x, 1) and (x, 1) belong to ?m', it follows that by applying synchronous-superposition operations to them we may obviously obtain the perfect disjunctive normal form with a unitary delay for any function cp(x1, ... , xn>• Therefore, the system ~Jt' is 1-complete.

Let us consider the logically possible cases for the pair (f, T).

1) (f, T) is a nonmonotonic function having a zero delay. Since the function f t;x1, ... , xn) is nonmonotonic, it follows that, as is well known, one can find collections a(a1, 0!2, ... , Q!i-1• o, Q! i+t• ... , O!n) and a' (0!1, 0!2, ... , Q!i -1• 1, Q!i +t• ... , O!n) which are adjacent in the i-th component and aresuchthat f(a) = 1, f(a') = o hold. Inthepair (f, T) let us replace the variable xj, j ~ i by the pair (a" 1) E \R 4 and the variable xi by the pair (x, 1) E \R~. It can easily be seen that as a result of the substitution we obtain a pair equal to the pair (x, 1). This case has been reduced to the basic one.

2) (j, T) is a nonmonotonic function having a unitary delay. This case obviously can be reduced to the basic one also, which is what it was required to prove.

Lemma 8. The set ffis=( 0A4U 1AU 1BU 1r) is 1-precomplete.

Proof. Letus showthat ?ffi=\R5 U{(f,T)} ,where (f,T)Et\R5 is 1-complete. Itisobvious that for this it is sufficient to show that either the pair (X, 0) or (i, 1) belongs to [?m].

Let us consider the logically possible cases for the pair (f, T).

1) (f, T) is a nonmonotonic function having a zero delay. From it, as has already been shown in Lemma 7, one can obtain the pair (X, 1) by means of the pairs (x, 1), (1, 1), (0 1)E?ffi.

2) (f, T) is a function identically eq ual to unity and having a zero delay. Substituting the pair (1, 0) for the variable x and the pair (x, 0) for the variable y in the pair (x· Y. 1) E m, we obtain the pair (i, 1).

3) (f, T) is a function which is identically equal to zero and has a zero delay. Substituting the pair (0, 0) for the variable x and the pair (x, 0) for the variable y in the pair (x V y, 1) o,n , we obtain the pair (X, 1).

4) (/, T) is a ö-function having a unitary delay. It is obvious that the pair (/ f;x, ••• , x), 1) coincides with the pair (i, 1), which is what was required to prove.

The validity of the following statement is similarly established.


Lemma 9. The set \R 6 =(0A4 U 1~U 1BU 1r) is 1-precomplete.

Lemma 10. The set ffi 7 =(0A1 U1L) is 1-precomplete.

11

Pro of. Let us show that the set m = \Jh U (/, -r) , where (/, -r) ~ \Jh, is 1-complete. For the case in which the pair (f, r) is equal to the pair (x, 1), the system m will obviously be 1-complete .. The sorting of all remaining possible pairs (j, r) shall be reduced to this basic case.

Let us consider the logically possible cases for the pair (j, r).

1) (f, T) is a nonmonotonic function having a zero delay. From it one may obviously obtain the pair (x, 1) using the pairs (0, 1), (1, 1), (x, 1) E mc.

2) (j, T) is not a negation of a monotonic function having a unitary delay. If the function f (x1, ••• , xn) is either a ß-function, an a-function, or a y-function, then it is obvious that [ID1] contains the pair (x, 1). If the function f (x1, ••• , Xn) is a o-function, then one can obviously find two collections ä = (a 1, ••• , a 11 ) and Oi1 = (a~, ... , a~1) suchthat Oi < ä1 and j(a) < j(ä1).

Let us partition the variables (x1, ... , x11 ) by collections Oi and ä1 into no more than three groups. To the first group we assign the variables x i for which ai = ai = 0, while for the second group we assign the variables x i for which a i= aJ = 1, and to the third group we assign the variables Xk for which O!i = 0, ar = 1. In this case in the pair (j, T) let US replace the variables from the first group by the pair (0, 0), the variables from the second group by the pair (1, 0), and the variables from the third group by the pair (x, 0). It is obvious that as a result we obtain the pair (x, 1), which is what it was required to prove.

Lemma 11. The set lRs=( 0D3 U1D3) is 1-precomplete.

Pro of. Let us show that the set m = \Rs U {(!, -r)} is 1-complete, where (j, -r) ~ \R8 •

Let us consider the logically possible cases for the pair (j, r).

1) (j, T) is a nonself -dual function having a zero delay. Since the set D3 is precomplete in P 2, it is obvious that the system m is 1-complete.

2) (j, T) is a nonself-dual function having a unitary delay. Since from any nonselfdual function one may obtain 1 or 0 by substituting x and x for certain variables, it is obvious that [IDCJ contains the pairs (1, 1), and (0, 1). Note that for any function cp(x1, ... , x 11) one can find a self-dual function f (x, x1, ... , x11 ) in the form (x V cp (x1, ... , xn)) · (x V ij) (x~> ... , xn)). Let us consider the pair (f, 0) E \R8• In this pair let us replace the variable x by the pair (0, 1), and the variable xi by the pair (xi, 1). As a result it is obvious that we obtain the pair (cp, 1), which is what it was required to prove.

L e m m a 12 • T h e s e t \R9 = (0 Da U 1 Y) i s 1 - p r e c o m p l e t e .

Pro o f. Let us show that the set m = \R9 U (!, -r) , where (f, -r) Et \R9 is 1-complete. For the case in which the pair (j, T) coincides with the pair (x, 1), the system m, will obviously be 1-complete.

Let us enumerate the logically possible cases for the pair (f, T).

1) (j, T) is a nonself -dual function having a zero delay. In this case the system m, is obviously 1-complete.

2) (j, r) is an odd function having a unitary delay. Since the function f (x1, ... , xn) is not even, it follows, as is wen known, that one can find at least two collections a = (a 1• ... , O!n)

and a' = (a 1• ... , ä;1 ) on which the function j (x1, ... , X 11) takes opposite values c and c. Let US

partition the variables x1, ... , xn into two groups in accordance with the collection ä. To the


first group let us assign the variables xi for ai == 0, while to the second group let us assign the variables xi for which a i == 1. In the pair (j, T) let us replace the variables from the first group by the pair (x, 0), while the variablesfrom the second group are replaced by the pair (x, 0). As a result we obtain the pair (g(x), 1), where g(x) is either x or x; i.e., [lmJ contains the pair (x, 1). Thus, this case has been reduced to the one considered above, which is what it was required to prove.

Lemma 13. The set ffi 10 =(0D1 UIAU1ß) is 1-precomplete.

Proof. Let us show that the set lm=ffi 10 U (f, <), where (f, •)1\R 10 is 1-complete. First let us establish the fact that for the case in which [lm] contains the pairs (0, 1) and (1, 1), the system lm is 1-complete. Note that for any function cp(x1, ... , x 11 ) one can find a self-dual a-function f (x, y, x1, ... , x 11) in the form (cp (x1, ••• , xn) V x) y V ((Ji (x1, ••• , xn) V y) x. In the pair (j (x, y, x1, ... , x 11), 0) let us replace the variable x by the pair (0, 1), the variable y by the pair (1, 1), and the variable xi by the pair (xi, 1). Each of these pairs belongs to [lm]. As a result we obviously obtain the pair (cp, 1); i.e., the system lm is 1-complete in this basic case.

Let us consider the logically possible cases for the pair (j, T), and let us show that each of them will reduce to the basic case.

1) (f, T) is a nonself-dual function having a zero delay. From this pair one can obviously obtain either the pair (0, 1) or the pair (1, 1) by means of the pairs (x, 1) and (x, 1). Since the pair (x+y+z, 0)E 0D1, then,byreplacing the variable x by the pair (1, 1), the variable y by the pair (x, 1), and the variable z by the pair (i, 1) in it we obtain the pair (0, 1); or, by replacing ~e variable x by the pair (0, 1), the variable y by the pair (x, 1), and the variable z by the pair (x, 1) in it we obtain the pair (1, 1).

2) (j, T) is a ö -function having a zero delay. In this case the system lm is 1-complete, since it contains the 1-precomplete class ffis and does not coincide with it.

3) (j, T) is a y-function having a unitary delay.

4) (j, T) is a ß-function having a unitary delay.

Cases 3) and 4) can immediately be reduced to the first case. The lemma has been proved.

Lemma 14. The set lRu=(°F!U-1AUifU 1ß) is 1-precomplete.

Let us Show that the set lm = ffiu U (/, 't) , where (f, <) ~ \R 11 is 1-complete. Let us first establish the fact that for the case in which the pair (j, T) is equal to the pair (1, 1) the system lm will be 1-complete. Sorting of all the remaining logically possible cases for the pair (j, T) will be reduced to this basic case. Let us show that the system lm' = \R 11 U (1, 1) is 1-complete. Note that for any function cp(x1, ... , ~1 ) one can find a ß-function of the form cp(x1, ... , x 11) V (z+y+ 1) andafunction f(x,y,z,x1 , ••• ,xn)EF~ oftheform x(cp(x1 , ••• ,xn)V(z+y+1)). In the pair (/ (x, y, z, xi> ... , xn), 0) E \R 11 let us replace the variable x by the pair (1, 1), the variable y by the pair (x, 1), the variable z by the pair (x, 1), and the variable xi by the pair (xi, 1). It is obvious that as a result we obtain the pair (cp, 1); therefore, in this basic case the system is 1-complete.

Let us consider the logically possible cases for the pair (j, T):

1) (j, T) is a ß-function having a unitary delay,

2) (j, T) is a y-function having a zero delay,

3) (j, T) is a ß-function having a zero delay,

4) (f, T) is a ö-function having a zero delay.


Each of these cases can obviously be reduced immediately to the basic case.

5) (j, T) is an a-function having a zero delay, which does not belong to °Fg. In this case it is obvious that a-functions belong to [lm] , which have a zero delay and among which there are a-functions which do not satisfy the condition (A2), a nonself-dual a-function, and, consequently [2], a a-function which does not satisfy the condition (a 2); then [2] [lffi] contains the pair (x V y, 0) , and from it we obtain the pair (1, 1) by substituting the pairs (x, 1) and (X, 1). Consequently, this case also can be reduced to the basic one, which is what it was required to prove.

Lemma 15. The set ffit2=( 0AU 111U 1f) is 1-precomplete.

Pro o f. Let us show that the set lm = \Jt11 U (f, <), where (!, <) ~ \J\, is 1-complete. First let us establish the fact that in the case in which the pair (j, r) is equal to the pair (1, 1), the system lm will be 1-complete. This case is called the basic one.

Thus, let us show that the system lm' =ffi 12 U (1, 1) is 1-complete. Note that for any function cp(x1, ... , X11 ) one canfindana-function j(x, y, x1, ... , x11 ) of the form xV yrp(xt. .. . , "Xn). In the pair (j, 0) let us replace the variable x by the pair (0, 1), the variable y bythe pair (1, 1), and the variable xi by the pair (xi, 1). Allthesepairs belang to [ID1']. It can easily be seen that as a result we obtain the pair (cp, 1).


1) (j, r) is a ß-function having a unitary delay,

2) (j, T) is a o -function having a zero delay,

3) (j, T) is a y-function having a zero delay,

4) (j, r) is a ß-function having a zero delay.

It can easily be seen that [lm'J contains the pair (1, 1) in each of these cases.

5) (j, r) is ana-functionhavingaunitary delay. In this case the set [lffi] is 1-complete, since i!J1 contains the 1-precomplete set \J\10 and does not coincide with it; this is what it was required to prove.

Lemma 16. The set \Jt 13 =(0A2 U111U 1fU(1, 1)) is 1-precomplete.

Pro of. Let us show that the set lm = \Jt 13 u (f, <), where (!, •) ~ \Jt 13 , is 1-complete.


1) (j, T) is a ß -function having a unitary delay, which differs from a constant. In this case it is obvious that the pairs (x·y, 0), (x V y, 0), (x, 1), (x, 1) belang to (lffi]; consequently, for any function cp (x1, ... , x 11) one may obtain a perfect disjunctive normal form having a unitary delay (i.e., the system lm is 1-complete).

2) (j, T) is an a-function having a unitary delay. This case is analogaus to case 1).

3) (j, T) is a nonmonotonic function having a zero delay. In this case it is obvious that the pair (x, 1) E [lm], and consequently it is analogaus to case 1).

4) (j, T) is a function which is identically equal to zero and has a zero delay. In this case the system m is 1-complete, since it contains the 1-precomplete set ffi 7 and does not coincide with it; this is what it was required to prove.

The validity of the following postulate is established similarly.

Lemma 17. The set ffi14=(0A2LJlALJlBLJ(O, 1)) is 1-precomplete.


Since the classes ffits- ffit9 are dual relative to the classes ffia. ffiu, ffit2• ffi14, ffita , it follows that by virtue of the duality principle its 1-precompleteness derives from Lemmas 6, 14, 15, 17, 16.

§ 4. The Basic Theorem on 1-Completeness of Finite

Systems of Functions Having Delays

We shall show that the following basic theorem on 1-completeness holds.

Theorem 1. In order for the finite system l8s;; 1P2 tobe 1-complete it is necessary and sufficient that it does not belang entirely to any 1-precomplete class from the system N, and among the functions having a zero delay there are contained the pairs

The necessity of the Statements of the theorem derive from Lemmas 2 and 3 and the fact that the classes of the system N arenot 1-complete. Let us show that the conditions of the theorem are sufficient for the system l8 tobe 1-complete. Thus, assume the system l8 has the form { (/p 0), (cpi' 1)} and satisfies the conditions of the theorem.

All such systems l8 will be partitioned into several groups as a function of the prop-· erties of the functions having zero delay which appear in m.

I. The system l8 includes a o-function having zero delay.

II. The system l8 includes only o-functions having zero delay.

III. The system l8 contains only a- and ß-functions having zero delay.

IV. The system 18 includes only a- and y-functions having a zero delay.

V. The system 18 includes only a-, ß-, and y-functions having zero delay.

These five groups completely define all possible forms of the system 18. The closure of the system of functions having zero delay from l8 coincides with some specified closed class 0p2• In each of these specific cases it will be shown that 18 is a 1-complete system.

The proof of sufficiency will derive from a series of lemmas.

Let us consider each of the groups I-V separately.

I. The system l8 includes a o-function with zero delay. From a consideration of 1-precomplete classes of the system N it follows that in order for the system 18 not to belang to any 1-precomplete class it must satisfy one of the following two conditions:

1) among the functions having zero delay in 18 one can find a nonself-dual function, while among the functions having a unitary delay one can find a function which düfers from a constant;

2) among the functions having a unitary delay in l8 one can find an odd function and a nonself-dual function.

Lemma 18. If the system ~ contains a o-function having a zero delay and does not belang to any 1-precomplete class from the system N, then 18 is a 1-complete system.

P r o o f. It is sufficient to consider separately the two cases in each of which the system 18 satisfies one of the conditions 1), 2).


Ao The system m contains a nonself-dual function having a zero delayo It is obvious that in this case [ { (jh O)}] coincides with the class 0p2; further, having the pair (cp, 1), where cp differs from a constant, and the pairs (0, 0), (1, 0), (X, 0), one may obtain the pair (x, 1) by means of synchronous-superposition operations, and this in turn involves the relation [Q)J ~ 1P 2•

Bo The system ~ contains only self-dual functions having a zero delay o There exist only three closed classes of logic-algebra functions of such a type: 0 4, L5, and D3 [4]o It is obvious that [ {Uio 0)}] may coincide only with 0D3o The pair (x, 1) can be obtained from an odd function having a unitary delay by means of synchronous-superposition operations using the pairs (x, 0), and (X, 0), whichevidently belong to 0D3o Therefore, it may be assumed that the set 1D3 c [Q)j. Thus, the system Q) is 1-complete, since it contains the 1-precomplete class IRs and does not coincide with ito

Ilo The system m includes only a-functions having a zero delayo From a consideration of the 1-precomplete classes of the system N it follows that in order for the system m not to belong to any 1-precomplete class the system must satisfy one of six conditions:

1) among the functions having a zero delay in m one can find a function which does not satisfy the condition (A2), a function which does not satisfy the condition (a 2), and a function which does not belong to the class D1; among the functions having a unitary delay one can find a o -function and an a-function;

2) among the functions having a zero delay in m one can find a function which does not satisfy the condition (a 2); among the functions having a unitary delay one can find a 6-, a-, and ß -function;

3) among the functions having a zero delay in m one can find a function which does not satisfy the condition (A2); among the functions having a unitary delay one can find a o-, a -, and y-function;

4) among the functions having a zero delay in Q) one can find a function which does not belong to the class A4; among the functions having a unitary delay one can find an a-,ß-, and y-function;

5) among the functions having a zero delay in m one can find a function which does not belong to the class A4; among the functions having a unitary delay one find a o -, ß-, and yfunction;

6) among the functions having a unitary delay in m one can find an a-, ß -, y-, and ofunctiono

Lemma 19o If the system m contains only an a-function having a zero delay and does not belong to any 1-precomplete class 1Rt-1Rt9• then Q) is a 1-complete systemo

Pro of o Let us consider six cases separately, in each of which the system Q) satisfies one of the conditions 1)-6)o

Ao The system m contains a nonself-dual function having a zero delay, a function which does not satisfy the condition (A2), and a function which does not satisfy the condition (a 2)o In this case (see [2]) the pairs (x o y, 0) and (x V y, 0) belong to [Q)J, and [ {(fi, 0)}] coincides with the class 0A4 or 0Ao It is sufficient to consider the case in which [ { Ui, 0)}] coincides with the class 0A4o In this case the pairs (x, 1), (X, 1), (0, 1), (1, 1) belong to [Q}], and consequently for any function <p (x1, ... , xn) EP2 one may use synchronous-superposition operations (just as in Lemma 7) to obtain the perfect disjunctive normal form having a unitary delay o Thus, the system lE is 1-completeo

16 L. A. BIRYUKOV A AND V. B. KUDRYAVTSEV

B. The system ~ contains a function having a zero delay which does not satisfy the condition (a 2). In this case [ { (j i, 0)} 1 may coincide with one of the classes: 0D1, °F~, °F~ ( f1, = 2, 3, ... ) •

Let us consider each of these three cases.

a) [ {(ji, 0)} 1 coincides with 0D1• Note that for any function cp(x1, ••• , x11 ) one can find a self-dual a-function f (x, y, x1, ••• , X11 ) and (qJ (x11 Xz, ... , xn) V x) · y V(~ (xb x2, .. ·.• xn) V y) x. In the pair (j (x, y, x1, ••• , x11), 0) let us replace the variable x by the pair (0, 1), the variable y by the pair (1, 1), and the variable Xi by the pair (xit 1). Each of these pairs belongs to [Q3].

As a result we obviously obtain the pair (cp, 1), i.e., the system ~ is 1-complete.

b) [{<fit 0)} 1 coincides with one of the classes °Ff (p, = 2, 3, ••. ), for example, with the narrower class °F 5. Note that for any function cp (x1, ••• , x 11) one can find a function f (x, y, z, x1, ••• , x 11) from the class F5 of the form x·(qJ (x1 , .•• , xn) V (z+y+ 1)). In the pair (j(x, y, z, x1, ••• , x 11), 0) let us replace the variable x by the pair (1, 1), the variable y by the pair (x, 1), the variable z by the pair (x, 1), and the variable xi by the pair (xi, 1). Each of these pairs belongs to [~]. As a result we obviously obtain the pair (cp, 1), i.e., the system m is 1-complete.

c) [ {(ji, O)} 1 coincides with one of the classes °F~ (p, = 2, 3, •.. ), for example, with the narrower class °F 6. Note that for any monotonic function cp (x1, ••• , x11 ) one can find a function f (x, x1, ••• , x11 ) from the class F 6 of the form x · cp (x1, ••• , Xn).

Substituting the pair (1, 1) for the variable x, and the pair 6{i, 1) or ~, 1) for the variables xi in the pair (j (x, x1, ••• , x11), 0), one may obtain the perfect disjunctive normal form for any function having a unitary delay; i.e., the system m is 1-complete.

C. The system QJ contains a nonmonotonic function having a zero delay. In this case [ {(fi, 0)} 1 obviously coincides with one of the classes: 0A, 0D1, °Ff, °Ff (p, = 2, 3, .•• ).

Let us consider each of these four cases.

a) [{(fit 0)} 1 coincides with the class 0A. Since the pairs (x, 1), (x, 1), (0, 1), (1, 1) obviously belong to the class [QJ]' it follows that the system m is 1-complete.

b) [ {(fp 0)}] coincideswith 0D1• A system m of this form, as was shown in B, is 1-complete.

c) [ { (fi, 0)} 1 coincides with one of the classes °Ff (p, = 2, 3, •.. ). A system IE of this form (as was shown in B) is 1-complete.

d) [ {<fi, 0)}] coincides with one of the classes °Ff (p, = 2, 3, .•• ). Since the classes °Ff are respectively dual with respect to the classes °Ff (p, = 2, 3, ... ), it follows that by virtue of the quality principle the system m is 1-complete.

D. In this case the system m contains functions of the type a, ß, y, and o having a unitary delay. It is evident that [{(fit 0)} 1 [2] coincides with one of the classes: 0A, 0A4, 0D1,

OFf, OF~, OFf, IPf (p, = 2, 3, ••• ), OD2. All of these subcases, with the exception of the two latter ones, have already been con

sidered above; therefore in each of them the system QJ is 1-complete. Since the classes °F~ are respectively dual with the classes °Ff (p, = 2, 3, .•. ), it follows that by virtue of the duality principle the system \8 is 1-complete in this case also. If [ { <Ii, 0)}] coincides with 0D2,

then since for any monotonic functions cp (x1, x2, ••• , x 11 ) the pair (x · y V;. yqJ (xb ... , xn) V x · Y· cp (xj, ... 'X"n), 0) enz, one may obviously obtain the perfect disjunctive normal form for any logic-algebra function having a unitary delay (i.e., the system Q3 is 1-complete in this case also).


The cases in which the system m satisfies conditions 3) and 5) can be considered in dual fashion, which is what it was required to prove.

III. The system 9\ includes only a- and ß-functions having a zero delay. From a consideration of 1-precomplete classes of the system N, it follows that in order for the system not to belong to any 1-precomplete class from the system N it must satisfy one of eight conditions:

1) among the functions having a zero delay in f{) one can find a nonmonotonic function; among the functions having a unitary delay one can find a y-function which is nonzero;

2) among the functions having a zero delay in Qi one can find a nonmonotonic functions; among the functions having a unitary delay one can find a o-function;

3) among the functions having a zero delay in ~ one can find a nonmonotonic function; among the functions having a unitary delay one can find a ß-function which differs from 1 and the constant O;

4) among the functions having a zero delay in ~ one can find a nonmonotonic function; among the functions having a unitary delay one can find an a-function and the constant 0;

5) amongthe functions having a zero delay in \B one can find only monotonic functions; among functions having a unitary delay one can find ß- and y-functions which are respectively different from 1 and 0;

6) among functions having a zero delay in ~ one can find only monotonic functions; among the functions having a unitary delay one can find Cl!- and y-functions, the y-function being nonzero;

7) among the functions having a zero delay in ~ one can find only monotonic functions; among the functions having a unitary delay one can find ß- and o-functions, the ß-functions being different from 1;

8) among the functions having a zero delay in ~ one can find only monotonic functions; among the functions having a unitary delay one can find only a- and o-functions.

Lemma 20. If the system ~ contains only a- and ß-functions having a zero delay and does not belong to any 1-precomplete class from the system N, then ~ is a 1-complete system.

Pro o f. Let us consider separately the cases in each of which the system ~ satisfies one of conditions 1-8).

The system m, satisfying one of the conditions 1-4) contains a nonmonotonic function having a zero delay, and therefore [ { (/i, 0)}] coincides with the class °C2 or with one of the classes °Ff (JJ. = 2, 3, ... ) (for example, with a narrower one of them- °F;'). It can easily be seen that the pairs (x, 1), (X, 1), (0, 1), and (1, 1) belong to the set [Q}J. If [ {(ji, 0)}] coincides with 0c2, then the system 18, is obviously 1-complete.

Assurne [ {(ji, 0)}] coincides with °F;'. Note that for any function cp(xh ... , xn)EP2 one can find a function f (x, x1, ... , x11 ) of the form x V cp (xb ... , xn), which satisfies the condition ( a"" ). Substituting in the pair (j (x, x1, ... , x11), 0) the pair (0, 1) E [Q}], for the variable x and the pair (x;, 1) E [Q}J, for the variable xi, we obtain the pair (cp, 1) (i.e., the system >S is 1-complete).

The system ~' satisfying one of the conditions 5-8) contains only monotonic functions having zero delay, and therefore [ { (ji, 0)}] coincides with the class 0A2 or with one of the classes °Flf (JJ. = 2, 3, ... ) (for example, with a narrower one of them, °F;'). It can easily be seen that the pairs (x, 1), (X, 1), (0, 1), and (1, 1) belong to the class [Q}].


If the system [ { Ui, 0)}] coincides with 0A2, then for any function cp (x~> · · ·, xn) E P2 one may use synchronous-superposition operations (just as in Lemma 7) to obtain its perfect disjunctive normal form having a unitary delay.

Assurne [ {(fi, 0)}] coincides with °F:i. Note that for any monotonic function cp(x1, ••• , xn) which differs from a constant, one can find a function f (x, x~> ... , xn) E r; of the form x V cp (xb ... , xn). Substitutingthe pair (0, 1) for the variable x, and the pair (xi, 1) or 6ci, 1) for the variable xi in the pair f (x, x1, ••• , Xn), one may obtain the perfect disjunctive normal form or a function from P 2 having a unitary delay. Thus, the system \B is 1-complete.

By virtue of the duality principle the following lemma is valid:

Lemma 21. If the system Qs contains only a- and y-functions having zero delay and does not belang to any 1-precomplete class from the system N, then ~ is a 1-complete system.

Thereby the case IV has been considered fully.

V. The system 18 includes only a-, ß-, and y-functions having zero delay. From the consideration of 1-precomplete classes of the system N it follows that in order for the system m nottobelang to any 1-precomplete class it must satisfy one of the following two conditions:

1) among the functions having a zero delay in )!) one can find a nonmonotonic function; among the functions having a unitary delay one can find a function which differs from a constant;

2) among the functions having a unitary delay in m one can find a nonmonotonic function and a function which is not a negation of a monotonic function.

Lemma 22. If the system m contains only a-, ß-, and y-functions having zero delay and does not belang to any 1-precomplete class of the system N, then ~ is a 1-complete system.

Pro o f. If the system ~ does not belang to any 1-precomplete class, then it is obvious that [ {Ub 0)}] coincides either with 0P 2 or with 0A1• Obviously, the pairs (0, 1), (1, 1), (x, 1), (X, 1) belang to [QJ], and now it is not difficult to show that the system [\8] will be 1-complete in each of these cases, which is what it was required to prove.

This completes the proof of the sufficiency of the conditions of Theorem 1.

From the proof of the basic Theorem 1 the following statement derives.

Theorem 2. From any finite 1-complete system ~ one may iso-1 a t e a 1- c o m p 1 e t e s u b s y s t e m Qi' c o n t a in in g n o m o r e t h an f i v e f u n ct i o n s having delays.

As will be shown below, the requirement of finiteness of the complete system in Theorem 2 is essential.

§ 5. Certain Properties of Systems of Pairs

fr om 1P2

In this section the following proposition, in particular, will be established.

Theorem 3. In the set of pairs from tß2 there do not exist 1-precomplete classes which differ from the classes \R1 -\R19 •

For proof of this theorem we shall need certain auxiliary statements.


Assurne IR={(/;, 0), (<pj, 1)} is a closed class from 1F2, which is not contained in any of the classes llt1, ... , \J1 1g. Then the following statements hold.

Lemma _23. If in IR the system { (fi, 0)} coincides with some one of the classes 0P 1 , 3P 3 , 0P 5 , 0P 6 or with some one of the classes 0s 1 , 0S 3 , 0S 5 , 08 6 , then \R is not a 1-precomplete class.

Pro of. By virtue of the duality principle it is sufficient to consider, for example, the case in which { Ui, 0)} coincides with some one of the classes 0P 1, 0P 3, 0P5, 0P 6• In ordertobe specific let us assume that the system IR is a 1-precomplete class, and (again tobe specific) that the set {(fi, 0)} coincides with the class 0P 6 (the remaining cases may be considered to be analogous). The function g = x~r V x~2 V ... V x7,n shall be called an ( m, n) - f u n c t i o n, where m denotes the number of ones in the collection (a 1, ••• , a 11). From the fact that IR is not a 1-complete system and IR~ 0P6, it obviously follows that there exists a constant k such that for any (m, n)-function for which m < n, the condition m < k holds, and the function g 1 = x 1 V x 2 V ... V xk V xk+! V .Xk+a does not belong to IR. Let us consider the system IR U {g1}.

By definition it must be 1-complete. Weshall show, however, that the function g2 =x1 V V x2 V ... V xk+! V xk+z V i;,+a cannot be contained in [IR U {g1}], from which the validity of the statement of the Iemma will follow.

Let us assume that gz E [IR U {gl}J. Then g2 = f 1 • f 2 • ... • fs holds, where a certain function fi ~ c is derived from the function g1 by substituting functions from P6 into it. After transformations, fi may obviously be represented in the form W1 V ~! 2 V ... V Wq V xh V xi2

V ... V xiP• , where q < k + 2, and each 2lu is not a cofactor of any other Wv and is independent of the variables xil' ... , Xj 1,.

Assurne q = 0. Then f i (1, .•. , 1) = 0, which would invol ve the vanishing of the function g2 on the set (1, 1, ... , 1); however, g2 (1, ... , 1) = 1.

Assurne q > 0. Since q < k + 2, it follows that one can obviously find two collections from which f i will vanish, and thereby the function g2 must vanish on these same collections; however, this is impossible, since g2 vanishes only on one collection. Thus, g2 1 (IR U {g1}], which is what it was required to prove.

Lemma 24. If in IR the system { (fi, 0)} coincides with some one of the closed classes of linear functions and the degrees of functions having a unitary delay from the system IR are bounded in aggregate, then the system \R is not a 1-precomplete class.

Pro of. Let us suppose that the system m is a 1-precomplete class. Let us use r to denote the highest degree of functions from the system {(<Pi, 1)}. Assurne that the pair ( 1/J (x1, ••• , x 11), 1) does not belong to the set IJl , and that the degree of 1/J is equal to r + 1. Then, obviously, the set IR does not belong to the sequence of functions having a unitary delay and degrees r + 2, r + 3, ••• , from which the function I/J(x1, ••• , x11 ) can be derived by identification of variables. From the Supposition of 1-precompleteness of the system IR it follows that the system {lli U (1p (x1 , ... , xn), 1)} must be 1-complete; however, this is not so because from pairs of this system we obviously may not obtain functions having a unitary delay which are of degree r + 2 and higher using synchronous-superposition operations; this is what we were required to prove.

Below we shall consider only those systems m, in which the degrees of functions having a unitary delay arenot bounded in aggregate.

Lemma 25. If in m the system { (ji, 0)} coincides with some one of the classes 001, 002, .•. , 00o, then IR is not a 1-precomplete class.


The proof of this lemma is completely analogaus to the proof of Lemma 23.

Lemml!_~~ If in \R the system { (fi, 0)} coincides with the class L 3 , then IR is not a 1-precomplete class.

Pro o f. More precisely, we shall establish that in the case considered the set \R is 1-complete. For proof of this fact it is sufficient to show that \R contains a pair (1, 1) and all pairs of the form (x1 • x2 • ••• • xr, 1), r = 1, 2, .•..

Actually, having these pairs and the pair (x + y, 0), which by stipulation belongs to the set \R , one may use synchronous-superposition operations to obtain the pair (P f, 1) for any pair (j, 1), where P f is the Zhegalkin polynomial of the function f.

Let us begin by showing that {1, 1) E IR. From the fact that \R does not belang to the set IR!, ... ' IR19 it follows that I}( must include the pair (cp, 1), where cp is either aß- or a o

function.

Let us identify the variables in the pair (cp, 1); we obtain either (1, 1) (if cp is a ß-function) or (X, 1) (ü cp is a o-function). Substituting the pair (0, 0) from \R, into 6c, 1), we likewise obtain the pair (1, 1).

Let us now show that any pair (x1 ·x2 • ••. ·Xr, 1) E \R. Assurne the pair (ljJ (x~> ... , xn), 1) E \R and the degree of ~ is equal to k ::::: 1. Without loss of generality it may be assumed that the Zhegalkin polynomial of the function ~ has the form

s n

X 1·X2 • ... ·Xk+ ~ ~i+ ~ CjXj+C, (1) i=1 i=1

where ~; are conjunctions of degree no higher than k, while c, ci E {0, 1}. Let us show how, starting from the pair (~, 1), one can use synchronous-superposition operations to obtain the pair (x1 • x 2 • ••• ·xk, 1). In the pair(~, 1) let us replace the variables xk+ 1, ••• , xn by the pair (0, 0). As a result we obtain the pair (~', 1). The Zhegalkin polynomial of the function ~· is obviously the sumofall those terms of the polynomial (1) which do not contain variables düfering from x 1, ••• , xk. as cofactors, plus a free term. In the pair (~', 1) we replace the variable x1 by the pair (0, 0). As a result we obtain the pair (~<1 >, 1). In the pair (Yd-Y2, O)Effi let us replace the variables y1 by the pair (1/J', 1), and the variable y2 by the pair (1f;( 1), 1). As a result we obtain the pair (1fJC2 >, 1). The Zhegalkin polynomial of the function zp( 2) is obviously the sum of all those terms of the polynomial (1) which contain x1 as a cofactor. In the pair (1f;C 2>, 1) let us now replace the variable x2 by the pair (0, 0). As a result we obtain the pair (~ < 3 >, 1). In the pair ( (y 1 + y 2), 0) let us replace the variable y 1 by the pair (~ < 2 ), 1), and the variable y2 by the pair (~C 3 >, 1). As a result we obtain the pair (~< 4 >, 1). The Zhegalkin polynomial of the function ~ <4 ) is obviously the sum of all those terms of the polynomial (1) which contain x 1 and x 2 as cofactors. Having carried out analogaus constructions for each xi, i = 1, ... , k, we obviously obtain the pair (x1 • ... • Xk, 1).

Let us now note that since the set of degrees of functions having delays which are included in IR, is not bounded in aggregate, one can find a sequence

(/1, 1), (/2, 1), ... , (/n, 1), ...

(h 1) E IR , such that the condition i = 1, 2, ... , applies to the degrees s 1, s 2, ... of the functions f 1, f 2, .... From this, by virtue of the constructions described above, we derive the fact that all pairs of the form {x1 • .•. ·Xs,. 1). belang to the set IR • Since, obviously, any pair (x1,· ...

Xr, 1) may be obtained from a pair (x1· ... ·x._, 1), where j > r, by identifying certain variables, J

it follows that all pairs (x1 • ... • Xu 1), r = 1, 2, ... belang to \R, which is what it was required to prove.


By virtue of the duality principle the following statement holds:

Lemma 27. If in \R the system { ( fi, 0)} coincides with the class L 2 , then lR is not a 1-precomplete class.

Lemma 28. If in lR the system { ( fi, 0)} coincides with the class L 1 , then \R is not a 1-precomplete class.

The proof of this statement is completely analogaus to the proof of Lemma 26.

Lemma 29. If in ffi the system { (/i, 0)} coincides with the class L 4 , then ~R is not a 1-precomplete class.

21

Pro o f. Let us assume that ;)y is a 1-precomplete class. If ffi is not contained in any of the functions ffi1, ffi2, ... , \1\ 19 , it follows that ffi contains either a-, ß-, and y-functions or ß-, y-, and o-functions, or a- and o-functions having a unitary delay. In the first and second cases one can evidently obtain the pairs 6{, 1) and (x, 1), respectively, by identifying the variables and substituting the pairs obtained into the pair ((x + y + z), 0) E \R • For the case in which the system \R contains o- and a-functions having a unitary delay, it follows that by identifying the variables and substituting the previously obtained pairs (x, 1) and (x, 1) in special fashion [2] into the pair (f, 0) E L 4, where f is a nonself-dual function, we obtain the pair (c, 1), where c = { 0, 1}. Replacing the variables by the pairs (c, 1), (x, 1), (x, 1), respectively, in the pair ((x + y + z), 0), we obtain the pair {c, 1) as a result. Consequently, the system ffi contains the pairs (0, 1), (1, 1), (x, 1), (x, 1).

From the supposition that \R is a 1-precomplete class it derives that ffi is not 1-complete. By virtue of this there must exist a natural number n0, which is such that for any pair from \R of the form (x1, ••• , Xro 1) the condition r :::; n0 holds. Otherwise, starting with these pairs and also with the pairs (X, 1), (x + y + z, 0), (1, 1), and (0, 1), we could construct the pair (P f• 1), where P f is the Zhegalkin polynomial for f.

Assurne n0 is suchthat in \R there exists a pair (x1, ... , xllo, 1) but not a pair (x1, ••• , xllo + 1, 1). It is obvious that n0 ~ 1. By virtue of what has been said above, it may be assumed that \R contains all functions of degree no higher than n0 having a unitary delay. From the assumed 1-precompleteness of \R it follows that the set IDl = \li U {(x1 ... Xno-i 1, 1)} must be 1-complete. Weshallshow that if ~m is complete, then \R must necessarily contain a pair (x~> ... , Xno+J, 1). This will contradict the choice of the number n0, and thereby the lemma will be proved.

Thus, assume IIDfl ~~ 1P2. Then, in particular, [ID1] must contain a pair (x1· ... · Xno+2, 1).

It is evident that one can find functions f of degree n0 + 2 and g of degree n0 + 1 which are such that x1 • x2 • .... x n+ 2 = f + g, it being true that the pair (f, 1) E \R. Let us consider the function f in greater detail. Assurne f' is a function stipulated by a Zhegalkin polynomial which is obtained from the Zhegalkin polynomial of the function f by eliminating all terms of degree n0 + 2 and n0 + 1. It is obvious that (/', 1) E lJL Further, it is obvious that the pair (f+ f', 1) E\R, it being possible to assume without lass of generality that the Zhegalkin polynomial of the function f + f' has the form

s

X1·J'2 · · · .Tno+2-~ ~ ~;, i= 1

where each term of ~;, that is the product of certain variables from the set {x1, x2, ... , Xno+ 2}

has the degree n0 + 1 and the number of terms s ~ 1. Let us consider two cases. s

1. s > 1. Assurne that the sum .2J ~; contains the terms x1x2· ..•... ·:~.·u- 1Xu-r~· ... ·Xno-J-2 2=1 •

and x 1x 2 · ... ·xv_1xv+1' ... ·Xno+Z• u < v. Let us identify the variables xu and xv in the function f'.


Assurne f" is the function which is obtained under these conditions. It is obvious that the Zhegalkin polynomial of this function has the form

r

X1X2 • . · · · Xu-tXuXu+l' . · . • Xv-!Xvu· ..• • Xno+2 + ~ bj, i=l

T

where the degrees of all terms b1, ••• , br are less than n0 + 1. Since [{f", 1), ( ~ bh 1)) s ffi r i=l

and Since f" + ~ bj = x1x2 ... Xu-IXuXu+t' ... · Xv-!Xv+l -j- ... + Xno+2, it follOWS that the pair i=l

(xl • Xz • · · · · Xu-IXuXu+l' · . · · Xv-!Xv+l' · · · · Xn0+2, 1) E \R.

2. s = 1. Assurne 2f1 does not contain the variable xi. Let us identify any pair of variables Xu and xv, u ~ j, v ~ j, u < v in the function f'. As a result we obtain a certain function f". Obviously, the Zhegalkin polynomial of this function is the sum of two terms, one of which has the degree n0 + 1, while the second has the degree n0• In exactly the same way as in case 1 we construct a pair (x1· ... ·Xno+l, 1), which must belang to \R.

Thus, \R contains the pair (x!' ... ·Xno+t, 1), and this contradicts the choice of n0; this is what it was required to be proved.

Lemma 30. If in \R the system { ( /p 0)} coincides with the class L 5 , then \R is not a 1-precomplete class.

The proof of this statement is analogaus to the proof of Lemma 29.

Proof of Theorem 3. Assurne ffi is a 1-precomplete class. Let us show that \Jr coincides with one of the classes of the system N. Let us assume that the latter is not fulfilled; more precisely, ffi does not coincide with any of the classes indicated and is not contained in any of them. Let us consider the possible cases for a system { fi, 0 }, \R • If the system {/i, 0} were to contain the pairs (j, 0), (<f', 0), and (lf;, 0), such that f ~ L1, cp ~ Ps, and 1jJ E Ss , then from the fact that the set \R does not belang to the class of the system N it would follow by virtue of Theorem 1 that \R is a 1-complete system. Thus, (see [2]), the system { (ji, 0)} may only coincide with one of the following classes: Si, Pi, Li, Ok, i = 1, 3, 5, 6, j = 1, 2, 3, 4, 5, k = 1, 2, •.. , 9. However, by virtue of Lemmas 23-29, \R is not 1-precomplete in any of these cases. The theorem has been proved.

Theorem 4. There is a continuum of closed classes in 1Fz. Pro of. Let us construct an infinite sequence of functions having unitary delays, in

which any pair does not belang to the closure of the set of the remaining pairs. Thereby düferent subsequences of our sequence will correspond to düferent closed classes which are closures of these subsequences. In order to complete the proof it remains for us to note that one may choose a continuum of düferent subsequences.

Following [2], let us consider the function

hll (x~> ... , x!l+l) = x 2 ·X3 · ... ·x!l+l V x 1x 3 · ... ·X11+ 1 V .. . . . . V Xj' ... ·Xi-!Xi+l' ... ·X!l+l V ... V Xt• ... ·XI', !-1 > 2.

We choose

(h2 , 1) (h3 , 1), ... , (h1,, 1), ...

as the sequence about which we spoke above. Only the operation of identüication of variables is applicable to the terms of this sequence.


Let us show that none of the pairs of this sequence can be expressed in terms of the remaining ones by means of the operation of identification of variables. The latter fact derives from the fact that the function h!l has the property (All ) but does not have the property ( Af.J+i) (see [2]), while at the same time in the identification of any pair of variables of the function hp a function is obtained which has the property ( A co ); this is what it was required to prove.

As was established in § 4, from each 1-complete finite system one may isolate a subsystem consisting of no more than five functions having delays, as well as a 1-complete function. However, in the general case this statement turns outtobe untrue. Moreover, the following statement holds.

Theorem 5. In 1P 2 there exists a countable 1-complete system no finite subsystem of which is 1-complete.

As such a system one may, for example, choose the set 1P 2•

The set 1P 2 likewise has one more interesting property. Assurne IDL=:::; 1P2 and [.IDf] = 1P 2 ;

assume further that ~J(' c ~)t and the difference 1))/' JJl' contains only a finite number of functions having delays. Then [IDI'] c= 1P 2 . Actually, assume ID("I)Jt' consists of the pairs

where r i :s r i + 1 for i = 1, s - 1. Let us consider the pairs (cp 1, 1), ••• , (cp s' 1), where cp i = ~l's

f; (x1, ... , x,..) + Y x, . .• Obviously, (<p;, 1) E [~Jl']. Moreover, z j:;:-1 r+J

(/; (xt. ... , x,), 1) .c (cp; (x 1, ••• , x,.i, x,. 1, ••• , x,.), 1).

From this it follows that [IDl'] ~-.• 1P2 .

A 1-complete infinite system of functions having delays from 1P?, for which no intrinsic subsystem is 1-complete is called a b a s i s . Constructions analogaus to those carried out in the proofs of Lemmas 23-30 allow us to establish the validity of the following statement which is a generalization of the property just considered for infinite 1-complete systems.

Theorem 6 • No b a s es e x ist in 1ß 2 •

The validity of the following statement likewise derives from a description of the system N of 1-precomplete classes from Theorem 1.

Theorem 7. In 1 P 2 there exist closed classes which arenot 1-complete and are not contained in any 1-complete class.

As such a closed class one may take, for example, the class 0 L 1 U IJl" , where IJl'. is the set of all pairs of the form (j, 1), while the function f has a degree no higher than r 2: 1.

The statements proved above allow another formulation of the basic Theorem 1 to be presented.

Theorem 8. In orderforafinite system ':1Sr;:;; 1F2 tobe 1-complete, it is necessary and sufficient that it does not belong to the classes of the system N or to three rigorously increasing chains of closed classes

[ 0PHUIJJF]c[ 0P6UIDl 2]c ... c[0P6 l)~Jl']c ... ,

[ 0S6 U ~J/ 1 ] c [0S6 U ID/2] c ... c.: [0S6 U ~Ji'J c ... ,

[ 0LtUIJF] c[ 0LtlJ9( 2 ] c ... c[ 0L1 l)IJl']c.: ... ,


where ffiF is the set of all pairs of the form (f, 1) such that the rank of f does not exceed r, r = 1, 2, ... , any element of any chain not being contained in any 1-precomplete class and the union of all elements of any of the chains being 1-complete.

The requirement of finiteness in this theorem is essential, since the system 1A U 1B U 1 f u {(x, 1)} obviously satisfies the conditions of nonappearance in the sets indicated in the theorem.

but, as is easily seen, is not 1-complete.

Literature Cited

1. V. B. Kudryavtsev, "Completeness theorem for a certain class of automata without feedbacks," in: Problemy Kibernetiki, No. 8, Fizmatgiz, Moscow (1962).

2. S. V. Yablonskii, G. P. Gavrilov, and V. B. Kudryavtsev, Logic Algebra Functions and Post Classes, Nauka, Moscow (1966).

3. S. V. Yablonskii, Functional Constructions in k-Valued Logic, Transactions of the V. A. Steklov Mathematics Institute of the Academy of Sciences, Vol. 51 (1958).

4. E. L. Post, Two-Valued Interactive Systems of Mathematical Logic, Princeton (1941).

ASYMPTOTICALL Y ST ABLE DISTRIBUTIONS OF CHARGE ON VERTICES OF AN n-DIMENSIONAL CUBEt

V. K. Leont'ev

Novosibirsk

We consider the set En of binary sequences of length n. Let M = {A1, A2, ••• , A5} be any s-subset from En. Consider the nurober

(1)

where p(Ai, Aj) is the Hemming distance in En. t S. V. Yablonskii has posed the problern of finding an s-subset M s En, in which the functional H (M) has a minimum. Physically the set M can be interpreted as a stable position of s like charged particles placed on vertices in E 11 •

In [1] this problern was completely solved for the case s(n) = 211 - 1• In this case it turned out that there exist two extremal sets, both of even parity. For other s, however, the question of the structure of the sets remained open. In [2] an asymptotic formulation of the problern was considered. It consists of the following. Suppose Hs (n) = min H (M), itisrequiredtofindase-quence { M11 } of s -subsets from E n such that JI r;;; En

It was also remarked in that paper that since log s (n) ~ n, all s-subsets from En have asymptotically identical energy § and hence in this case the asymptotic formulation of the problern turns out to be devoid of interest.

t Original article submitted February 5, 1968. t The so-called Hemming distance between points A = (a1, a 2, ••• , a 11 ) and B = (ß1, ß2, ••• , ß11 )

is the nurober

§ The function H (M) ~ ~ proceeding from the physical analogy introduced above, p (A,, A 1)'

1~t<r::s

is called the energy of the set M. 25

26 V. K. LEONT'EV

The basic results of the present work consist of the following:

1) the discovery of an asymptotic expression for H8(n), where s == s(n) is an arbitrary increasing function;

2) a proofthat "almost all" s-subsets of E 0 have asymptotically minimum energy and the discovery of an estimate of the deviation of H(M) from H8(n) for almost all s-subsets M from E 0 ;

3) the construction, in the case of functions of the form s (n) = 2cp(nJ and with <p (n) an arbitrary increasing integral function, of s-subsets of asymptotically minimal energy.

Besides, it is shown that the MacDonald equidistant code [3] has absolutely minimalenergy. That is, instances of sets with absolutely minimal energy- even parity and MacDonald code- are at the sametime sets with maximal minimum distance among all sets of the same cardinality.

§ 1

In this section the asymptotic mm1mum energy is found and the distribution studied for sets with asymptotically minimal energy. Here it is convenient to consider immediately more general functionals than energy, which introduce no additional difficulty in the obtaining of the needed results and which give supplementary information about the distribution of the minima of other functionals.

Let {M1 , M 2, ..• , M es } be all s-subsets of vertices of the n-dimensional unit cube E0 and zn

let <p (x) be an arbitrary function defined on the set of natural numbers. Let M == { A1, A2, ... , As} and

In the sequel we shall consider a probability space whose points are the C~n subsets of car-

dinality s, and to each subset we assign the probability The function ~ will be con-

sidered a random quantity defined in this space.

The values of the mathematical expectation and the variance of the random quantity ~ are set out in the following lemmas.

Lemma 1. n

Mt= s (s-1) ~ Ck (k). 'o 2(2n-1) L.J n<Jl

k=1

Proof. Suppose M 1, M 2 , ••• , Mc, are all the subsets of cardinality s from E0 • In-troduce the function 2n

k { 1, ~ij= 0,

if (A;, Ai) EMk,

if (A;, Aj) E Mk·

Obviously, with the help of the function ~~i we can express M~ as follows:

C~n

M~ = 2;. ~ ~ ~~i<Jl [p (A;, Ai) I zn k= 1 ;,Fi

(2)

ASYMPTOTICALLY STABLE DISTRIBUTIONS OF CHARGE 27

(the summation extends over all pairs consisting of distinct points of the set E 11 ). Changing the order of summation in (2) we set

c;n

s Czn

M~= zds ~ cp [p (A;, Ai)l ~ ~~i· 2n i"'j k= 1

(3)

It is obvious that Aii = ~ ~~i is equal to the number of s-subsets from E 11 containing the pair k=1

(Ai, Ai), i.e., is equal to c;;~ 2 independently of the points Ai and Ai. Hence:

(4)

By the symmetry in E11 we can write

(5)

where Ai is an arbitrary point of E11 • Choose as Ai the origin. Then p (Ai, Ai) == II Ai II· Considering that in E11 there are. exactly C~ points with norm k, we get from (5)

n

S=2n ~ C~cp(k). (6) k=1

Finally, from (4) and (6) we get

n

Mt= s(s-1) ~ C~ (k). 'o 2 (2n -1) LJ cp

k= I

Lemma 1 is proved.

Let n n

W (n) = ~ C~cp (k); k=1

W(n)~= ~ C~cp2(k); w(s, n)=2s2 -2s(2n+l_f)+2n+1(2n-1). k=l

Lemma 2. D= w(s,n)s(s--1) [wn-1J!2(n)J

S 4 (2n -1) (2"- 2) (2"- :3) ( ) 2"- 1 .

P r o o f • By the relation

and by Lemma (1), it is enough, in calculating variance, to find

. s L zn

2lft2 = - 1- '5', l s (M k) J2. "' C' .._

2n k= 1

As in the previous case, webring M~2 to a more suitable form:

C~n

M1;2 = +.- ~ { ~ ~ S~j(jl [p (A;, Aj)] V· 2n k=l iJj

28 V. K. LEONT'EV

(Henceforward weshall abbreviate cp[p(Ai, Ai)] simply by 'Pij•) Further,

8 8 8 C2n C2n C 2n

M~2 = 4C18 [ ~ ~ ~~j{[Jfj + ~ ~ ~~j~~i{[!ij{[!ji + ~ ~ ~~j~~. r([!ij([!8r J • 2n k= 1 i=j k= 1 iof'i k= 1 (i, i)"fo(8, l')

Noting that cp ii = 'Pii, we can write:

From Lemma 1 there follows

Now we calculate the sum

Changing the order of summation, we get

C~n

s2 = 4;8 ~ {[!ij{[!sr ~ ~~j~~r· 2n (i, j)*(s, r) k= 1

C~n Let R (A;, Ah A., Ar)= ~ ~~i~~,.. Clearly, this sum depends only on the number of different

k=i

points among ~. Ai, Ar, AS' This number can be equal to 3 or 4. Denote the cardinality of the set M by I M j. Then

S2 = 4;. ~ ([!u([!srR (A;, Aj, A., Ar) + 4;. ~ ([!u([!srR (A;, Ai, AsAr)· zn IA;UA;UAsUAt!=4 zn IA;UAjUAsUA,.f=3

Clearly, in the first sum R (A;, Aj, A., Ar) is equal to the nwnber of s-subsets containing the

four points Ab Aj, As, Au i.e., is equal to c~;~ 4 , while in the second sum R (A;, Aj, A., Ar) =C~;~ 3 • Thus

We start by calculating the second sum. Since I A; U Ai U As U Ar 1 = 3, there are the four possibilities: Ai= A 8; Ai= Ar; Aj = A8 ; Aj =Ar. Corresponding with this, the second sum breaks up into four equal sums. We calculate one of them, for example, the first. Wehave

s~:)1 = ~ {[!;j{[!ir· i, j, r


Clearly

(2) ~ ~ ~ ~ • S 2, 1 = ..:::.J <Jlij<Jlir = ~ <jlij ~ <Jlir- ~ <Jlij·

i,j r i,j

Wehave

n

~ <p;,. = ~ C~<p (k) = 1Jf (n) r k= 1

(as remarked above, this sum does not depend on the point Ai because of the symmetry of the set E 11).

Analogously,

~ <jl~j = znw (n). i. j

Accordingly we get for the second sum in (7) the closed expression

We now calculate the first sum in (7). We get

- ~ <jlij<jlsj- ~ <jlij<Jl;r- ~ <Jlij<Jljr- ~ <Jlij<jlj;- ~ <Jlii· i,],S i,j,r 'L,J,T 1,) i,j

The sum of the second, third, fourth, and fifth were calculated above. The sum of the first, sixth, and seventh will also be calculated according to the demonstration of Lemma 1. Thus we get

Since

cs-3 cs-4

(8)

(9)

~2"-.3 s (s--1) (s-2)

c~n = 2" (2 11 -1) (2n -2)

and 2"-4 = s (s-1) (s-2) (s-3) C 2n (2--1) (2n --2) (2n- 3) '

(10) zn

it follows, taking account of (7)- (10), after simple transformations, that we have

J)t _ w (s, n) s (s-1) [w (n) _ '!'2 (n)J "_c'J(Z"-1)(2"-2)(2"-3) 2''--1.

Wenowapplytheresult just obtained to find the asymptotic minimum of the random quantity ~. Let

H~ (n) = min ~ (M), Mt;En

where I M I = s. We shall suppose that cp (x) lies in a class of functions R = { cp (x) } with the following properties:

30 V. K. LEONT'EV

1) cp (x) is convex in [0, + oo ],

2) cp (x) is decreasing in [0, + oo ],

3) there exists a A. > 0 and a constant c such that

limrp(n)n'-=c. n-+:::o

Theorem 1. If rp(x)ER and s (n) :S 2n is an arbitrary increasing function, then

Theorem 1 shows that a minimum of the random variable ~ for the indicated class of functions asymptotically coincides with its mathematical expectation.

Rem a r k. The class of functions R = { cp (x)} unquestionably does not exhaust the functions for which the assertion of Theorem 1 holds. In particular, Condition 3) obviously can be significantly weakened.

It would probably be of interest to find the conditions on the class of functions R under which Theorem 1 holds.

W e introduce two lemmas for the proof of Theorem 1.

Lemma 3.

Proof. Clearly

From Lemma 1 we get

n

H s ( ) s(s-1) ~ c" k) 'I' n < " .LJ nfJl ( .

2(2 -1)"-=1

We proceed to find the asymptotic expression

n ~ C~cp (k)

L (n) = "-"=....:1.,--2n-1

To do this we first show that for any p > 0 the asymptotic equality

holds. This follows without difficulty from noting that the Bernshtein polynomials n

Bn (x) ~ C~ ( ~ ) P x" ( 1- x)n-k approximate the function j (x) = lp/ at the point X = 1/2. "-= 1 J!"urther, using Condition (3) we find A. > 0 and c such that

lim rp (n) n" = c. n-HD

(11)

(12)

ASYMPTOTICALLY ST ABLE DISTRIBUTIONS OF CHARGE

Consider the equality

n n ,h

s ( n) c 1 }. ~ c~ er ( k) - ;n ~ ~;: 1· Re~ 1 k= 1

Wehave

k n

s (n) < ). ~ c~ I cp (k)- ;,.j = 2~, · ~ c~ r (k), n=1 1<=1

where r (k) = j cp (k)- ;~-.j· We have r (k) = o ( :~-.) , i.e., there exists an e > 0 such that

1 r (k) < k'-+e •

Hence we get, using (12),

S (n) = o (T,. (n)).

That is,

Substituting (13) in (11) we get

as was to be proved.

Lemma 4.

Pro o f • Since, by Condition l), cp is convex downwards on [ 0, + oo ], it follows, using Jensen's inequality, that

[ 2 ~ .. p (A;, Aj)J "" [ {A A)] s(s-1) i!St<J"S ~ 2 s (s-1) [ ns J "' 2 cp 2(s-1) ·

31

(13)

(14)

(15)

(16)

32 V. K. LEONT'EV

Using the condition on the increase of the function S(n) and Condition 3) we get

s s2 ( n ) Hq;(n) ;:;;.Tcp 2 '

as wastobe proved.

The following theorem gives useful information about the character of the distribution of the minimum of the random quantity ~ for the indicated class of functions.

Theorem 2. Let cp (x) ER. Then for almos t all s- s ubsets M of E n the inequality

I ~ (M) - s; !p ( ~ ) I ~ ~ !p ( ~ )

holds.

Pro o f • From Lemmas 1 and 2 and the definition of the class R = { cp (x)} it is not difficult to obtain

where lim !l (n) = oo. Substituting now in Chebyshev's inequality 71->00

D~ P{I~-M~I> t}<t2'

t = ~ cp ( ~ ) , and noting (17) and (18), we get what is required.

(17)

(18)

For the case cp(x) = 1/x~ Theorem 3 can be formulated as follows. Almostall positions of s identically charged particles at vertices of a unit n-dimensional cube are asymptotically stahle. Here, for almost all distributions M there holds the following estimate of the deviation of the energy H(M) from the minimum:

I s2 I s H(M)-r;: ~n.

§2

In this section, sets are constructed having asymptotically minimal energy. For the construction of such sets there serves certain simple information from the theory of group codes. The entirety of this information can be found in [3].

The subset G ~ E" t is called a group code if Gis a subgroup of the group En. It is at the same time clear that G is a linear space over the field GF (2). If the dimension of the code G as a linear space is equal to k, then G is called a group (n, k) code. The metric structure of a group (n, k) code is completely determined by the set of weights of its code points, i.e., by the vector M(G) = (a 1(n), a 2(n) , ... , a 11 (n)), where a i(n) is the nurober of code points of weight i. The vector M(G) = (a1(n), a2(n), ... , an(n)) is called the spectrum of the code G. A group code G* is called dual with respect to the group code G if the subspaces G

tThe set En is an Abelian group for the operation of positional addition mod 2.

ASYMPTOTICALLY STABLE DISTRIBUTIONS OF CHARGE

and G* are orthogonal. It is obvious that the dim.ension of the code G* dual to the (n, k)code Gis equal to n- k, i.e., the code G* is an (n, n- k) code.

33

Let M(G) = (a1(n), a 2(n), ••• , au(n)) be a spectrum of the (n, k) code G and M(G*) = (b1(n), b2(n), •.• , bn (n)) be a spectrum of the dual code G*. Suppose also the dimension of the code G satisfies the following inequality

k(n) > ~"' (n) =log n +log log n + ffi (n),

where w (n) is an arbitrary increasing function of n. Under these circumstances the following assertion holds.

Lemma 5. For the energy of the (n, k(n))-code Gn there holds the following asymptotic equality

Pro o f. We use the MacWilliams formula [6] which connects the spectrum of the code Gn with the spectrum of the dual code

n n L bi(1 + xt-i (1- x)i = 2n-h ~ a;x;.

i=O i=O

From (19) there follows

n n

~ bt J (1+x)n-: (1-x)i dx= 2n-k ~ ~~ xi. i=O i=O

Transforming the integral on the left into two parts, we have

The first and the second integrals in (21) we take individually:

i i

uHx)= J (i-:-x)i dx=lnx+ J (~ (-1)'Cixr-1 )dx =lnx+ ~ (-i)'Ci x; ;

Further:

It is known [4] that

r=i r=i n-i n-i

ui(X)= J [~ c~-iX8-1 (1-x) 1]dx= ~ C~-i J X 8 - 1 (1-x)1 dx. S=f S=f

i ( ) J S-1 (1 )i d ~ 1 m cm xm+s V;, s X = X -X X= (- ) i -- . m+s

m=O

i i

u?>= ~ (-i)'Ci+= ~ +· r=i r=i

(19)

(20)

(21)

(22)

34 V. K. LEONT'EV

We calculate Vi,s (1). Wehave

1

v;, 8 (1)= J x•-1 (1-x)idx=B(s-1, i). 0

where B (s - 1, i) is the E uler integral. It is equal to

B( _ 1 ') _ (s-1)1 i! S ' ~ - (s+i)!

Using (20)-(23) we get

n n i n n-i

zn-k"" ~ ="" b· ~ ..!... +"" b· "" C! . (s-1)! i! ~ i ~ '...;.J r ~ '~ n-t (s+i)! i=1 i=1 r=1 i=1 •=1

Further:

c· . (s-1)1 il = _1_ c~+i n-t (s+ i)! ci , .

n

From (24) and (25), setting s + i = r, we get

Since

n

H(Gn)=2"-1 ~ ~;, i=i

it follows that

n i n n 22k-1 "" "" 1 22k-1 b . er

H(Gn)=~ ~ b; ~ r-+21> ~ cf- ~ r_:i • i=i r=i i=O n r=i+i

(23)

(24)

(25)

(26)

(27)

We now find the asymptotic expression for the second sum in (27). Changingorder of summation we have

Let

Then

n n r n r-1 - 22k-l ~ b; ~ cn 2k-1 ~ r ~ b; 11> n --·- 7,- --=-- C 2 ( ) 2n ...;.J ci r-i 2n . n ci (r....:. ')

i=O n r=i+1 r=1 i=O n 1

r-1

"" b; 'P (n, r) = k.l ci (r-i) • i=O n

n 2211-1 "" r 11>2 (n) =~ ~ Cn( ) th (Zr-n)2 th t ee , r - n > nu n , en n2ö2 (n) > 1 , so a

1 "" r 1 (2r--.-n)2 ~ 21' ..:::.J Cn (n, r) < F n2ö2 (n) L.J

J2r-nJ>nö(n) j2r-n[>nö(n)

n 4 maHp (n, r) 1 ~ 2 ( n ) 2 4ln n n ln n

< n2&2 (n) zn L.J Cn r- 2 < n2ö2 (n) . T = n2ö2 (n) ' (29) r=O

since

n 1"" r( n)2 n zn- ..:::.J Cn r--z = 4. r=O

Now in (29), putting o(n) = lnn/ln, we get (28), as wastobe proved.

We note, moreover, that for cp(n) :$ {Ü ln n

[-%-]+cp(n)

l . "" bi 0 lffi ..:::.J -. =. n-.oo i= [TJ c~

(30)

Hence,

since b i !l w (n) it follows that

I G* j- 2n-k < Zn n - h ln n~ (n) '

where !l (n) = 2w(n). Hence

I IJl~ I Vn v- zn Vn t qJ (n) zn < n ln n 2nn ln n~ (n) - ~ (n) .

Since lim w (n) = oo, we have proved (30). ,...,.""

36 V. K. LEONT'EV

Moreover,

Using (30) it is easy to show that the second sum tends to zero as n goes to infinity. Hence

[~] cp (n, [ ~ ]+cp(n)) ~ ~

i=O

i.e.,

It follows now at once from (31) that

21n ~ C~cp (n, r) ~ (jl { n, ~ ) .

j2r-nj<nö(n)

Hence

Using the condition k(n) > .6.w(n), we get

n i n 2211-1 ~ ~ 1 2211-1 ~ . . ln n n-k 11_ 1 { 22h) - 2- b; J -<-2 - b; ln~<---2 =2 lnn=o - . n .. r n 2n-2h+l n

i=i r=i i=i

From (32) and (33) we get finally

as wastobe proved.

(31)

(32)

(33)

The following theorem shows that the presepce in the code Un of a code Gn of sufficiently large dimension having asymptotically minimal energy is a sufficient criterion that the code Un too has asymptotically minimal energy.

Theorem 3. Let the group (n, k(n))-code Gn with k(n) ::::: .6. w (n), is an arbitrary increasing function, have asymptotically minimal energy, and let G belang to some other group code U where w (n) is an arbitrary increasing function, have asymptotically minimal energy, and let G n belang to some other group code Un, i.e ., let it be a subgroup of Un. Then the code Un also has asymptotically minimal energy.

Pro o f • It follows from Lemma 4 that

22h H (Gn) d<-. n

(34)


We get from Lemma 6

lil 22k 22k ""' bi H(G) ~--1-- .LJ

n n ' n i~l C~ { 1- ~ )

That is, it follows from (35) and (34) that the group (n, k(n))-code G11 with k(n) 2: ~w(n) has asymptotically minimal energy if and only if

lim W (G~) = 0, n-.oo

where

lil ""' bi W (G~) = .LJ 2 i=! c~(t--f)

37

(35)

(36)

Since by hypothesis the code G11 has asymptotically minimal energy, it follows that (35) is fulfilled. From the direct inclusion Gn ~ Un follows the inverse inclusion for the dual codes, i.e., G~ 2U~. Hence

which means

W (U~) < W (G~).

Now from (35) we get

lim W (U~) = 0, n-.oo

i.e., the code U11 has asymptotically minimal energy, as wastobe proved.

Theorem 3 shows that in learning to construct an asymptotically extremal (n, k(n))-code G 11 with k(n) 2: ~w(n), we could construct an asymptotically extremal (n, cp (n))-code G11 for any function cp (n) 2: k(n). This theorem illustrates the huge diversity in methods of constructing these extremal codes.

It is worth noting another interesting fact about the "universality" of a group (n, k(n))code G 11 with asymptotically minimal energy.

Theorem 4. For any increasing function s (n) -s 2 k( 11 ) in the code G 11 there exists a set of cardinality s(n) with asymptotically minimal energy.

Pro of. Choose as a probability space the set of all subsets of cardinality s (n) from the code G11 and consider the random variable

where M = {A11 A2, 0 0 0, As} ~ Gno As in Lemma 1, using symmetry of the code G11 , the mathematical expectation of ~ is readily calculated:

38

'-.r--' d(n)

r(n) r-"-----1

V. K. LEONT'EV

n MG- s(s-1) "..!!:!!._

- 2 (2k<nl -1) LJ k ' h=i

n

Fig. 1 where {a1(n}, a2(n), ... , a 11 (n)} is the spectrum of the code G 11• Since by hypothesis

it follows that

Hence

n " ..!!:!!._ ~ 2k+l LJ k n ·

I<= I

s2 Mt.~-

"' n ' (37)

which proves the theorem, since from (37) there follows that among s-subsets of the code G11

there mustexist at least one whose energy does not surpass M~.

Accordingly, in the construction of sets of any "increasing cardinality" with asymptotically minimal energy, it effectively suffices to have only one group extremal (n, k(n))-code with k(n) 2:: ß w(n), where w(n) is an arbitrary increasing integral function with natural argument.

We use this circumstance in §3 to construct sets with asymptotically minimal energy.

§ 3

Now consider a special class of group codes in which we can successively construct an extremal set.

We Iet n be the length of a binary sequence and Iet d(n} be a natural number: 2 :s d(n) < n. We divide n by d(n):

n = d (n) s (n) + r (n),

and correspondingly we separate our sequence into s (n} blocks of length d (n) and a block of length r (n) (see Fig. 1).

Each block ui is filled either entirely with single zeros or entirely with single ones. It is clear that the thus obtained set of "block sequences" (points) is a group (n, s + 1)-code. The spectrum is readily calculated of the code Gl1' Actually each point of the code G 11 has weight either vd (v = 0, 1, ... , s) or vd + r (v = 0, 1, ... , r). Here it is obvious that avd = avd + r = C~ and a r = 1 (for r ~ 0). We have the following result.

Theo_rem 5. The code G 11 has an asymptotically minimal energy for S(n) 2:: log n + w (n), where w (n) is any increasing function.

Proof. We calculate the energy of the code G 11 • Wehave

S cv S V

H (Gn) = 2'" -" +2' ~ _s__+~. LJ vd ~ vd+r r V=i v=O


Consider the asymptotic behavior of H(G11 ) as n-oo and s(n) 2: log n + w(n), where w(n) is some increasing function. We have

s ~v ,s u , ~ C, __ 2s ~ C, 22s+1 2 ----- -~--

vd d u sd v~l v=l

and

I ')2s+" ~)s

since 0 < r d < 1. Hence H(Gn)- 1-ld.l l -,--1-; however, it is clear that s n n r n)

n= s (n) d (n)

[ 1 + s (:) (;)(n) J -~ s (n) d (n), since lim s (n) = oo 0 Hence, n-HX)

39

(2s+l )2 2s H(Gn)~---1--- o

n ' r (n) ' (38)

however,

r ~:) < 2' = 0 ( 2~s )

on the strength of the condition s (n) 2: log n + w (n). Thus we get finally

H (G ) ~ (2s+l)2 n n '

i.e., the code G11 has asymptotically minimal energy.

With the aid of Theorem 5 we can construct an asymptotically extremal (n, s(n))-code for the function s(n) having the form

(39)

where 2 < d (n) <;:: 1 n ( for arbitrary increasing function w (n). There are two unsatisfac-ogn-;-w n)

tory things about this method. First, the sequence s(n) = [n/d(n)] does not represent allintegral functions s (n) ~ n and second, there are bounds on the growth of the function s (n), i.e .,

1ogn-j-w(n)<;:S(n)<;: ~ 0

Our approximative problern will be free of these defects. Theorem 3 shows that in the construction of asymptotically extremal (n, k(n))-codes with an arbitrary function (kn) 2: .0-w(n) it suffices to construct only one (n, k(n))-code with any particular function k(n) satisfying the inequality

k (n) > ~ro (n)o

By Theorem 5 we obviously are able to do this. Thus only the case k(n) < .0-w(n) remains to be cleared up.

40 V. K. LEONT'EV

This case can conveniently be broken down into three subcases:

I. log n+w (n) < k (n) < ßro (n)o Ilo log n< k (n) <log (n) +w (n)o IIIO k (n) <logno

The first subcase presents no difficulty. We need only follow the preceding construction

using, as d(n), the function [ k (n~ _ 1 J 0 All the foregoing arguments remain in force.

In the second case we try to change the construction in such a way that in (38), r(n) will be an increasing function, whichwe then choose inthe capacityof w(n). [Itis easyto imagine that r(n) = o; the above introduced construction would need no changes.] That is to say, we let

log n < k (n) < log n + w (n)o

As above, we divide sequences of length n into blocks of length d(n) of such a formthat the number of all the blockswill be k(n). If, meanwhile, the unique block of length r(n) = n- [k(n) - 1]d(n) increases with growing n, then all the previous constructions, in order, proceed without change. If this is not the case then we introduce the following point of view: an increasing sequence of blocks in some growing length is counted as a sequence of blocks, but of such form that in that sequence of blocks the next to the last block had a growing length. Obviously this can be done. Clearly, the dimension of the code does not change because of this. Its spectrum will emerge as follows:

Ck_2 points of weights vd, v = 0, 1, ... , k- 2;

Ck _2 points of weights vd + r 1, v = 0, 1, ... , k - 2 (r 1 is the length of the last block);

Ck _2 points of weights vd + d1, v = 0, 1, ... , k- 2 (d1 is the length of the nex:t to last block);

Ck-2 points of weights Vd + d1 + r1, V= 0, 1, ... , k- 2;

one point of weight r 1;

one point of weight d1.

We now calculate the energy of this code. Wehave

R-2 CV k-2 cv k-2 cv R-2 V " 2 k 2 k 2 C "-2 2k-1 2k-l H(Gn)=2k-1~_x--+2k-1"" - -+-,·2"-1"" -, +21<--1"" x +--+--..G:J vd ..G:.I vd -f- d1 ..G:.I vd ~- r 1 ..G:.I -v-,-d -j-,---r 1---:-t--,d,.-1 r 1 d 1 °

v=l v=O v=O V=O

As in the preceding case, it is not difficult to get an asymptotic expression for H(Gn):

(40)

where now lim r1 (n) = oo and lim d1 (n) = ooo On the strength of the condition n--+oo

k (n) >log n it follows from (40) that

i.e., the code Gn has asymptotically minimal energy, as wastobe proved.

ASYMPTOTICALLY ST ABLE DISTRIBUTIONS OF CHARGE 41

In case III we need only use the equidistance of the MacDonald [3] or the Plotkin [3] codes, which, between arbitrary pairs of points, have distance asymptotically equal to n/ 2. Any of the subsets of these codes with increasing cardinality is an asymptotic extremal subset, as required.

We can, incidentally, easily get from Lemma 5 the following precise inequality of the energy of s-subsets

H (M) > (s-1)2 0

n

Choosing s = n + 1 and, in the role of M, that MacDonald code all code points of which have weight d = (n + 1)/2, we set

2 1 H (NI) = Cn+i 0 d = no

From (41) and (42) it follows that the code M has absolutely minimal energy.

Literature Cited

(41)

(42)

1. B. S. Zil 'berman, "On the distribution of charge in the vertices of the unit n-dimensional cube," Dokl. Akad. Nauk, Vol. 149, No. 3 (1963).

2. T. N. Kruglova, "On asymptotic methods of solving problern on charges," in: Problemy Kibernetiki, Vol. 13, Moscow (1965) •

• 3. W. Peterson, Error Correcting Codes, Wiley, New York (1961). 4. G. Polya andG. Szego, Aufgaben und Lehrsätze aus der Analysis, Springer, Leipzig

(1935). 5. D. D. Joshi, "A note on upper bounds for minimum distance codes," Inf. Control,

1(3):289-295 (1958). 6. J. MacWilliams, 11The structure and properties of binary cyclic alphabets, 11 Bell. Syst.

Techn. Journ., 44(2):303-332 (1965).

ON NETWORKS CONSISTING OF FUNCTIONAL ELEMENTS WITH DELA YS t

O.B.Lupanov

Moscow

§ 1. Statement of th e Problem and Formulation

of Results

As is weil known, in "traditional" methods for the realization of logic-algebra functions (by contact networks, Il-networks, networks consisting of the functional elements, formulas) the so-called "Shannon effect" holds: "almost all functions" of n arguments have "an almost identical" complexity which is asymptotically equal to the complexity of the most complex function of n arguments. The hypothesis on this effect was stated by C. E. Shannon in 1949 (see [11]) and was subsequently proved by the author of the present paper (see, for example, [5, 3]). In certain cases (for example, for disjunctive normal forms) the "weakened Shannon effect" holds- "almost all functions of n arguments have almost identical complexity"; true, this complexity is less than the complexity of the most complex function [1, 7, 8].

In the present paper a certain natural class of supervisory systems is consideredproper networks consisting of functional elements with delays, for which these effects generally do not hold.

We shall consider networks consisting of elements of the following form. Each element Ei is juxtaposed with a certain logic-algebra function cpi (x,, ... , xn) (which depends essentially on k i arguments) and two positive numbers: Pi, the "weight of the element," and Th the "delay of the element"; these numbers arenot assumed tobe integers. If ki 2: 2, then the numbers Pd (k i - 1) and T /log ki t shall be called the red u c e d w e i g h t and the red u c e d d e 1 ay of the element E ü respectively. It is assumed that the number of inputs of the element Ei for ki 2: 1 is equal to ki, while for ki = 0 (i.e., if the element realizes a constant) it is equal to 1.

The networks are constructed over a finite set '15 = {E,, ... , Er} of elements of the indicated form in accordance with the rules of constructing networks from functional elements [5, 4].

t Original article Submitted December 30, 1968. t Here we have in mind binary logarithms throughout.

43

44 O.B.LUPANOV

We shall call the chain between the elements E (i) and E< 2 l the sequence of elements E;1 , .•. , E;., having the properties:

1) a certain input of the element E;i is connected to the output of the element (2 :s j :s s);

2) E;i-i (2-<.j<.s); 2) Em = E;1 , E<2> = E; •.

We shall call the number T; 1 + ... + T;. the de 1 ay of this chain. We shall call a chain pr in c ip al if a certain input of its first element E;1 is an input of the network, while the output of its last element E;. is an output of the network. The network S shall be called proper if the delays of all of its principal chains are equal. t This common number shall be called the delay of the networ k and denoted by the symbol T (S). The function realized by the network and the complexity L (S) of the network are defined in the same way as they are for networks consisting of functional elements.

We shall assume that the system ~ of original elements is complete in the sense that for each logic-algebra function f one may construct a proper network which realizes f (with a certain delay); i.e., we have in view "completeness in the second sense" in the terminology of V. B. Kudryavtsev [2]; however, here commensurateness of the delays of the basis elements is not assumed.

A complete basis shall be called r e g u 1 a r if networ ks in this basis may be used to realize all four functions of one argument x, :X, o, 1, t having one and the same delay. Otherwise, the basis shall be called irre g u 1 a r. From the results obtained by V. B. Kudryavtsev [2] it follows that in the case of commensurate delays (for example, integer delays) the basis is regular. §

The logic-algebra function f (x1, ••• , xn) shall be called an x-function (correspondingly an X-function, a 0-function, a 1-function) if the function f (x, ••• ,X) is equal to X (correSpondingly X, 0, 1)~ ; X- and X-functions shalllikewise be cailed «<J-functions, while 0- and 1-functions shall be called c-functions.

Let us consider the following Shannon functions:

L(f) - the minimum of the complexities of (proper) networks which realize the function f;

T(f)- the minimum of the delays of networks which realize the function f. Assurne SRn (SR~, SR~), respectively) is the set of all functions (correspondingly, c-functions, «P-functions) f tx1, ••• , xn), and assume

L (n) = max L (!), fE'inn

Lc (n) = max L (!), !E'in~

Lw (n) = max L (!), !E'in~

T (n) = max T (!); fE'inn

Tc (n) = max T (!); fE'in~

rw (n) = max T (!). IE'in~

t From this definition it follows that for any network elements the delays of all chains traveling from the inputs of the network to the inputs of this elementare equal.

t Below (§ 2) another (equivalent) definitionwill be given. The existence of irregular basis will be established in §4 (Lemma 14).

§ See likewise below, p. 4 7. ~In E. Post's terminology (see [10, 9]) these are correspondingly a-, ö-, y-, ß-functions.

NETWORKS OF FUNCTIONAL ELEMENTS WITH DELAYS 45

Assurne CI> is the set of all basis functions qJi (:r1, ... , xk), which depend essentially on at least two arguments, while CI>* is its subset consisting of all rJl -functions. Assume, finally, that

. Pi pc=ffilll~ l

'PiE<l> '

o* c= min ____!_i__ I cpiE<l)* lq~1 '

T=min _T_i -· 'PiE<!> log k; '

T* = min _T_; -<Jl,E<l>* log k;

It is obvious that if the numbers p* and T* are defined, t then

p*;:>p, T*;:>T.

The following statements hold.

Theorem 1. If the basis ~ is regular, then

2" 1) L (n) ~ Lc (n) ~ L 0 and any function f (x 1, ... , Xn) of a fairly large number of arguments there exists a network S which realizes f and is such that

2" L (S) < (1 + e) p n, T (S) < (1 + e) w.

Theorem 2. If the basis ~ is irregular, then

la) 2" Lc (n) ~ p n;

lb) 2" L (n) ~ L 0 and any c-function (correspondingly, <P -function) f (x 1, ••• , x 11 ) of a fairly large number of arguments there exists a network S which realizes f and is such that

2" L(S)<(1+e)p-, T(S)<(1+e)w n

(respectively, L (S) < (1 +e) p*_r_, T (S) < (1 +e) T*n) . n

These theorems show that although p and T (and likewise p* and T*) may be attained on various elements, it is nevertheless true that for almost all functions it is possible to construct networks which are asymptotically optimal simultaneously with respect to both complexity and delay. Theorem 2, moreover, indicates that in the case of an irregular basis and

t From Lemma 2 proved below it follows that in the case of an irregular basis CI> is non-empty.

t Actually, for "almost all" functions f (x1, ... , x11 ) of n arguments, L(f) "' p(2n/n) and T(j) "' Tn.

§ For "almost all" c-functions (correspondingly, <P-functions) j(x1, ... , Xn) of n arguments, L(f) "' p(2 11/n) and T(n) "'T (n) [L(j) "' p*(2n/n) and T(j) "' T*n, respectively].

46

N

2z" ----------r-----------1 I I I I

O.B.LUPANOV

N

22" ---------------

I 2" -z2 ----------11~----....L~----- -------

Fig. 1

1 I

/J·t I "

L

Fig. 2

struct networks which are asymptotically Qptimal simultaneously with respect to both complexity and delay. Theorem 2, moreover, indicates that in the case of an irregular basis and p* > p "almost all" functions can be partitioned into two subsets, each containing "almost half" of all the functions; these subsets are realized with different complexities. This phenomenon may be more conveniently expressedas follows. Let us use N 11(L) to denote the nurober of functions f (x1, ••• , x 11) for which L (j) ::::; L. The behavior of the functions N 11 (L) is depicted approximately in Fig. 1. fu the case of "traditional" methods of realization, the approximate behavior of the corresponding function N 11 (L) is depicted in Fig. 2. The increase of the bounds for Lrfl (n) and Trfl (n) in the case of an irregular basis occurs due to the fact that in this case networks for cp-functions may consist of elementsbelanging only to apart of the basis ( 11 cp -elements" - Lemma 2).

The first statement of Theorem 1 (for the condition that the delays of the elements are integers) is an almost trivial corollary of the theorem on the asymptotic behavior of the Shannon function for networks consisting of functional elements ([5]; see also [6], Theorem D.12). The second statement (for the same condition) is likewise almost obvious. The third statement is somewhat less trivial (if p and T are attained on different elements). It turns out to be possible due to the fact that the "complexity of the network" and the "level of the network" are created by different parts of the network, and these parts of the network may be constructed from elements of different ldnds.

Plan for Proving This Theorem. In§3thelowerboundofthefunctionsL and T will be established. In obtaining the bounds for Lrfl (n) in the case of an irregular basis the property mentioned above of such basis (§2) is used. The lower bounds for T(n) and TC(n) are likewise actually known. Their proof is presented solely for completeness of the picture.

The methods of network synthesis mentioned in the third statements of both theorems (these methods give the upper bounds for the asymptotic relationships in the first two statements) are described in §§8-10 (§§4-7 contain auxiliary propositions).

Three synthesis methods will be presented:

1) for realization of any functions in an arbitrary regular basis;

2) for realization of 0-functions in an arbitrary irregular basis;

3) for realization of x-functions in an arbitrary irregular basis.

It is obvious that special methods of network synthesis (in an irregular basis) for 1-functions and for x-functions are not required - the corresponding networks can be obtained from networks for 0-functions and x-functions as a result of attaching networks realizing negation to their outputs.


§ 2. Certain Properties of Regular Basis

We shall call the set of functions m j o in t (relative to the basis ~) if all functions from m can be realized by networks with identical delays. By virtue of the completeness of the basis, the following statement is valid.

(*) Assurne m is a joint set and cp (x1, ... , Xm) is an arbitrary function. Then the set of functions of the form cp(cp 1, ... , <Pm), where Cft E ~m, is joint.

Lemma 1. In order for the system of functions {x, x, 0, 1} tobe joint (i.e., in order for the basis tobe regular) it is necessary and sufficient that just one of the pairs

{.c, .x}, {x, O}, {x, 1}, {x, o}, {x, 1}

be joint.

Proof. The necessity is obvious.

Sufficiency. Assurne that the pair {x, x} is joint. Let us consider the function g(x, y, z) which satisfies the conditions

Then

g(O, 0, O)=g(O, 0, 1)=g(1, 1, 1)=0,

g(O, 1, 1)=g(1, 0, O)~g(1, 1, 0)=1.

g(x, x, .:r)==O, g(x, x, x)=x, g(x, x, x)=x, g(x, x, ~=1,

and by virtue of (*) the system { x, x, 0, 1} is joint.

Assurne that the pair {.x E8 a, ß} t is joint (any of the four latter pairs in (1)). Let us consider the function cpt (x) = x E8 a E8 1. By virtue of (*) the functions cp2 (x) = cp1 (x E8 a) = x and cp3 (x) = cp1 (ß) = a E8 ß E8 1. are joint.

Further, the functions

(jl2 (cpz (x)) = x, (jl2 (cp3 (x)) = a E8 ß, (jl3 (qJ3 (x)) = a E8 ß E81

are joint; i.e., x, 0, 1 are joint.

Finally, let us consider the function h(x1, x2, x3, x4) which satisfies the conditions

Then the functions

h(O, 0, 0, O)=h(O, 0, 0, 1)=h(1, 1, 1, 1)=0,

h(O, 0, 1, 1)==h(O, 1, 1, 1)=1.

h(O, 0, 0, x')=O, h(O, 0, x, 1)=x, h(O, :r, 1, 1)=1, h(x, 1, 1, i)=x

are joint.

The lemma has been proved.

C o r o 11 a r y ( s e e [ 2 ] ) • If all elements of the basis have rational delays, then the basis is regular.

t The symbol EB implies addition modulo 2.

(1)

48 O.B.LUPANOV

Actually, networks 81 and 82 exist which realize x and 0, respectively; assume that their (rational) delays are equal to p/q1 and p2/q2• Having connected q1p2 copies of the network 81 in cascade, we obtain the network for x having the delay p1p2• Analogously, from p1q2 copies of the network 82 we obtain the network for 0 having the delay p1p2• In this way the pair {x, 0} is joint, and by virtue of Lemma 1 the basis is regular.

Assurne * E {x, x, 0, 1, .p, c}. The basis element realizing the *-function is called a *element.

Lemma 2. Assurne ~ is an irregular basis and 8 is a network in ~. which contains at least one c-element. Then 8 realizes a certain c-function.

Pr o o f . Assurne 81 is the network obtained from 8 as a result of identification of the inputs. Let us consider the principal chain A = [E;1, ••• , Ei), passing through a certain celement E;i. Then either constants are applied to all of the inputs of this element, or the function x is applied to all of its inputs, or the function x is applied to all of its inputs (since otherwise one of the pairs (1) would be joint, which is impossible for an irregular basis). In all three cases a constant is realized at the output of this c-element. Therefore (again by virtue of the irregularity of the basis) constants are likewise applied to the inputs of all the next elements of the chain Ei. 1, ••• , E; , and the network 8 1 realizes the constants.

J+ s


Let us consider two properties of a basis.

I. Two chains with identical delays exist which consist solely of .P-elements, one of which contains an even number of X-elements, while the other contains an odd number of Xelements.

II. Two chains exist with identical delays, one of which consists solely of .P -elements, while the other contains at least one c-element.

The chain E;1 , .•. , E;. shall be called special if all inputs of the element E;1 are attached to one input of the network, while all inputs of the element E;1 (2 <;_:: l <;_:: s) are attached to the output of the element E;1_1 (Fig. 3). Each special chain (treated as a network) realizes a certain function of one argument, say of x.

Let us now give a different definition of a regular basis, which is more convenient for checking.

Lemma 3. In order for a basis to be regular it is necessary and sufficient that it have one of the properties (N 1 ) and (Nu)·

Pr o o f. 8 u f f i c i e n c y • If property I is fulfilled (property II, respectively ), then having formed the corresponding Special chain we find that the pair {x, x} (a certain pair {x EB a, ~ }) , respectively) is joint; therefore, the basis is regular by virtue of Lemma 1.

Ne ce s s i ty. Assurne that the basis is regular, and assume that T* is the minimal positive delay with which functions of a certain pair (1) may be realized (see Lemma 1). Assume 81 and 82 are two networks having the delay T* which realize the functions indicated. By virtue of minimality of the number T*, one of the following possibilitiest holds for each element E of these networks:

t Compare with the proof of Lemma 2; there an analogous situation arose due to the incompatibility of any pairs at all from (1).

Af. V 1s

Fig. 3


a) constants are applies to all inputs of this element;

b) the function x is applied to all inputs of this element;

c) the function x is applied to all inputs of this element.

Assurne S is one of those networks which realize the function x ffi cx, and E;1 , .•. , E;8 is its principal chain. Then for all elements of this chain b) or c)

is fulfilled (since if constants are applied to the inputs of the element Eij a constants will appear at its output; therefore, constants are applied to all inputs of the element E; , etc.). Therefore, the chain consists only of <P -elements •

.7+1

Let us now consider two cases.

1) The networks Si and S2 realize x and x. In this case their principal chains consist of <P -elements, an even number of x elements being contained in the first and an odd number in the second (i.e., the property I is fulfilled).

2) The networks Si and S2 realize x ffi cx and ß. In this case the principal chain ßi of the network Si consists of <P -elements. Let us consider the principal chain 1'1 2 ~· [E;1 , •.•

of the network s2• Assurne E;i is the first element of this chain, which produces a constant (since E;, produces a constant, it follows that such an element exists). Then b) or c) is

fulfilled for it. Therefore, E;i is a c-element. For the chains ßi and ß 2 the property II is fuilfilled.

The Iemma has been proved.

§ 3. Lower Bounds of Shannon Functions

A. Complexity Estimates

Lemma 4. The following relationships are valid:

2n c 2n <ji zn L (n) d p- . L (n) d p- , L (n) d p- , n n n

while in the case of an irregular basis,

"' Zn L"'(n) d p*-. n

Actually, it is well known (see [5, 6]) that for "conventional" networks consisting of functional elements (without matehing of the delays of the circuits) the Shannon function L 0(n) satisfies the condition

0 2n L (n) d p-, n

where p is the minimum of the reduced weights of the basis elements. In the case when only proper networks are used the Shannon functions may only increase. Therefore, in our case we have

2n L(n) ~ P-;;-.

This same inequality is also valid separately for <P-functions and c-functions (since the logarithms of the numbers of functions from IJl* and from IJl~ are asymptotically equal to 2n).

50 O.B.LUPANOV

In the case of an irregular basis the lower bound for Lw (n) is raised: by virtue of Lemma 2 one may use networks over only a portion of the basis consisting of <P -elements in the realization of <P -functions; therefore, the constant p in the lower bound may be replaced by p*.

B. Delay E stimates. Let us use M(T) to denote the maximum number of elements in irreduciblet proper networks in the basis ~. which have one output and satisfy the condition

T(S)<.T. T

Lemma 5. M(T)<.C1T2-r:, where T is the minimum of the reduced delays of the basis elements .t

Pr oof. Assume T 0 is the least of the delays of the basis elements. Let us partition the domain of variationofT into half-intervals Ji == [(i -1)T 0, iT 0). Weshall use induction with respect to i - the number of the half-interval to which T belongs - to prove the inequality

T

M (T)<. ~0 27 (2)

If i == 1, i .e ., 0 :s T < T 0, then M (T) == O, and the inequality (2) is obvious.

The Inductive Step. Assume TEl;, i-:>2. InthiscaseT~T 0,andM(T)is attained on the network S having at least one element. Assume E is an output element with k inputs and the delay T* •

If k == 1, then we have, on the basis of using the inductive proposition and the inequalities T ~ T 0 and T* ~ T 0,

T-T* T T T T

Jf(S).<:i+M(T-T*)<l+_!_;T* 2-'-<.1-+- T-TTo 2-;; =1.J__TT 21 -21 < TT 2r. 0 0 0 0

1' T* T* If k ~ 2, then • <- log k log k <. --=t, k <. 2 -r: , and by analogy we have

1'-T·• T T T T . T- T* -- T- T* - 1' - T* - T -!Y!(S)<.1+kM(T-T*)<1+k-1,-2 T <.1-J--7-2-r: =1+-r 2-r: --r 2" <.-1. 2T.

n o o o o


Lemma 6. The relationships

T (n) J. Tn, Tc (n) 'j· Tn, Tq, (n) ;c Tll

are valid, while in the case of an irregular basis, T'P (n) ;e. T*n.

t A network is called irreducible if for each element the following condition is fulfilled: its output is either an output of the network, or it is attached to the input of a certain element. It is obvious that for each proper network Sone can indicate an irreducible proper network S' which realizes the same function as S does and is suchthat L(S') :s L(S) and T(S') == T(S).

t Throughout this paper the letter C with its subscripts is used to denote constants (in general, constants which depend on the basis).


Pr oof. Assurne e is an arbitrary positive number. Let us consider networks whose delay does not exceed (1- e)rn. By virtue of Lemma 5 the nurober of elements in each such network does not exceed

Therefore, for n > ne it follows by virtue of Lemma 4 that not just any function of n arguments may be realized by such a network. Thus, for n > ne we have

T(n)>(1-c)Tn.

In exactly the same way the following inequalities arevalid for fairly !argen:

Tc (n) > (1- c) w, TtP (n) > (1-c) Tn.

Finally, in the case of an irregular basis the constant r may be replaced by r * in the lower bound for Trp (n) (for the same reason as it is in the proof of Lemma 4).

The Iemma has been proved.

§ 4. Certain Auxiliary Statements

Lemma 7. There exist networks S 1 , S 2 , S3 , which have one and the same delay T< 1 l and realize the functions x, xy, xVy, respectively.

Pro o f. By virtue of the completeness of the basis, a proper network S0 exists having a certain delay T 0, which realizes the function 6 (x, y) = x V y. Therefore, the function 6 (6 (x1, x2), 6 (:c3 , x4)) = x 1x2Vx3x~, may be realized with a delay 2T0•

Identifying the inputs for this function in the network, we obtain the required networks:

when x1 =x, x2 =x, x3 =x, x4 =x-function for x; when x1 = x, x2 = y, x3 = x x4 = y- function for xy; When Xt=X, X2=X, Xao~y, x,=y- function for xvy.

Lemma 8. For any r and m, r ~ m, there exists a network Kr,m which realizes x 1x 2 ••• xr, and is such that

L (Kr, m) <: C3m, T (Kr, m) =]log m [T(i)_

Pr oof. Assurne tJ. =]log m [. The network Kr,m is obtained from a tJ.-story dichotomic tree which is constructed from the network s2 (and realizes a conjunction of length 2J.l) as a result of identifying certain inputs. Let us note also that m ~ 2J.l <2m.

The following analogaus Iemma is valid.

Lemma 9. For any r and m, r ~ m, there exists a network Dr,m which realizes x1Vx2V ... Vxr, and is suchthat

Lemma 10. There exist networks S4t S5 , s 6 , s 7 having one and the same delay T< 2 l and realizing the functions 1, o, xy, xVy, respe cti vely.

52 O.B.LUPANOV

Pr o o f. By virtue of the completeness of the basis, proper networks Sex and 88 exist which realize the functions a (x, y) =xVy and ß(x, y) = iy, respectively, having the delays T cx and T 8. Therefore, the following functions may be realized with a delay Tex + T 8:

a(ß(x,x), ß(x,x))=i, ß(a(x,x),a(x,x))=O,

ß(a(x, x), a(y, x))=xy, a(ß(x, x), ß(y, x))=xVy.

Assurne K'ln (x1, ... , xm) is the system of all conjunctions that (a 1, ... , am) ~ (0, ... , 0), (a 1, ... , am) ~ (1, ... , 1).

which are such

Lemma 11. For any conjunction from K~ there exists a network S which realizes this conjunction and is such that

L (S) <C5m, T (S) = r< 2> +]log m [T< 1>.

Proof. The network S is obtained from the network Km,m as a result of attaching m s6 networks for X'y to its inputs (under these conditions a conjunction of the form u1 ... um v1 ...

v m is realized) and identifying certain inputs.

Along with the networks S1-S1 the network S8 (which exists by virtue of the completeness of the basis) will be used further on; this network has a certain delay T (3) and realizes the function 6(x, y, z, w)=x(yzVW).

The functions x1x2 0 0 0 xmVxaqtxaq2 0 0 0 xaqz where l::::; m, 1::::; q1o::::; m, and the collection qi q2 ql ,

(crq1 , 0 o 0 , <Jq1) differs from the zero collection, shall be called A-functions. From this definition it follows that any A-function is an x-function.

Lemma 12. For any A-function a(x 1, ... , xm) there exists a proper network S which realizes this function and is suchthat

Pr o o f . It is easy to check the fact that

We shall denote this function by w (u1, ... , um, v 1, ... , v m, w 1, ... , w m).

oa (] Assurne now that a = x1 0 • 0 Xm V x qt 0 0 0 x qz is an arbitrary A-function. Its variables

qi ql

can be partitioned into three groups:

1) variables appearing in the second conjunction without negations (i.e., appearing in both conjunctions); assume these are x;1, 0 0 0 , Xia;

2) variables appearing in the second conjunction with negations; assume these are

3) variables which do not appear in the second conjunction; assume these are xh1, 0 0 0, Xhc·

It is clear that if 1 ::::; a ::::; m, 0 ::::; b :s m - 1, 0 ::::; c ::::; m - 1. Therefore, the function a may be derived from w as a result of a certain Substitution of variables (if a = m, then we assume that v 1 = ••• v m = w 1 = ••• = w m = x1; if c = 0, then we assume, for example, as a preliminary step,thatw1= ... =wm = v1).

NETWORKS OF FUNCTIONAL ELEMENTS WITH DELAYS

The network S for a is derived from the network Km m and m s7 networks attached to inputs of the network Km m as a result of the identification 'of inputs described above.

'

53

Finally, a proper network S9 exists having a certain delay T( 4), which realizes the function ~(x, y, z)=xyVxz.

The networks s1-s9 shall be called c an o n i c a 1 •

Rem ar k. Instead of the function ~(x, y, z, w) one could take a "simpler" function 'YJ(x, y, w)=x(yVw), since 11(x, y, w)&'YJ(x, z, w) =~(x, y, z, w), or the function ?;(x, y, z) just introduced above, since Uw, x, y) & ~ (y, x, x) = 11 (.1,', y, w).

Lemma 13. The system of elements realizing the functions ö(x,y) =xVy, ß(x, y)=xy, ~(x, y, z, w)=x(yzVw) having any delays (in particular, noncommensurate delays) is complete.

Actually, any 0-function f (x1, ... , x 11) may be realized as a disjunction of conjunctions from K~; under these conditions only elements for ß and ö are used (see Proof of Lemmas 7, 8, 9, 11). Any 1-function can be realized as a negation of a 0-function (negation realizes one additional element for ö with identified inputs). Any x-function can be realized as a disjunction of A-functions; under these conditions the elements for ö and ~ are used. Any xfunction can be realized as a negation of an x-function.

Lemma 14. Irregular bases exist.

Actually, an example of such a basis may consist of the system of elements for the functions ö, ß, ~ (Lemma 13) with the delays 1, 12, /3, respectively. Each of the conditions I and II is fulfilled, since the numbers 1, 12, f3 are linearly independent over the field of rational numbers.

Lemma 15. Assurne E is an arbitrary element of a basis which realizes a function that differs from a constant and has a delay T. Then the networks ZE and CE exist which have the properties:

1) ZE has three inputs and one output; its function g(x, y, z) satisfies the condition g(O, 1,z)=zffia (for a certain a);

L(ZE)=C7, T(ZE)=T.

2) CE has two inputs and two outputs; the functions g 0 (x, y) and g 1 (x, y) which realize it satisfy the condition g 0 (0, 1) = 0, g 1 (0, 1) = 1;

Proof. Assurne E realizes the function cp(x1, ... , xk). Constantst exist: a 1, ... , ak_1,a which are such that cp (a1, ... , ak _1, z) = z E8 a. Assurne i 1, ... , i 5 are the numbers of the zero components of the collection (a1, ... , ak_1), while j 1, ••• , h are the numbers of its unitary components. The network Z E is obtained as a result of attaching the inputs of the element E having the numbers iv ... , i 5 to the first input of the network, the inputs having the numbers h' ... , jt to the second input of the network; and the k-th input to the third input of the network.t

The network CE is obtained from two ZE networks, in one of which theinputz is identified with x, while in the other the input z is identified with y.

t For k = 1 the constants a 1, ... , ak _1 are missing. t If a certain constant is not used, then the network input corresponding to it is not attached

to the inputs of the element E.

54 O.B.LUPANOV

Rem ar k. A chain oft cascade connected networks CE realizes the constants 0 and 1 with the delays 0, T, ... , tT if the constants 0 and 1 are applied to the inputs of the chain. A chain of Z E networ ks of even length t (in the presence of constants produced by the chain of networks CE) realizes the function f (z) = z with a delay tT.

§ 5. Pieces and Principal Pieces

We shall consider collections of zeros and ones having a certain length m. Let us m

juxtapose each collection u = (a 1, ... , a m) with a nurober I 0: I = .2j cr;2m-i. The set of whole t=l

nurobers l which satisfy the condition l 1:::::; l < l 2 will be denoted by [l 1, l 2) and called a pie ce. A piece of the form [A.2i, (A. + 1)2i), where A. and i arenonnegative integers, will be called a p r in c i p a 1 p i e c e • The terms "piece tt and "principal piece tt shall likewise be used to denote sets of corresponding collections.

Lemma 16. Any piece I (of collections of length m or of their corresponding numbers) is a union of no more than 2m principal pieces.

Proof. Let us use o(Z) to denote the largest nurober o suchthat 2ö divides l [if l = 0, then o (Z) = ao ]. Assurne I= llt. l2), ö (/) = max ö (Z; t and assume l 0 is the nurober for

11<:::1<:::12

which o (l ~ = o (I). The piece [l 1, l 2) shall be called left if l 0 = Z1, right if l 0 = Z2, and middle if Z1 < l 0 < l 2• It is clear that the middle piece [ l 1, l 2) is the union of the right piece ~ 1, l 0) and the left piece [ l 0, Z2).

Let us now show that any left (right) piece is a union of no more than m principal pieces. Let us begin by considering the particular case I = [ 0, l). Assurne l = 2 ;1 + 2;2 + ... + 2is (ij arenonnegative integers; i 1 > i2 > ... > i8 ; s:::::; m). Then

I= [Ü, 2i1) U [2il, 2i1 + 2i2) U ... U [2il + ... + 2is-1, 2i1 + ... + 2is-l + 2is) =

= [Ü· 2\ 1· 2i) U [2i1-i2. 2i2' (2i1-i2 + 1) 2i2) U ... U [ (2ii-is + ... + 2is-1-is) i•, (2ii-is + ... + 2is-I-is + 1) 2is),

i.e., I is a union of no more than m principal pieces.

Assurne now that Z1 > 0 and I = [l 1, l 2) is a left piece. Wehave l1=(2s+ 1) 26<11>. Further,

(3)

since otherwise l*=lt+26<11>E[lh ld,on the one hand, while l*=(2s+2)2ö(Zil=(s+1)2'~<Zil+t, on the other hand; i.e., o (l*) > o (l 1), and this contradicts the fact that max ö (l) is attained

h<O:l<O:l2

for l = Z1• From (3) it follows that from the representation of the piece [0, Z2 - Z1) in the form of a sum of (no more than m) principal pieces one obtains the corresponding representation of the piece [l, l 2) (it is sufficient to attach Z1 to all ends).

Analogaus statements are also valid for right segments.

Thereby the lemma has been proved.

Rem a r k. The collection corresponding to numbers from the principal piece I = [A.2i, (A. + 1)2i) constitutes all possible collections (a1, ... , am-i' am-i+t' ... , am) for which

t The right end is taken into account!


m-i

a portion (CJ 1, ••• , CTm-i) is specified and A= S ai2m-i-i. Therefore, the characteristic funci=t

tion cp1 (x1, ••• , x m) of this piece is the conjunction <t ... x:~-r Assurne I(f) [I0(f), respectively] is the minimal number of pieces (principal pieces,

respectively) whose union makes up the set of collections on which f goes to unity.

Lemma 17. Assurne the function f depends on m arguments, while the functions j+ and f- are derived from it by adding and eliminating, respectively, one conjunction in the perfect disjunctive normal form. Then

10 (f) <.2m!(!), (4)

10 (j+) <.10 (f) + 1, (5)

10 (t) <.10 (!)+2m. (6)

Pr oof. The inequality (4) derives from Lemma 16. The inequality (5) is obvious. The inequality (6) derives from the following concepts. For elimination of one collection, the principal piece containing this collection is partitioned into no more than two pieces, one left and one right. Each of them is a union of no more than m principal pieces (see proof of Lemma 16).

§ 6. Principal Blocks and Their Properties

A • An ( E , t) - b 1 o c k. Assurne E is an element of a basis ha ving at least two inputs, and that t is a nonnegative integer. We shall call an (E, t)-block a proper network constructed from elements E and constituting a t-story tree (Fig. 4). Assurne E realizes the function cp (x1, ••• , x,J and has the weight P and the delay T. Then an (E, t) -block consists of (~-t t - 1) I (~-t - 1) elements, has a delay tT, a complexity [(~-t t - 1) I (~-t - 1) ]P, N == ~-t t inputs, and realizes a certain function F(y 0, ••• , YN _1) which depends essentially on N arguments (the arguments are numbered in natural order). In particular, a network consisting of one pole is an (E, 0)-block. (E, l)-blocks (0::::; l::::; t) which areentered in the composition of an (E, t)-block and are attached to its inputs shall be called its principal subnetworks.

The set of variables assigned to the inputs of any (E, l)-block entered into the composition of an (E, t) -block (and likewise the set of numbers of these variables) shall be called an inter v a 1 (having a height t -l) (of variables and numbers of variables, respectively). It is obvious that an interval of numbers of variables is a piece in the sense of § 5.

L e m m a 1 8 . A s s um e S i s an ( E , l ) - b I o c k; S 1 , ••• , S x a r e i t s p r i n -cipal subnetworks, which are (E, l- 1)-blocks; J 1 , ••• , Jx are the intervals of the variables of these subnetworks, and F', F 1 , ••• , Fx are functions realized by the networks S, S 1 , ••• , Sx, respectively. Then for any i (1 ::::; i ::::; ~-t) one can find a collection of values of arguments which do not appear in J i and which when substituted into the appropriate place in the function F' cause the latter to go over to the function F; EB a;, where ai is equal to 0 or 1.

Pro o f. Since depends essentially on all of its arguments, there exists a collection of constants a 1, ••• , a i _ 1, a i• a i + 1, ••• , a x which is such that

(7)

Further, for each function F i (j ~ i) there exists a collection of values of its arguments (i.e., arguments from Ji) which converts it into a i. The statement of the lemma derives from this and from (7), since F' = cp(F1, ••• , Fx>·

56 O.B.LUPANOV

!lo

Fig. 4

Lemma 19. Assurne F(y 0, ••• ,

YN _ 1) is the function realized by the (E, t)-block 8. Then there exist constants c 11 ,c (0 ::::; ry, ~ ::::; N - 1) which have the following properties.

1) Assurne I is an arbitrary interval of numbers of variables, and aET. Then all constants ca cx

t for fixed a and any ß from I are equal to one another.

2) F (c11 , 0, c11 , b ••• , c11 , 11-t. y11 , Cn, n+t. . · ·,

cTJ, N-1) =Y11 EiJc11 , 11 ('I'J=Ü, ... , N-1).

Pro o f . Assurne 81, ••• , 8x are principal subnetwor ks of the networ k 8, which are (E, t -1)-blocks; F1, ••• , Fx are the functions realized by them; and J 1, ••• , Jx (!1, ••• , Ix, respectively) are the intervals of variables (the numbers of variables, respectively) of these subnetworks. By virtue of Lemma 18 there exists for any i (1 ::::; i ::::; x) a collection of constants c~> (y~l;; i .• e., rE!(i-1)xH ix1- 1)), which when substituted in place of the variables xy cause the function F to go over to F; ElJa; (a; is a certain constant). Let us now assume that cTJ,b=cti>. for all pairs (ry,t;) which aresuchthat 'I'JE/;, ~Eli and i ~ j. It is obvious that

these constants c11 ,c (these are part of all required constants) satisfy condition 1).

We deal analogously with "smaller" intervals. Assume, for example, that 8w ... , 81x are principal subnetworks of the networks 81, which are (E, t- 2) blocks, while J 111 ••• , J 1x (Iw ..• , I1x, respectively) are intervals of variables (numbers of variables, respectively) of these subnetworks. Applying Lemma 18 to the network 81, we obtain the result that for any i (1 ::5 i ::5 x) there exist constants cv· i) ("\'Eh 'V~ /ii), which when substituted in place of Xy

cause the function 1 to go over into F 1i E8 aii ( F 1i is the function realized by the networ k 81 i;

au is a certain constant). Let us now assume for all (ry, !;) suchthat 'I'JE/1;, ~Elu i=!=j;

that cTJ, b ==c~i, i), etc. In the final analysis for all pairs (TJ, (;) suchthat 17 ~ (; there will be defined constants c 11,, which satisfy condition 1) and are suchthat for any 17

where c 11 , 11 is a certain constant.


B. Partitioning of Collections

Lemma 20.t Assurne that the element E has x inputs; F(y 0 , ••• ,

yN_ 1) is a function realized by an (E, t)-block; m is an integer and

M = J ~ [ . T h e n th er e e x i s t

the numeration ~ of all sets of length n consisting of zeros and ones by pairs of subscripts: a1 .~ (0-<l<M-1; 0<~-<N-1)t;

t This lemma characterizes the construction of generalized expansion functions l/J for the case in which "an external expansion function" is a "homogeneous Superposition" of functions which depend on a limited number of arguments.

t 0 ::5 ~ ::5 2 m- 1 - (M - 1)N for l = M - 1.


the system of functions 1/Jzh(Yh, :r~> ... , xm), U<:;:l<:;:M-1, U<:;:h<:;:N-1, which have the properties:

1) F('ljlz,o(Yo, Gz,~;), 1/JI,I(Y~o Gz,<), ... , 1/JI,N-dYN-I•al,~)) = Y~;; if a~Z:z, then

2) any function l/J from the set 'P'={1jJ1 ,h(ö,x~> ... ,xm)} (O<:;:Z<:;:M-1, O<:;:h<:;: N-1, 6=0,1) satisfies the condition

I (tp) < xt + 2.

Pr o o f. Let us juxtapose each collection ä of length m consisting of zeros and ones withapairofnumbers (A., J.L)whicharedefinedasfollows: lcri=I..N+~-t; 0<1..; O<~-t<N.

57

It is obvious that A. and J.L are defined uniquely according to the collection a. Let us nurober the collection ä by means of the indicated pairs of subscripts. Thereby the partitioning of the set of all collections ä into ordered subsets l::z will be defined. Let us note that the collections from I:z form a piece.

Assurne now that clJ,!: (0 ::s Y], t ::s N - 1) is a collection of constants which have the properties 1) and 2) (Lemma 19) and that (e 0, ••• , eN_1) is a collection such that

(8)

Let us now define the functions l/Jzh (yh, x1, ••• , xm) as follows:

}, =1=-l;

1.. = l and~-t =1=- h; 1.. = l and fl = h.

Then the property (1) of the functions lf!z h derives from the definition of these functions, the ' property 2) of the constants (see the formulation of Lemma 19), and (8).

Let us establish the property 2). The partitioning of the numbers of variables 0, 1, ... , N- 1 into intervals, which is defined by an (E, t)-block, induces the partitioning of each of the sets 2:: z into subsets whi eh shall likewise be called intervals t; in particular, the sets 2:: z themselves will be intervals. It is clear that each of the intervals of the sets is a piece. Let us now consider the following system of partitionings of the sets of all collections ä "relative to the collection uz,h":

0) 2:: 0' 2:: 1' • • ., 2:: M - 1;

1) l::z is partitioned into the intervals 2:: z1, ••• , I:zx having the length 1;

2) assume cr1, h E l.:!i1 • The interval :Z.:ti1 is partitioned into the intervals :Z.:!i11 , ...•• l::li",

having a height 2, etc. Finally,

5) assume GthEl::t,i1 , ... ,i1_ 1 • The interval :Z.:t,i;, ... ,i1_ 1 is partitioned into intervals

l::t, i 1, •.·. ,i1_ 1, ~, ••• , l::t, i 1, •.. ,i1_ 1," having the height t. Each of these intervals contains exactly

one collection; assume :Z.:z, i1 ... ,i1 = {ath}·

From the definition of the function l/Jz,h (yh, x1, •.• , x ml and the properties of the constants clJ, 1: it follows that the function l/Jz,h (o, x1, ••• , x m> is constant on each of the intervals

t In the set 2:: M _1 the images of certain intervals may be shorter or may be absent altogether.

58 0. B. LUPANOV

~~ (i:i=Z), ~~~ (i"'=it). ~~~~~ (i:Fi2), etc., it being equal to eh on all intervals I:i (i ~ l). Thus, lflzh (ö, x1, ... , xm) is equal to the disjunction of the characteristic functions of those of the enumerated intervals on which it is equal to 1. The nurober of such intervals, whose height is at least 1, does not exceed xt. Intervals of height 0 make up no more than two pieces ~~ U ..• U ~~-~ and ~l+1 U ••. U ~11- 1 (or are not needed at all). Finally, we have

Fig. 5 I (Wlh (ö, Xt, ••. , Xm))-< xt + 2.

The lemma has been proved completely.

From the second statement of Lemma 20 and from Lemma 16 the following corollary derives directly.

Corollary.

C. An (E, t, m)-Block and the Expansion of Functions Which Are

Associated with It. Assurne m: is an (E,t)-block,and M=] Z:: [. The (E, t, m)

block l8 consists of M copies of the block m:, which are attached to the inputs of the network DM,M (Fig. 5); the inputs of the l-th m: block are assigned the variables

Yl,o,,· · ., Yl,N-i (l=O, 1, ... , M-1).

It is obvious that the function ~ (y0, 0, ... , YM-i,N-i) realized by an (E, t, m)-block is equal to F (Yo. o •... , Yo. N-t) V ... V F (YM-t, o •... , YM-t, N-t)· From Lemma 20 the following property of the function ~ derives.

Assurne a is an arbitrary collection of length m, and a = az h· Then '

(1> (Wo. 0 (Yo. 0• a), ... 'Wo. N-1 (Yo, N-h cr), .•.• WM-1, 0 (YM-i, o. o), ... , wM-1, N-1 <Y.'If-1. N-1· cr>> = Ylh·

(9)

It is clear that

xtM L(l8)=ML(m:)+L(DM,M)< x-i P+C9M, (10)

T (l8)=T (21) +T (DM, M) < tT+C10 1og M. (11)

Assurne ~ is a function realizedbyan(E,t,m)-block, while I: and ll1 are the corresponding numerations ofthe collections of length m and the system of functions l/JZ,h (ö, z') (Lemma 20). Let us introduce a certain abridged notation which will often be used hereafter. Assurne G is the system of functi.ons gz h (v1, ... , vk) (0 :::5 l :::5 M- 1, 0 :::5 h :::5 N - 1) and that the function ,.., ,.., ' gG,E (z, v) is defined in the following manner:

The function

(1> (1Jlo. 0 (go, 0 {v), z), Wo. I (go, I (v) z), ... ' WM-i, N-I (gM-1, N-1 v), 'i'))

is denoted by [~, I:, ll1, G].

Lemma 21.


The statement of the lemma derives directly from the definitions of the functions gG ,.E; [, l; , '11 , G], and from (9).

D. A Certain Generalization. Weshall consider logic-algebrafunctions which depend on x = (x1, ... , xw). Assurne ID~- is a certain set of collections of the values x. Two functions f {x) and g {X) shall be called e q u a 1 o n m, if f (a) = g (a) for any collection a from m (use the notation f (x) IDI g (x)). For certain sets equality on them will be denoted in

a special manner. For the set ~Jlo, consisting of all collections with the exception of 0 = (0, ... , 0) and 1 = (1, ... , 1) we shalllikewise use the symbol ~ in place of the symbol = .

IDio For the set IDca. b of collections (a, ß') having the length a + b (a has the length a, ß has the length b) which aresuchthat a=i=O, a=i=l ß=i=ü, ß=i=l we shalllikewise use the symbol -a, b

in place of the symbol = . SJJia. b

The following analog of Lemma 21 holds.

Lemma 22. Assurne m is a certain subset of the collections of values (u, v), U = (ui, ... , Um), V = (vi, ... , Vk ). Assurne the systems of functions '11* = {1/!t h(Yz h' u)} and G* = { g*(a, v)} satisfy the con-ditions ' '

Then

Remark 1. If

then

Remark 2. If

'l''t, 11 (Ö, u) IDI 'l'z, 11 (Ö, u); g* (a, v> = g (a, 17). ~JI

[<D, l;, '1'*, G*] IDI [<D, };, '1', G].t

'PT.~~ (ö, u) ~ 'Pz. h (ö, u), g* (a, v> ~ g (a, v),

[<D, };, '1'*, G*] = [<D, };, '1', G]. m,l<

'l't. h (ö, u) ~ '~"z. h (ö, u), g* (cr, u) ~ g (a, li>

[the functions g* (a' u) depend on the same variables as those on which the functions 1/Jz,h(ö, u) depend~), it follows that

r<D, };, '1'*, c1 ~ g (u, u).

§ 7. The Expansion of Functions

Assurne f (x1, ... , x 11) is an arbitrary logic-algebra function. We shall partition its variables into three groups:

t Since on collections from ~JI all functions "corresponding to one another" which are entered in [ , l; , '11 * , G *] and in [, l;, '11, G] take eq ual values.

60

0 ßt

Xt ... Xh 0 ßn

0 ... 0

Gtt ... ah 0 t

I~ (ai, y

1 ... 1

TABLE 1

... , ak, ßt. ... , ßu)

O.B.LUPANOV

1

1

At } .

AP } s' <, s

;; = (xt. ... , X~t), y = (xh+t. ... , Xh+u),

Z = (xh+U+h ... , Xn);

let us denote n - k - u by m. Let us use f-v (;;, y) to denote the function t (x, y, ,Y).

Assurne E1 is an element of a basis on which T is attained. Assurne this element has the weight P 1, the delay T1, and a number of inputs equal to k1 • Then T1 = T log k1 •

Assurne t1 is a certain parameter which satisfies the condition (for n- oo)

(12)

Let us now consider an (EI, t 1)-block (and the corresponding partitioning ~ 1 of the collections y of length m- Lemma 20), the (E 1, t 1, m)-block l8' (and the function

 1 and system of functions 'lr 1 which correspond to it), and the system F 1 of functions t(x,y, Yl', h'). Then the function t (x, y, z) allows the representation (Lemma 21)

t(x, !i,z)=[ID', ~~. 'l'',F'J. (13)

Assurne M'- J 2m [ h b f ) h - (k'{ . T en y virtue o (12 we ave

(14)

and fortheblock l8' we have [see (10), 11)]

L (lB') < (~}~~· P' +CgM' <.C112m, }

T (l8') = t'T' +]log M' [T(l) = -r:t' log k' +]log M' [T<u. (15)

Each function !-:.; (x, ii) may be stipulated by a table having two inputs (Table 1).

In this table the value of the function on the collection (a1, ••• , akt ß 1, ••• , ßn) is placed at the intersection of the row corresponding to (a1, ... , ak), and the column corresponding to (ß 1, ... , ß n>• Let us partition the matrix which determines the values of the function !-:;,(;;, Y) into strips A1, ... , Ap, each of which (except, perhaps, for the last) contains s rows, while the last contains s 1, s 1 ~ s rows. The number p of strips satisfies the condition

211 p<.-+1. s (16)

Let us use fvz·. h'• i to denote the function which coincides with f-y1 •• h' on the strip Ai and is

equal to 0 outside the strip Ai. It is clear that

-- p - -

fvz·, n' (x, Y) = i':!.t hz·. n'• i (x, y). (17)


Assurne E 11 is an element of the basis on which p is attained. Assurne that this element

has the weight P 11 , the delay T 11 , and a number of inputs equal to k". Then P" = p (k" - 1).

Assurne t11 is a certain parameter which satisfies the conditions

t" --7 oo, (18)

(19)

Let us now consider an (E n, t") -block (and the corresponding partitioning ~" of steps of length u- Lemma 20), the (E 11 , t 11 , u)-bloek )8" (and the function 11 and system of functions >lF 11

corresponding to it), and the system F'i· h',; of functions

tion f-v .(i, y-) allows the representation l', h't 1

Assurne M" =] 2u1" [. (k")

and for the block ln" we have

1-v ; (.;:, Y) = [1:l", L:", 'Y", F'i· h' i]. l'' h'• • '

Then by virtue of (19), (18) we have

L (Q·") < (k"(M. "P'' +C,M" ~ p2"' } ) k"-·1 !-) ~ '

T (\b") = t"T" +]log M" [Tm <C12 (t" + u).

§ 8. The Method of Synthesizing Networks

i n th e C a s e o f a R e g u 1 a r B a s i s

Then the func-

(20)

(21)

(22)

A network will have several "layers" (separated from one another by the horizontallines in Fig. 6). Within eaeh layer the network eonsists either of identieal elements or of eanonieal networks having identieal delay. The network will eonsist of bloeks of two kinds:

Blocks whieh ealeulate "intentional" functions (denoted by the letter A with subseripts);

Matehing bloeks which are used to mateh the delays of various parts of the network. Blocks whieh transmit eonstants (for the produetion of delays in aeeordanee with Lemma 15) willlikewise be assoeiated with matehing bloeks. Matehing bloeks will be denoted by the letter Z with subseripts.

In describing bloeks their complexity, delay, and (in eertain eases) number of outputs (denoted by the letter Q will be indieated.

In deseriptions of this and the two subsequent methods of synthesis, bloeks whieh realize identieal (or similar) functions have identieal numbers. This explains the gaps in the numeration of the bloeks.

If a certain bloek is situated inside a layer in which the elements (canonical networks) have a delay T, while the entire layer has a delay tT, it follows that the expression 11the bloek realizes cp with any delay" implies that for any i (0::::; i::::; t) there exists an output of this bloek on whieh cp is realized with the delay iT (relative to the inputs of this bloek).

Let us now go over direetly to the deseription of the blocks of the network (see Fig. 6).

62 O.B.LUPANOV

Fig. 6

The block A0 realizes the constants 0, 1, the variables x1, ... , Xn, and their negations x1, ... , Xn· Since the basis is regular, this can be accomplished with one and the same delay T 0:

L (A0) < C100n, T (A 0) = T 0 •

The b 1 o c k A1 realizes the conjunctions x~' ... x~k (2 k of them) and the constant 0 on the basis of the functions 0, x1, ... , Xn, x1, ..• , Xn already realized by the block Ao. Each conjunction is realized by a Kk,k network (Lemma 8); the constant 0 is transmitted to the output of the network by means of a chain of canonical networks 81• Therefore,

The b 1 o c k A2 realizes all conjunctions x~t+l ... x~ttu and the constant 0,

The b 1 o c k A4 realizes all conjunctions x~~:-t-\ ... x~'!_:;:'-f:f, 0 :s j :s m (for j = 0 the conjunction is empty and identically equal to 1; the number of all functions is equal to 2m+ 1) and the constant 0;


The b 1 o c k A5 realizes all possib1e different functions fvz·. h'• i (x, ß1 .. , ~t•). Each of them

is obtained as a disjunction of no more than s conjunctions which have been realized by the b1ock A1 using one network Dr,s for a certain r. Wehave

further, by virtue of Lemma 9,

The b 1 o c k A6 realizes all possib1e functions <tp"l'', h" (ö, i/), which are obtained from the functions of the system \lf" as a resu1t of substituting the constants 0 and 1 in place of the first argument. Each of them may be obtained as a disjunction of no more than 2u conjunctions, t which are realized by the b1ock A2 using one network Dr, ~u for a certain r. We have, taking account of (19),

Q (A6) c= 2M" (k"( <Cto52";

L (AB) <C106M" (k"(2u <C10722 "; T (A 6) = uT<0 •

The b lock A8 realizes a11 possib1e functions '(li·. h' (ö, z). By virtue of the corollary from Lemma 20, each of these functions is a disjunction of no more than (k't' + 2)2m conjunctions realized by the b1ock A4, and may be obtained using one network Dr,(k't'-!-2) 2m. We have, taking account of (14),

Further, since k' is a constant,

it follows that

Therefore,

Let us introduce the notation

T* =]log m [ +-Jlog ((k't' +- 2) 2m)[.

We shall assume that the conditions (for fairly large n)

]log k [+-]log s ( < T*, ]log u ( + u < T*,

k ~~ O(log n)

are fulfilled. From (24) it follows that

T* ·< C 112 log n.

t Here it is sufficient to use a very rough bound for the complexity.

(23)

(24)

(25)

(26)

(27)

64 O.B.LUPANOV

fu order to equalize the delays of the networks Ac.As, A2-As, ArA8, we attach systems of chains of canonical networks S1 of appropriate length to the outputs of the blocks A5 and A6•

Assurne z1 and Z2 are blocks consisting of these systems of delays:

L (Z1)<C113p2' logn,

L (Z2) < C1142u log n.

The block A9 realizes all possible different superpositionst ljJ"(j~v.Jx, ß), y) on the basis of the relationship

One network s9 is used for the realization of each of them:

L (A9) < C us2u p2', T (A9) = T<4>.

The b 1 o c k A10 realizes all functions fv1,, h'• i (;;, Y) in accordance with the representation (20) (see likewise p. 59). One (E", t", u)-block m". is used in the realization of each of these functions. Therefore [see (22)]

L (A!O) = M' (kf pL (m"),

T (A 10) = T (>S") = t"T" +]log M" [Tm.

Let us require that the condition

2h -----? 00

s

be satisfied. From (14), (16), (28), (22) we have

For convenience we shall assume that the block A10 consists of two parts: The block Ag' containing all elements E"

T (Ag>)= t"T",

and the block AW containing the remaining elements

From (19), (25), (21), (27) it follows that

t" = 0 (log n),

log M" = 0 (log n).

(28)

(29)

(30)

The block A11 realizes the functions f (;, y, y1•• h') in accordance with (17). One network Dp,p is used for one function. Therefore [see (25), (26), (27)]

L (Au)= zmL (Dp, p) <Cu6p2m,

T (Au)= ]log p [T'll < C117 log n.

tWe drop the indices ß', h', ß ", h".

(31)


The b 1 o c k Z3 "transmits the constants 0 and 1" to the 2-2' level. It consists of two chains of s1 networks. From (27) it follows that

The b 1 o c k z4 transmits the constants 0 and 1 to the 3-31 1eve1 (see Lemma 15)

L (Z4) <;:C119 •

The b 1 o c k Z5 realizes the constants 0 and 1 with al1 the de1ays between 3-3' and 4-41

(see remark of Lemma 15). We have, taking account of (29),

We shall assume the following condition is satisfied:

t' and t" are even numbers. (32)

The b 1 o c k Z6 transmits the functions of the b1ock A8 to the 3-31 1eve1 (see remark of Lemma 15). Taking account of (14), we have

The b 1 o c k Z 7 transmits these functions to the 4-4' 1eve1

The b 1 o c k Z8 transmits these functions to the output 1eve1 of the b1ock A11• This b1ock consists of chains of S1 networks. Wehave [see (30), (31)]

The b 1 o c k A14 realizes the functions 1(;;,, "' (f (.i, y, y;,, "'), z) on the basis of the re1ationship

and is arranged similar1y to the b1ock A9

L(Au)<C12G2m, T(Au)=T~'>.

The b 1 o c k A16 realizes the function f (x, y, ~) in accordance with the representation (13). This block is the (E', t', m)-block ~)'. Therefore [see (15)]

L (A 16)<;:C1272m, T (A 16) =Tt' log k' +]log M' [Tw.

Let us now ass ume that

k=[2logn], u=[2logn], s=[n-5logn], }

t ' 2 [ 1 ( r: l l] t" = 2 [~] . = 2 log k' n- J og n ' 2 log k" • (33)

Then the conditions (12), (18), (19), (25), (26), (28), (32) are fulfilled. It is likewise easy to check the fact that in this case L(A10) :;( p(2n/n) and L(A1) = o(2n/n) for i ,c 10; L(Z1) = o(2n/n). Further, for i ,c 16, we have T (A1) = O(log n);

log M' <;,m-t' log k' +0 (1) =log n+O (1).

66 O.B.LUPANOV

Therefore,

T (A 16) < r;t' log k' + 0 (log n) "' Tn.

Thus, for the entire network S we have

2n L (S) ~ p - , T (S) ~ Tn. n

Thereby the upper bound of Shannon function is established in the case of a regular basis.

Rem a r k • For the particular case of a regular basis, for which all of the elements have integer delays and there is a unitary delay in the basis, the synthesis method is somewhat simpler than in the general case. In this particular case:

1) It is not necessary to equalize the delays in each layer (equalization before union of the networks only is sufficient);

2) Instead of the networks based on Lemma 15 which simulate chains of delays one may use 11 conventional11 delays. Therefore, instead of eight blocks which in one way or another are associated with the execution of the delay function (ZcZ 8) it is sufficient to have only two (consisting of chains of unitary delays);

one for equalizing the delays in the networks AcA5 and A2-A6,

the second for equalizing the delays in the networks (A1, A2, A11) and A4-A8•

§ 9. Method of Synthesizing Networks for

0-Functions in the Case of an Irregular Basist

A. Auxiliary Functions and the Additional Expansion of 0-Funct i o n s • Let us introduce the notation:

Aa (u1, •.. , Ua) = Xo (u!l ... , ua) x1 (ub ... , U0 ),

fla, b (u1, ... , Ua, V1, ••• , vb) = 'Aa (u~o ... , U 0 ) /,b (vb ... , Vb). t

It can easily be seen that

/,a (u!l ... , ua) = (ul V ... V ua) (u1 V ... V Ii") = u1U2 V ~u3 V ... V ua-1Ua V uau1. (34)

Let us determine the following functions of the variables x1, ••• , Xn (the variables will be assumed to be partitioned into three groups x, y, ~ which contain k, u, and m variables, respectively; see §7).

1. The functions of x, y, ~ (Table 2):

e!(x, y, z) = !lk+u, m (x, y, z) = "Ak+u (x, y)/'m (z), ez (x, y, z) = Xo (x, y) Am (z), e3 (x, y, z) = X1 (x, y) },m (Z),

e4 (x, y, z) = Xo (z) Ak+u (x, Y), e5 (;-, y, Z) =X! (Z} Ah+u (x, y), ea (;, Y, z) = X1 (;:, y) Xo (i') = X1 ... Xk+uXI<+u+l .•• :;,

e7 (x, y, z) = Xo (x, y) X! (z) = ~~ ... Xk+uXk+U+l ... Xn.

tThis method is also applicable for a regular basis; however, it is intentional only in the case of an irregular basis.

~ This function is the characteristic function of the set IJJ/a, b, introduced in § 7.


TABLE 2 TABLE 3

-- 0 1 x, y - 0 1 X -y

u, 0 I e2 I e7 0 I d2

-

e" e 1 e:; dt

I

·I; 1 e6

I ea

I I 1 dz

I

2. Functions of x, y (Table 3):

d1 (X', y) ~' ~~k." (x, y) = Ak (~) /,u (y),

d2 (;:, y) = Xo (.Z} %~ (y) V Xo (!/) Xo Cl V Xt (X') Xt (y) V Xt (y) X1 (;:).

Assurne further for the arbitrary 0-function f (x, y, z) that

Since f (o, o, O) = j(l, 1, l) = o, it follows that

For each function f;:; (x, y) = f (x, y, y) (see § 7) we define the functions

·(i) ~. - .~ - ~ -Jv (.<, y) = h:; (x, y) di(..t, y), i = 1, 2,

f(~) (:-;:, y) = tV) (;·, y) V/~)(~. Y).

The following properties derive from the definitions given above: (0) - - ~ -

1) Yr- (x, y) ~ f;,; (f, y);

2) if a certain function g (~·, y) satisfies the condition

21) analogously' if a certain function g (x, y, z) satisfies the condition g (x, y, z) = f k+u, m

(x, y, z), then f(l) (x, y, ;) = g (r, y, z) e!(x, y, z);

3) (4) - - - (0) - - - - -f (x, y, z) = fo (x, y) e4 (x, y, z),

j(5) (x, y, ij = t<;) (x, y) e5 (x, y, ;).

B. The Method of Synthesizing Networks. Thismethodismorecumbersome than it is in the case of a regular basis. In order to simplify its description we shall indicate only the blocks which execute intentional functions, "Delay blocks" used to match delays are depicted by thick lines in Fig. 7. The complexity of delay blocks will be estimated

68 O.B.LUPANOV

8---------------------------------------8'

Fig. 7

at the end. In describing the method the notation from §7 will be used. The values of the parameters here are the same as those in §8. Therefore, the conditions (12), (18), (19), (25), (26), (28), (32) are fulfilled.

Let us go over to a description of the blocks.

The b1ock B0 (Fig. 8) realizes the auxiliary functions: 0, 1, ei (1 :s i :s 7), d1• It consists of the b1ocks B00-B04.

The b 1 o c k B00 realizes constants (for their use in de1ay b1ocks). It consists of the canonica1 84 and 85 networks (Lemma 10)

L (Boo) = C2oo. T (B00) = r< 2>.

The b 1 o c k B01 realizes the functions 'Ak (,;), 'Au (ij), 'Ak+u (x, ii) and /,m {Z) in accordance with (34)

T (B01) = T< 2> +]log n [T<o.


The b 1 o c k B02 realizes d1 (~, Y) = Ak (~)Au (i/) and e1 (X', y, z) = Ali+u (X', Y) Am (z) ,

Fig. 8

L (Boz) = Czo2• T (B02) = r<!).

The b 1 o c k B03 realizes the functions e 2-e5 on the basis of the representation

and analogaus representations using Lemma 11 and networks

The b 1 o c k B04 realizes the functions e6 and e 7

Thus,

L (B0) <;:: C205n2,

max(T(Boo), T(Bo1-B02), T(B03 ), T(B04))~2T( 1 )logn.

The blocks B1 and B2 realize the systems of conjunctions K0(X') and K0G'), respectively, and likewise the constant 0. Wehave (see Lemma 11)

L (B!) <;::C20ak2", T (B1) = r< 2 l --;~]log k [T(!),

L (B2) <;:: Cz07u2", T (B2) ~ r< 2 l +]log u [T< I).

The b 1 o c k B3 realizes the constant 0 and all possible conjunctions :t~1 ... x~kx~~~ ... x~'i"' which correspond to the collections (a, ß> = (a 1, ••• , ak, ß 1, ... , ßu ), satisfying the condition

(~ =, o & ß =F O) V(~== f & ß =F 1) V (ci' =F o &jJ == O) V (ci' =F f & ß = 1).

The number of such conjunctions is smaller than [see (32)] 2(2k + 2u) = 2u+2• Therefore (Lemma 11),

L (B3) <;:: C208u2",

T (B3) = r< 2l +]log (k + u) [T< o.

The blo ck B4 realizes all conjunctions x~~-;;:~~1i 1 ••• :J::~'_';,f-j·i, 1-<:j-<rn, containing at least one variable with negation, at least one variable without negation, and the constant 0:

L (B,)<;::C20gm2m,

T (B4) = r< 2l +]log rn [T(I).

The b 1 o c k B5 realizes the set of all functions h(x) which have the properties:

a) h(O) = 0, h(i) = O;

70 O.B.LUPANOV

b) h(x) outside of a certain band (see Table 1) is equal to 0. It is obvious that for each

function 1::;1,_ ,. •• ; (;;, ßt", n") there exists a function h(x), which is suchthat f-:;; 1 •• ,. •• i (:t; ßz", ,."):::::::; h .(x);

this function h(X') will likewise be denoted by ffl . (x, ß1", ,.,-). The system of all functions Yt•, h'''

o - - •o fvz·. ,. .• i (x, ßz", n") for fixed l', h', i shall be denoted by Fz·. h', i· It is clear that each function

h(X') is a disjunction of no more than s conjunctions from K0(x). By virtue of Lemma 9,

The b 1 o c k Bs re~lizes the functions 'ljl;.?_ "" (ö, y) = '1Jl7". h" (ö, y) x0 (y) x!(y) as disjunctions of conjunctions from K~ (y)

It is obvious that

"0 - ~ 'IJlz",h" (ö, y):::::::; '1Jl7", h" (ö, y).

The system of functions

'\jl~·?. h" (0, y) Yl", h" V 'P;,?, h" (1, y) Yl", h" = ~ (Yl", h", 'ljl;·.?. h" (0, y), 'ljl;.?, h" (1, y)) shall be denoted by llF " 0•

The b 1 o c k Ba realizes the functions 'ljl;?!,.. (ö, z) = 'IJli·. "' (ö, z) x0 (z) x!(z), which are an

alogous to the functions 'ljl;2, "" (ö, y). Therefore,

'0 - -'IJlz·. h' (ö, z):::::::; 'ljl/•, h' (ö, z). (36)

The function 'ljl;?, "' (ö, z) differs from 'ljli,·, h' (ö, z) by no more than two collections. Thereby, by virtue of Lemma 17 and the corollary of Lemma 20

Io('IJli?, h' (ö, Zl)<:;;:J0 ('1jli, n•(Ö, Z)) +4m<:;;:(k't' +4) 2m.

Since 'ljl/9, h' (ö, 6) = 'ljl[9, ,., (ö, t) = 0, it follows that for the formation of '1Jli9, h' (ö, z) only conjunctions realized by the block B4 are used. Therefore [see (14)]

L (Bs)<C213M' (k'{ (k't' +4) 2m<:;;:C2142mn2,

T (Bs) =]log ((k't' + 4) 2m) [T<o, Q (B8) = C2152m.


is denoted by llF' 0•

The b 1 o c k B7 realizes the function t1> (;;, y) (see p. 67) on the basis of the conjunctions realized by the block B3,


It is easy to check the fact that of the networks B 0, BcB5, B2-B6, B3-B7 ~ BrBa the network B4-B8 has the largest delay:

T(Z) _l_ (]log m [ + 1 log ((k't' -;- <1) 2m) [) T(J) ~ 31'( 1) log n.

This quantity determines the delay ofthe entire network between 1-1' and 2-2'.

The b 1 o c k B9 is analogaus to the block A9• It realizes the functions t

~ u~. ;ez, ßl, ~;"0 (o, i/), 1Jl"0 (1, YJ), L(Bg)<.C2172up2\ T(B9)=T( 4 l.

The b 1 o c k B10 is analogaus to the block A10• It realizes the system of functions

2n L (B10) -;;; p n, T (BIO)= t"T" + 1 log M" [1'( 1).

Since f':L i (~, ßz" I'")~ f- i (::C, ßz" h") and •h/2, h" (ö, y) ~ •1'1'", lt" (ö, -y), it follows that by virtue Yl',h" '' Yl',h" ' 'Y 'Y

of Lemma 22

[<D", L", 'Y"0, F'i9 "' i] =I- · (J;, y). ' ' k,u Vt',h'• 1

(37)

The b 1 o c k B11 realizes the functions

From (37) it follows that

~~~- h' (x, 'YJ k t:vz- ,. (x, 'YJ. ' 'n ' t.

(38)

The b 1 o c k B12 realizes the functions fl.!l (c, y) on the basis of the representation [see 'V

(38) and property 2) on p. 67] fVl (;:, y) = f~ (x, y) d1 (;:, y);

L (B12) <. C2192m, T (ß12) = 1'0 >.

The b 1 o c k B13 realizes the functions jr,!!l (-;:, y) = fl.!l (-;:, y) V j(ll (;:, y); 'V 'V 'V

It is clear that

t<!Jl (i, !IJ ~ h (x 'YJ. 1'

The system of functions Nl (::C, y) shall be denoted by F10• vz· h'

The blo ck B14 reali;es the function ~(!~ 1 •. ll' (;:, y), 1jJ[9, h'(O,z), 1Jli9, h' (1, z));

twe drop the indices l', h', zn, h".

(39)

72 O.B.LUPANOV

The b 1 o c k B16 realizes the function f* (;::, y, z) = [<D', L'. '!'"'0 , F'0 ) by means of one block fb' • We have [see (15)]

By virtue of Lemma 22, (36), and (39)

Let us use F~ (e = 0, 1) to denote the system of functions f (e, e, y) (this is a system of constants; the collection y runs the gamut of all collections of 1ength m).

The b 1 o c k B17 realizes the function f"' (~) = [<D', L', '!'"' 0 , F~] (by means of one \S' b1ock; we note that

i.e., it coincides with a certain function

By virtue of Lemma 22,

tö <~) ;:::; t <o. o, z).

The b 1 o c k B1a realizes f! ~) = [<D', L', '!'"' 0 , F~] ana1ogous1y;

t: (~);:::; t <f. I, :Z); L (Bis)<. Czz42m.

The b 1 o c k B19 realizes f (x, y, z) in accordance with the representation

(1) - - - - - - - - - (2) - - - . ·- - - -f (x, y, z) = f* (x, y, z) e1 (x, y, z), f (x, y, z) = fö (z) e2 (x, y, z), (3) - - - * - - - - (4) - - - - - - - - -f (x, y, z) = f 1 (z) e3 (x, y, z), f (x, y, z) = fo (x, y) e4 (x, y, z),

(5) - - - - - - - - - (6) - - - - - - - - -f (x, y, z) = f 1 (x, y) e5 (x, y, z), f (x, y, z) = f (1, 1, 0) e6 (x, y, z),

(7) - - - - - - - - - - - - 7 (i) - - -f (x, y, z) = f (0, 0, 1) e7 (x, y, z), f (x, y, z) =V f (x, y, z). i=1

The functions f* (x, y, z), fö (~). f! (z) are realized by the b1ock B16, B17, B18 ; the functions fo (r, y), fT (;::, y) are realized by the b1ock B13 (together with other functions f-;;; (x, y)); the constants f (1, l, 0) and f (0, 0, l) are realized by the b1ock B0; the functions ei (x, y, z) are realized by the b1ock B0;

(one story of the S2 networks and the network D7,8).

It is easy to check the fact (just as it is for the synthesis method from § 8) that for i "o 10, L(Bi) = o(2n/n) and L(B10) :e p (2n/n);


T(Bi)=O(logn) for j7"=1ß, 17, 18,

T (B16 ) = T (B17) = T (B18) ~ Tn.

Thus, the delay of the entire network does not exceed C226n; therefore, the sum of the complexities of all the delay networks (depicted by the thick lines) does not exceed

Thus, for the proper network S which has been obtained we have

2n L (S) '(: p ---n, r (S) ~ Tn.

§ 10. Method of Synthesis of Networ_~_s~~

x-Functions in the Case of an Irregular Basis

73

This method is analogaus to the method of synthesizing networks for 0-functions; however there are the following basic differences.

I. Instead of the elements E' and E" (on which T and p, respectively, are obtained) the elements Em and EIV are used (on which T * and p* are attained; see § 1).

rr. Instead of the auxiliary functions ei and di' their analogs ei (r, y, Z) V X! (x, y, z) and di (.i:, y) V X!(;:, y).

III. The construction of Lemma 15 for forming delays is not used.

Let us consider these differences in greater detail.

I. Assurne Em is an element of the basis on which T* is attained. This is a <P-element. Assurne this element has the weight pm, the delay Tm, and a number of inputs equal to km. Then Tm = T * log km. In the method described we shall use the (E m, tm, m) -block jz)"'

(instead of the block ~'), the numeration 1; "' of collections of length m defined by it, the function <P'" and the system of functions 'lF m corresponding to it, as well as the system of func-tion F'" (actually the "old" system of functions f (x, y, z), only the collections y in it are newly numbered by pairs of indices), and the representation

f (;, y, z) =[CD"', 2:'", 11'"', F"'] (40)

based on them. The system of functions f-::; 1",, h"'·; (x~ YJ. is defined in accordance with the new

numeration of collections of length m.

Assurne E1v is an element of the basis on which p* is attained. Assurne this element has the weight piV, the delay T rv, and a number of inputs equal to kiV. Then P IV = p * (kiV- 1). In the method described we shall make use of the (EIV, tiV, u)-block \E1v (instead oftheblock ~"), the numeration 1;IV specified by it, the function q.IV and the system of functions >lfiV corresponding to it, as well as the system of functions Ff;;, h"',; (actually they coincide with the system F'i·. h',; ) and the representations

(41)

based on them.

74 O.B.LUPANOV

The numbers

are defined analogously.

The values of the parameters are chosen in a similar fashion:

This time,

k=[2logn], u=[2logn], s=[n-5logn],

t'" = 2 [210~ k'" (n-5log n) J,

L (Q')"') < C3oo2m = o ( 2:) ,

tiV = 2 [ log n J . 2log krv

T (Q3"') = •*t"' log k"' +]log M'" [T(iJ "'--r;*n,

L (Q3IV) ~ p*2",

T (Q3rv) = trvrrv +]log Mrv [T<t> = 0 (log n).

II. Let us introduce the functions (see §9.A)

e10+i (X:, y, ~) =e; (x, y, z) V x1 (x, y, z), 1-<i-<7,

d!O+i (x, Y) = d; (X:, !;) v x1 (x, !/), 1 < i < 2.

(42)

Rem ar k. Assurne aii (x1, ••• , xa) are A-functions x1x2 ... xa V x;xi. From the definition of the function d11 and from (34) we derive the equation

du (x, !/) = ("-~< (x) v x1 (x, y)) & (A.u (!/) v x1 (;, i})) =

= (n12 (x, y) V a23 (x, y) V ... V a~<-1, k (x, y) V ak, 1 (~, y)) &

& (ak+!, ~<+2 (x, y) V a1<+2, k+3 (x, y) V ... V a1<+u-1, k+u (;, y) V ak+u, k+t (x, y)).

An analogaus representation is valid for the function e11 (x, y, z). For an arbitrary x-function f <X, y, Z) we place

fU>(x, !/, z)=f(x, !J, ~ej(;, i/, Z), H<i-<17.

Since f (o, Ö, O) = 0, i t follows that

Note in addition that

17 - - -f (x, y, z) = V j<il (x, y, z).

i=11

tw (T, T, 1) = 1.

For each function t- (x, y) = f (x, y, ~) let us place 'V

tW (x, ii> = h; (x, ii> dj (x, ii>, j = 11,

~~10) (x, ii> =tV1) (x, y) v tV2) (x, ii>·

(43)

(44)

(45)


The following properties hold (which areanalogaus to the properties from §9.A) (10) ~ - - ~

1) fv (x, y) ~ f-y (x, y);

2) if a certain function g{X, y) satisfies the condition g (x, Y) k,u f-y (;, y), then

(11) - ~ - - - -f-y (x, y) ~ g (x, y) d11 (x, y)

(note that here the conventional equation may not hold: on the collection l the left and right parts may take different values);

21) if f (X., y, z) is an x-function and if a certain function g(x, y, z) satisfies the con

dition g (x, y, z) = f (x, ii. z) ' then k+u, m

(11) - - - ~ - - - - ~ - ~ -f (x, y, z) = g (x, y, z) e11 (x, y, z) V x1 (x, y, z)

[see (44)];

3) (!lt) ___ (10)-:- --- --

1 (x, y, z) =~ fo (x, y) e14 (x, y, z) V xdx, y, z), (15) - - - (10) - - - ~-f (x, y, z) ~~ f-r (x, y) e15 (x, y, z)

(since tY0l (1, l) = 1).

75

III. Note in. addition that circuits of the elements E m and EIV with identified inputs having the length tm and tiV, respective1y, realize the function f (x) = x. The network S8 (for the function ~) likewise realizes x for identification of the inputs. This allows the necessary de1ays to be realized in al1 layers without using the corollary from Lemma 15.

Let us now go over to a description of the blocks (Fig. 9).

The b1o ck Do realizes the "conditional constants" e(O)(ZJ = 'X. 1{z), e(OO)(X, y, z) = )'(.1 (x, y' z), and e< 1) (z) = Xk+u+l V ... V Xn, and the functions ei (11 :S i :S 17) and d11. It consists of the b1ocks D00, D01, D02 , D03, D04 • These b1ocks are connected to each other similar1y to the blocks B00-B04 (see Fig. 8).

The b 1 o c k D00 realizes conditional constants. Since ~ (x, x, x, x) = x, it follows that D00 may be chosen so that

It is obvious that

(46)

The b 1 o c k D01 realizes the A-functions a ij (x, y) and aii (X', y, z) (see remark) which are used for the realization of d11 and e11 • Their number is sma11er than 2n. We have (Lemma 12)

(the delays of the network for various functions aii are equalized by means of circuits from S1

networks).

The block D02 realizes d11 in accordance with the representation (43) (p. 74), and e 11 in accordance with an analogaus representation;

T (D02) =(]log n [ + 1) Tm.

76 O.B.LUPANOV

8----~========~---g' Fig. 9

The blo ck D03 realizes e12-e15 • Wehave [see (35)]

e!2 (x, y, z) = (Xt ... Xn V X! ... ~-mXn-m+t-Xn-m+2) V (xt ... Xn V Xt •.. ;:n-mXn-m+2Xn-m+a) V

V (x! · · · Xn V~ · · · Xn-mXn-tXn) V (xt ... Xn V X'. · · · X";,_mxn"Xn-m+!J•

For e 13-e15 the analogaus representations hold. Therefore,

The block D04 realizes the functions e 16 and e 17 (these are A-functions);

The b l o c k s D1 and D2 realize A-functions of the form


respectively (analogs of conventional conjunctions). We have (Lemma 12)

L (D1) < C306k2", T (D1) = r<a> +]log k [T(l),

L (D2) .:(:C307u2", T (D2) = T(3) +]log u [Tm.

The b 1 o c k D3 realizes A-functions x 1 ... xk+u V x~1 ... x~kx~~ 1 ... x~~u, whose collec

tions a = (a 1, ... , ak), ß = (ß 1, ... , ßu) satisfy the condition

(a = o & ~ o:F o) v (ß = o & a o:F o) v (a = 1) v (ß = T);

L (D3) .:(: C308u2u, T (D 3) = r<a> +]log (k + u) [Tm= 0 (log n).

The b 1 o c k D 4 realizes all A -functions Xn-mH ... Xn V <"--~tl ... x~'::;:;t~ (i ::::; j ::::; m);

The b 1 o c k D5 realizes the set of all functions h1{X) which have the properties:

a) h1(0) = o, h1(i') = 1;

77

b) h1(x) outside of a certain strip is equal to 0 (with the exception of the collection I on which, as has already been said, h1 is equal to 1). It is obvious that for each function f- . (:;:, PziV 1 Iv) there exists a function h1 (~) suchthat 1-yu., h"'· i (:;:, fl11v hrv):::::: h1 (x). This

'V l"', h'"' t . ' L ' '

function h1(x) shall likewise be denoted by tL, ""'' i ~;, ß1rv, "Iv). The system of all functions

!~ i (x, ß,Iv hrv) for specified l"', h"', i shall be denoted by l'", h"'. It is clear that each fune-r [m, hm, ,

tion h1(x) is a disjunction of no more than s functions realized by the block D1. By virtue of Lemma 9,

IV 1 ~ IV -- - ~ The b 1 o c k D6 realizes the functions 'ljl1rv hrv (ö, y) = 1jJ IV IV (ö, y) x0 (y) V x1 (Y) as dis-, l 'h

junctions of the functions realized by the block D2,

It is obvious that


lV ' - - IV 1 - IV 1 - IV 1 -'\jl1Iv, 'hrv (0, Y) Y 1Iv, hiv V '\jl1Iv, hrv (1, y) Y1Iv, hiV = ~ (Y1Iv, hiv. '\jl1rv, hiv (0, Y), '\jl1rv, hrv (1, y))

shall be denoted by w IV' 1•

The B 10 c k Da realizes the functions 'ljl~:.:,\,., (ö,z) = 'ljl;;.,, h"' (ö, z) Xo (z) V Xj (z)' which are IV 1 -analogaus to the functions 'ljl1nf, hiv (ö, y). Therefore,

By analogy with the situation prevailing for the block Ba, we have here (using the functions realized by the block D4)

(47)

78 O.B.LUPANOV


shall be denoted by'lfm,t.

The b 1 o c k D7 realizes the functions f~ 2 > (;, y) as disjunctions of the functions realized by the b1ock D3

As in the case of the method of synthesizing networks for 0-functions, it is easy to verify the fact that of the networks D 0, DcD5, D2-D6, D3-D7, D4-D8 the network DrD8 has the 1argest de1ay:

T(3) +(]log m [+]log ((k"'t"' + 4) 2m)[) r<0 .

t 1 ~ - IV 1 _, IV 1 -The b1ock D9 realizes the functions ~(fv,i (x, ß), 1jJ ' (0, y), 1jJ ' (1, y));

The b1ock D realizes the functions [CDiv ~Iv 'VIv, 1 piV, 1 l Wehave [see (42)] 10 , "-' , , 1IV, hiV, i ·

L (D!o) = p2mL (Qlrv) :(; P~' n

T (D10) = T (Q3IV) = 0 (log n).

/ 1 - R (~ R ) IV 1 -By virtue of the fact that vz"'.""'' i (x, p 1rv, "Iv)::::; f:r 1",, ""'· i x, P 1Iv, hiv and 'ljl1Iv, "IV (ö, y)::::; IV -1f1Iv, "rv (ö, y) , it follows that by virtue of Lemma 22 and (41)

IV IV IV 1 IV, 1 ~ -[CD , ,S , 'I' · , F1," ""' i] = /:;; ,· (x, y).

' ' k, U r l"', h"'' (48)

The b 1 o c k 0 11 realizes the functions

t!t..* ~ - IV IV IV, 1 IV, 1 .., 1., ""' (x, y) =V [CD , ~ , 'I' , Fl"', ""'· d,

• i

From (48) it follows that

!!.* (; y) = t~ (;, y). i' l"' hm ' k u 'V l"' h"' . . . (49)

The b 1 o c k n12 realizes the functions f'!.** (c, y) = f!t..* (;, y) d11 (;, y), .., ..,

By virtue of the properties 2), (45), and (49),

(50)

t W e drop the indices l m, h m, l IV, hlV.


The b 1 o c k D13 realizes the functions tr** (i, y) = t~** (X', y) V /~12 ) (X', y),

From (45), the property (1), and (50) it follows that

The system of functions f!.*** (X' y) shal1 be denoted by pm* • 'V["'' hlll '

The b 1 o c k D14 realizes the system of functions

,_. ----: ", 1 --- f/1 1 ,__ ~ (!!*** (x, y), 'ljJ1": ""' (0, z), 'I)Ju•: h'" (1, z)).

Yl"', h"' ' ' '

L(Du)<C32o2171 , T(D 14)=T('>

The b 1 o c k D16 realizes the function f** (x, y, ~) = [<D"', ~'", 'V"'· 1, F'"*J by means of one b1ock IS"'. We have [see (42)]

By virtue of Lemma 22, (47), (51), (40)

f** (X', y, 'i') = f (~. y, ~). k-j-u, m

79

(51)

(52)

We use F~ to denote the system of functions eU(e, 'E, y)) (z) (the collection y runs the gamut of the set of all collections of 1ength m).

--- "'1 _, 111 1 .-.. The b1ock D15 realizesthefunctions ~(e<•l(z), 'I)Jz":,h"'(O,z), 'I)Jz";,h"'(i,z)),

The b 1 o c k D17 realizes the functions fö* (z) = [<D"', ~"', w"' · 1, F~J by means of one b1ock ffi"'~ Wehave

L (D 17) = o ( 2:) •

By virtue of Lemma 22, (47), (46),

Jo* fzJ ~ 1 (u, u, Z'>.

The b 1 o c k D18 realizes f!* (i) = [<D"', ~"', 'V"'· 1, Fi], in analogaus fashion,

/!* (z) ~ f (f, 1, Z),

L (Dia)= 0 { :n) .

Fina11y, the b 1 o c k D19 realizes the function f (X', y, ~) in accordance with the representation

(11) ~ ~ ~ ** ~ ~ - ~ ~ - - ~ ~ f (x, y, z) = f (x, y, z) e11 (x, y, z) V x1 (x, y, z)

80 O.B.LUPANOV

[see property 2'), (52)]

( 1'') - - -, - - - - - - -f - (x, y, z) = W (z) e1z (x, y, z) V x1 (x, y, z), (13) - - - ** - - - - - - -f (x, y, z)=f1 (z)e1a(x, y, z)Vx!(x, y, z), (14) - - - **** - - - - - - - -f (x, y, z)=fo (x, y)e14(x, y, z) Vxl(x, y, z) (see (51)),

( 15) ._, _, - **** - _. ~ - -- -' -- --! (x, y, z) = f-r (x, y) e15 (x, y, z) V X 1 (x, y, z),

<16)--- {xl(x,!i,:Z), if t(f,T,o)=O, f (x,y,z)= --- ---

· e1s (x, y, z), if f (1, 1, 0) = 1,

<17) _ -- _ [ x!(x, !}. z), if !(0, o, 1)=0, f (x, y, z) -- - - - . - - -

e17 (x,y,z), 1f !(0,0,1)=1,

- - - 17 (i) - - -f(x, y, z)= V f (x, y, z),

i=11

L (D19) = Cazz, T (D19) = 5T<u

(two stories for the realization of xy V z and the network D7 8). '

The complexity of all (complementary) delay networks is of the order of o(2n/n).

Finally, for the entire network S we have

2n L (S) '(; p*---;;:, T (S) '(; •*n.

Appendix n

Assurne ';- = (a 1, ... , an) is a collection of zeros and ones, and assume I ';-I = 2j 2n-ia;. i=l

Let us introduce the notation:

rr"· q is the class of all vector-functions (i.e., systems of functions)

(in [6] such vector-functions were called (n, q)-operators); rr~· q ( rr~· q, rr~· q)' respectively)

is the class of vector-functions from rrn, q, for which all components are c-functions (x-func

tions, x-functions, respectively).

From what has been said above (see §§ 1-2) it follows that in the case of a regular basis it is possible for all vector-functions from rrn, q, tobe realized (by a proper network), while in

the case of an irregular basis it is only possible for vector-functions from rr~· q u %~' q u rri' q

to be realized.

Assurne r ::::: 2n and

%n, q, r is the class of all vector-functions l from rrn, q' which satisfy the condition

if lcrl>r, then t(a)=(O, ... ,0);

%~' q, r ( rr~· q, r, rr~· q, r), respectively) is the class of all vector-functions J from rr~· q

( %~' q, %i' q, respecti vely) which satisfy the condition

if r<l al<211 -2, then 1 (a) = (0, ... '0).


As above, for the vector-function J (realized by a proper network in the basis considered) let us use L(]) to denote the least of the complexi.ties, while T(]) denotes the least of the delays of the proper networks realized in J. Assurne IJ1 is a certain class of vectorfunctions realizable by proper networks. Assurne

T (IJC) = max T (fJ· 7€~

The following statements hold which generalize Theorems 1 and 2 (see § 1) and are analogs of Theorems A,lO and A.12 from [6].

Theorem A.l. If the basis ~ is regular and lo~nqn__,.o , then

q q 2n L ('"n, qn),...., L (%n, qn) ,...., L (~n, qn) ,...., L (%~' n),...., p -+~1 , V c x x n og qn

T (~n, qn),...., T (%~' qn) ,...., Tm:· q") ,...., T (~;· qn),...., nt.

And what is more, for any e > 0 and any vector-function f from ~n,qn there exists a network S which for fairly large n realizes the function f and is such that

T h e o r e m A .2. If t h e b a s i s ~ i s r e g u 1 a r ,

and

log qn ___,. O zn ,

then

L ('"n, qn, rn),...., L (~"· qn, rn),...., L(~n,qn, rn),...., L(%~'qn, rn),...., p qnrn ' V c x x log (qnrn)

T (%n, qn, r n),...., T (~~· qn, rn),...., Tm:· qn, rn) ,...., T (%~' qn, r n),...., •n.

And what is more, the bounds of the complexity and the delay are simultaneously attained asymptotically (i.e., on one network; see the end of the formulation of Theorem A,l).

Theorem A.3. If the basis 'G, is regular and logq" 0 then -zn--i> ,

And what is more, the bounds of the complexity and the delay are simultane ou s ly attained asympto ti c ally.

Theorem A.4. If the basis ~ is irregular,

82 O.B.LUPANOV

and

then

L (g:,"'• qn, rn),...., qnrn L (g:,n, qn, rn),...., L (%:'' qn, rn) ,-.- p* qnrn ' Oe P log (qnrn)' Ox x log (qnrn)

T (%~' qn, r"),...., o:n, Tm:· qn, r"),...., T (%~' qn, 'n),...., o:*n.

And what is more, the bounds of the complexity and delay are simultaneously attained asymptotically.

The lower bounds of the complexity and delay (in all four theorems) are established by analogy with the corresponding bounds of Theorems 1 and 2. A certain difference resides in the fact that the lower bound of the delay of the system of functions is determined by the complexity of one function of the system - in the case given, of order 2 n / n.

The synthesis methods which yield simultaneaus attainment of asymptotically minimal complexity and asymptotically minimal delay are likewise analogaus to the corresponding methods for one function. The basic difference resides in the fact that the functions

1::;1 •• 11 ,, i (r, ß; .. , "") are realized directly as functions of k arguments with an asymptotically

minimal delay ("'Tk). t The nurober of strips p satisfies the condition p<. 2n~ks + 2 (compare

with [6], p. 102). The values of the parameters may be chosen, for example, in the following

way J Assurne ft =log (qnrn), A = min (log n, n- ~og 11 ). Then

k=[log[t-t-A], u=[2A], s=[rt-5log[t],

t'= 2 [2lo~k' (m-A)J, t"= 2 [2lo~k"'J'

In the case of an irregular basis the block B0 in realizing vector-functions from 'iJ~' q

additi.onally realizes the function e = x 1x2 ••• Xn V x1x2 ..• Xn in accordance with the representation e = (x1 V I;) (x2 V x3) ••• (xn-t V xn) (xn V x1) and using S7 networks. Then 0-functions coinciding with the components of the original vector-function on all collections with the excepti.on of 0 and l are realized. Finally, either 0 (if the corresponding component is a 0-function) or e (if the corresponding component is a 1-function) is added to each of the functions obtained.

Literature Cited

1. V. V. Glagolev, "Same bounds for disjunctive normal forms of functions of the algebra of logic," in: Systems Theory Research, Vol. 19, Consultants Bureau, New York (1970), p. 74.

2. V. B. Kudryavtsev, "Completeness theorem for a certain class of automata without feedbacks," in: Problemy Kibernetiki, Vol. 8, Fizmatgiz, Moscow (1962), pp. 91-115.

3. 0. B. Lupanov, "On the synthesis of certain classes of supervisory systems," in: Problemy Kibernetiki, Vol. 10, Fizmatgiz, Moscow (1963), pp. 63-97.

t One may indicate a direct (but more complex) construction which does not use Theorem 1. t In realizing the vector-functions from ~:· qn and from \Y;' qn we take k"' and krv, respec-

tively, instead of k' and k", in the expression for t' and t".


4. 0. B. Lupanov, "On a certain class of networks consisting of functional elements, 11 in: Problemy Kibernetiki, Vol. 7, Fizmatgiz, Moscow (1962), pp. 61-114.

5. 0. B. Lupanov, 110n a certain method of network synthesis," Izvestiya Vuzov, Radiofizika, 1(1):120-140 (1958).

6. 0. B. Lupanov, "On a certain approach to the synthesis of supervisory systems - the principle of local coding," in: Problemy Kibernetiki, Vol. 14, Nauka, Moscow (1965), PP• 31-110.

7. S. V. Makarov, "The upper bound of the average length of a disjunctive normal form," in: Discrete Analysis (Transactions of the Mathematics Institute, Siberian Branch, Academy of Seiences of the USSR), No. 3 (1964)~ pp. 78-80.

8. R. G. Nigmatullin, 11 The variational principle in logic algebra, 11 in: Discrete Analysis (Transactions of the Mathematics Institute, Siberian Branch, Academy of Seiences of the USSR), No. 10 (1967), pp. 69-89.

9. S. V. Yablonskii~ G. P. Gavrilov, and V. B. Kudryavtsev, Logic-Algebra Functions and Post Classes, Nauka, Moscow (1966).

10. E. L. Post, "Two-valued iterative systems in mathematicallogic, 11 Princeton Ann. of Math. Studies, Vol. 5 (1941).

11. C. E. Shannon, "The synthesis of two-terminal switching circuits, 11 Bell System Technical Journal, 28(1):59-98 (1949).

PROOF OF MINIMALITY OF CIRCUITS CONSISTING OF FUNCTIONAL ELEMENTS t

N. P. Red'kin

Moscow

Introduction

In the synthesis of circuits realizing Boolean functions it is important to construct minimal circuits. Some results in this direction were obtained qy Cardot [5], who proved the minimality of relay contact circuits for a sum of n variables modulo-2, whereas Soprunenko [3] obtained a minimal realization of conjunctions and disjunctions with the aid of circuits of functional elements in a base consisting of Sheffer's stroke.

In this paper we consider the realization of Boolean functions by circuits of functional elements in a base consisting of functions realizing conjunction, disjunction, and negation (the notations are: E&, a conjunctor; EV, a disjunctor; and E-, an inverter)J We present circuits for the realization of a linear function, and of the comparison operator and coincidence operator. We also prove the minimality of these circuits.

The upper bounds follow directly from these circuits realizing the functions just mentioned.

The idea of the proof of the lower bounds is as follows: For any minimal circuit realizing a Boolean function (or operator), we establish the possibility of removing a certain number of elements, thus obtaining (perhaps after changing the configuration of the remaining elements) a new circuit realizing a similar function (operator), but with a smaller nurober of variables. We find that it is not sufficient to consider circuits with a limited number of inputs (as is done, for example, in [5]), our approach being based on the separation of bounded "pieces" from the circuits under consideration (in the sense of the number of elements they contain).

Prior to a detailed analysis, let us introduce the concepts of complexity and minimality of circuits.

LetS be a circuit of functional elements realizing conjunction, disjunction, and negation. The complexity of a circuit [denoted by L(S)] is defined as the nurober of elements occurring in this circuit. The smallest of the complexities of circuits realizing f (or the oper-

t Original article submitted March 15, 1969. t The definitions of eertain often-encountered concepts and terms, not given in this paper, can

be found in [1], [2], and [4].

85

86 N. P. RED'KIN

x 1 x 2

Fig. 1 Fig. 2 Fig. 3

ator F) will be called the complexity of a function f (or operator F) anddenoted by L(j) (L(F)). A circuit S that realizes a function f (or operator F) is said tobe minimal if L(S) = L(j) (or L(S) = L(F)).

§ 1. Two Properties of Circuits of Functional

E 1 e m e n t s E&, EV, E-

Let S be a circuit of functional elements E&, EV, E-.

Pr o p er t y 1. Suppose that together wi th the variables x1, ••• , Xn we have also constants 0 and 1 that can be applied to the inputs of the circuit S. If we apply to the input of an element of S the identical zero or the identical unity, this element can be removed from the circuit in such a way that the function realized by this circuit does not change. t

Indeed, suppose for example that the circuit has a disjunctor EV (Fig. 1) to one of whose inputs we apply a constant (0 or 1). If, for example, cp 1 = 1, we have cp0ut = 1, and the element EV can be eliminated from the circuit, and we can apply cp 1 to the inputs of the other elements to which cp0 ut is applied. If, on the other hand, cp1 = 0, we have %ut = cp2, and we can lik.ewise remove EV from the circuit and apply cp 2 to the inputs of the other elements to which cpout is applied.

This property can be formulated in a similar way for a conjunctor and an inverter.

Pr operty 2. Suppose that together with the variables x1, ••• , Xn we have constants 0 and 1 that can be applied to the inputs of the circuit. Hence if any circuit element realizes a function which is identically zero or unity, it is possible to remove this element from the circuit in such a way that the function realized by the circuit does not change.

This property is evident.

Now let us introduce a concept that will be often used below.

Suppose that the variables x1, x2, ••• , Xn are applied at the inputs of a circuit S that realizes a Boolean function. We shall say that a variable xi is ob s tru ct ed by the variables xi, xk, •.. , xz if the application of certain constants at the circuit inputs correspondiJW to the variables Xj, XkJ ••• , xz will result in an output function ofthe circuit that does not depend on xi. In abbreviated form the obstruction of the variable~ by the variables xi, xk, ••• , xz (with respect to a circuit S) will be denoted by S (xi' xk, ••• , xz- ~).

t It is assumed throughout that the configuration of the elements in the circuit can be changed (after certain elements have been removed).

PROOF OF MINIMALITY OF CffiCUITS 87

X; For example, for the circuit of Fig. 2 we have S(x2, x3 - x1),

S(x1 - ~,and S(x1 - x3), since <flout(x1, 1, 0) = 0, <flout(1, x2, x3) = x3,

and <flout (0, x2, xa) = x2• On the other hand, for the circuits of Fig. 3 none of the variables is obstructed.

Fig. 4

Below we shall also use the notation

X (Ec) = { ~: where oE{V, &}.

if if

Eo- di sjunctor, Eo- conjunctor,

The proof of the lower bounds will be given for the case where at the inputs of the circuit elements we are allowed to apply together with the variables x1, ••• , x 11 also the constants 0 and 1. It

is evident that these bounds will be valid also for the case where only the variables Xto ... , x 11 can be applied at the circuit inputs (it is precisely this case that is referred to in Theorems 1-3).

§ 2. Minimal Realization of Sum of n Variables

Modulo-2

Th eo re m 1. If fn (xl, ... , Xn) =X1 ffi.x2 (f) ... ffixn, a /~ (x!> ... , Xn) = X1 (f)x2 (f) ~ .. (±)"X;., n >2, then L(j 11 ) = L(JA) = 4(n- 1).

Proof of Theorem 1. Upper bound. Since

fn (xb · · ·, :l'n) ~c fn-1 (xb · · ·, Xn-!) (f) Xn,

f~ (xl, · · ·, J:n) = fn-1 (xl, · · ·, Xn-1) ffixn

and (as is easy to see by "trial") L(f2) = L(j~), = 4, the upper boundcan be easily proved by induction on n (the minimal circuits realizing f2 and !2. are shown in Fig. 3).

The lower bound is likewise easy to prove by induction on n, by using the following lemma.

Lemma 1. Let sg1in be a minimal circuit realizing one of the functions j 11 , j 1~, n 2: 3. If it is allowed to apply at the inputs of the circuit s;pin the constants 0 and 1, there will always exist four or more elements that can be removed from Sfrin, thus yielding a new circuit S11 _ 1 which realizes any of the functions / 11 - 1 and / 1~_ 1 • t

Pr o o f o f Lemma 1. Suppose we have a circuit S Irin, Depending on the possible form of this circuit let us consider all the possible cases listed in Tables 1 and 2 (Table 2 lists all the subcases of Case 2.3 of Table 1).

1.1. To this case there corresponds Fig. 4J

Since the circuit under consideration is minimal, the output of at least one of the elements Ej or E'k (say, the output of the element Ej) must be applied to the input of some element E z (and not to the output of the circuit).

Suppose that the constant X(Ej) has been applied to the input of the circuit Sf.!Ü11 that corresponds to the variable xi. Then the circuit S Irin will realize one of the functions fn _1 and f ~ _1,

t The notations used in Theorem 1 and Lemma 1 will be retained in this section throughout. t The constraints corresponding to this case are not formulated in the text, since these con-

straints are listed in the tables.

88 N. P. RED'KJN

T ABLE 1. Splitting of Proof of Lemma 1 into Cases

0) General case (the circuit realizing a linear function is minimal;

it is allowed to apply the constants 0 and 1 at the circuit inputs)

1) At least one variable Xi is applied directly at the input of an inverter (for example, of the

inverter EI)

2) None of the variables xi, ••• , x 0 is applied directly to the input of an inverter

1,.1) 1bere exist at least two elements at whose input s we apply the output of the

element Ei

1. 2) Tb.e output of the element Ei is applied to th-e input of only

one element

2 ~~is[sh:~e least' one variable

which is applied to the inputs

of more than two elements

2.2) There exists at least one variable which is applied to the input

of only one element

2.3) Each variable is applied to the inputs

of precisely two elements

constants will be applied at the inputs of the elements Ei, E j, E k• and E z, and these elements can be removed from the circuit Si[lin (Property 1) in such a way that the conditions of Lemma 1 are satisfied.

1.2. Let us note right away that the element Ej (Fig. 5) cannot be an inverter, since the circuit under consideration is minimal. Furthermore, under our assumptions we cannot confine ourselves to applying xi only to the input of the element Ei. Indeed, if xi is applied only to the input of Ei, it is possible to find constants a-1, ... , <Ti _ 1, <ri _ h ••• , a-11 , such that when these constants are applied at the circuit inputs corresponding to the variables x1, ••• , xi _11 xi+i• ... , x 11 , weshall have the relation cp 1 = x (Ej) (this can be always done, since cp 1 does not depend by assumption on Xi and cp 1 cannot be identically equal to a constant, since the circuit under consideration is minimal). In this case cpi = x (Ej) and the output function /out of this circuit will satisfy the relation

fout (at. · · ., <Ji-h 0, <Ji+1• · · ., <Jn) =fout (a1, · · ., <Ji-t. 1, <Ji+t. • • ., <Jn)·

But this relation signifies that the variable xi is obstr~cted by the variables x1, x2, ... , xi _1,

xi +t• ... , Xn, which cannot be the case for a circuit Slf1111 that realizes one of the functions f 11

and f~.t

Hence, tagether with Ei there exists an element Ek at whose input we apply the variable Xi (Fig. 6).

Since neither the output of the element Ef nor the output of the element E'k can be the output of the circuit, and we cannot simultaneously apply the output of Ej to the input of Ek and the output of E'k to the input of Ej, there mustexist an element Ez to one of whose inputs we apply either the output of Ej or the output of Ek. Let us assume that at the input of Ez we apply the output of Ek (as shown in Fig. 6).

Let us write xi = x (E 'k) (if the output of E j is applied to the input of E z, we shall write xi = x<Ej) and reason in the same way as below). In this case the circuit will realize one of the functions f 11 _ 1 and f ~ _1 of the variables x1, ... , x i _1, x i +1, ... , x11 , constants will be applied at the inputs of the elements Ei, Ej, E'k, and Ez, and it is possible to remove these elements from the circuit (Property 1) without changing the output function. Hence Lemma 1 will hold also in this case.

t Below we shall not explain in similar cases in detail the obstruction of variables.

PROOF OF MINIMALITY OF CffiCUITS 89

Fig. 5 Fig. 6 Fig. 7 Fig. 8

2 .1. In this case it is possible to single out at least three elements Ej, Er, E'k (Fig. 7) at whose inputs we apply a variable xi.

Since none of these elements can be the output element of the circuit, and the circuit under consideration is minimal and has no cycles, there mustexist an element Ez to one of whose inputs we apply the output of at least one of the elements Ef, Ej, Ek (for example, the output of Ef, see Fig, 7).

Let us write x i = x (Ei). In this case we apply constants to the inputs of the elemen ts Ef', Ej, Ek, Ez, andthese elements can be removed from the circuit (Property 1) in such a way that the conditions of Lemma 1 hold.

2.2. The Case 2 .2 cannot occur.

Indeed, if xi is applied only to an input of Ei (Fig. 8), we have s~in (x1, ••• , xi-t• xi+t• x 11 - xi), which is impossible.

... ,

It remains to consider Case 2.3. It is evident that in this case there mustexist an element Ei to both of whose inputs we apply variables belanging to x1, ••• , x 11 • Without loss of generality it can be assumed that the variables x1 and x2 are applied to the inputs of E'f. The variable Xt is applied also to the input of an element E j. Let us consider all possible types of circuits Slflin corresponding to this case (see Table 2).

2.3. 1.1. Let us write x1 = x (Ej) (Fig. 9). It is then possible to remove (Property 1) the elements Ef and Ej, as well as an element Ez to whose input we apply the output of Er (such an element E z always exists, since the circuit under consideration is minimal and the output of Ej cannot be the output of the circuit, since otherwise we obtain for x1 = x (Ej) the relation four(X (Ej), x2, ••• , X11 ) =X (Ej)), and an element Ek, since for x1 = x (Ej) the value of cpi does not depend on cpk, and the output of Ek is not applied to inputs of elements other than Ej. Hence Lemma 1 holds in this case.

2.3.12. Let us note the following necessary property of the circuit considered in this case (Fig. 10): For x2 = x (Ej) we must have cpk = x (Ej).

Indeed, if this property does not hold, it is possible, for x2 = x (Ei) and by applying some constants CJ 3, ••• , 0" 11 to the circuit inputs corresponding to the variables x3, ••• , x 11 , to obtain the relations

CjJ~< =X (Ej), Cjlj =X (Ej}, Cj)i =X (Ei).

But in this case s;rin (x2, ••• , x 11 - x1), i.e., the circuit under consideration does not realize any of the functions fn and fi~ •

Thus for the case under consideration the above-formulated property must hold, and for x2 = x (Ei) constants are applied to the inputs of the elements Ei, Ej, and Ez (cpk = x (EjH).

90 N. P. RED1KIN

TABLE 2. Splitting of Case 2.3

2.3) To the inputs of an element Ef we apply the variables "~. and X:l• The variable ~ is applied also to the input of an element Ej

2.3.1. To one of the inputs of the element Ej we apply the output of an element Ek

2.3.1.1) The 2.3.1.2) There output of the exists (besides 2.3.2.1)

2.3.2) To the input of the element EJ' we apply a variable xi

2.3.2.2) To an Ta an input of the

2.3.2.3) To an

element Ek is E~ an element input of the element E~ we input of the

element E? we applied only to Er to whose in• element E~ we J apply a variable J J an input of put we apply the apply the

f'pply the variablE betonging to the element Ej output of the variable X:z element Ek

2.3.2,3.1) The outpul of at least one of the elements Ef and Ej(for example, Ef

is "branehed," i.e., it is applied to the inputs of tw5) or more elements

2,3,2,3,2.1) Nene of the outputs of the elements E? and E? is applied directly

1 J to the inverter input

Fig. 9

x, :xg, ~~ • • 0 I Xn

2.3.2,3.2) The outpul of the element Ef is applied te the input of only one ete .. ment, and the Output of Ej is-likewise

applied to the input of only one element

2.3.2.3,2.2) The outpul <tf at least one of the elements Ef and Ej(for example

the Output of E?) is applied to the Gut-1

put of an inverter (~)

Fig. 10

Thus by satisfying the conditions of Lemma 1, it is possible to remove the elements Ei, Ej, Ez (Property 1) and the elements Ek (Property 2).

2.3. 2.1. This case cannot occur, since the circuit s;pin is minimal.

2.3. 2.2. This case (Fig. 11) is likewise impossible, since for it we have s;pin (x2, xi- x1).

2.3. 2.3. 1. In this case (Fig. 12), for x1 = xo<Ei), it is possible to remove four ele-ments (Ef, Ej, Ek, and Ez), and satisfy the conditions of Lemma 1.

2.2. 2.3. 2.1. This cannot occur. In fact, the outputs of the elementsEi and Ej cannot be applied to the inputs of one and the same element, since otherwise the circuit is not minimal. Hence the output of one of the elements Ej' and Ej (for example, Ef) must be applied to the input of an element E'k (Ek is not an inverter~). to whose other input we apply one of the variables x3, ... , xn, or a function cp 1 (as is shown in Fig. 13) that does not depend on x1 and x2 (x1 and x 2 are applied only to the inputs of the elementsEi and Ej).

PROOF OF MINIMALITY OF CffiCUITS

Fig. 11 Fig. 12

Ef65

Fig. 13 Fig. 14

Let us write x2 = x (E j) and apply to the circuit inputs corresponding to the variables x3, ••• , xk constants a 3, ••• , a 11 suchthat cp 1 = x (Ej). Then the values of cpi and cpk will not depend on x1 (the output of Ei is not 11branched 11) and we finally obtain S ~in (x2, ••• , x 11 - x1),

which cannot be the case.

91

2.3. 2.3. 2.2. Suppose, for example, that the output of the element E'f is applied to the input of the inverter Ek (Fig. 14). In this case the output of Ek (it cannot be the output of the circuit, since the output function of the circuit cannot be identically equal to a constant when a constant is applied to the circuit input that corresponds to the variable x1) is applied to the input of an element Ez. Let us write x1 = x(Ej).

In this case cpi = X (Ei), cpk :== x(Ei), andconstant will be applied to the inputs of the elements E'f, Ej, Ek, Ez. Hence follows (by Property 1) the validity of Lemma 1 forthislast case.

Thus we have completed the proof of Lemma 1.

Rem a r k. In a similar way it is possible to show that 7 (n - 1) elements are needed for a minimal realization of j 11 (and also of j~) in a base consisting of a disjunctor and negation or of a conjunctor and negation.

§ 3. Minimal Realization of Comparison Operator

The comparison operator F11 is a (2n, 1)-operator (see [4]). It compares two n-digit binary numbers

-· ~ { 1 Fn (x, y) == 0•

' k-1

t If u = (ao, <T 1, ••• , <T k-1), then I cr I= L a,2i. i~O

if I ;: I IYI·

92 N. P. RED'KIN

6)

Fig. 15 Fig. 16

In the language of formulas, the comparison operator can be defined inductively as follows:

Fn (xil ... , Xn, y,, ... , Yn) =XnYn V (xn V Yn)Fn-1 (x" · · ·, Xn-1! y,, · · ., Yn-t); F!(x, y) =xy.

Theorem 2. For any n 2:: 1 we have

L(Fn)=5n-3.

Pr oof of Theorem 2. The upper bound follows directly from the inductive definition of the comparison operator (Fig. 15).

The lower bound is easy to obtain by induction on n with the use of the following lemma.

Lemma 2. Let s[flin be a minimal circuit realizing the function Fn, n 2:: 2. If it is allowed to apply constants 0 and 1 to the inputs of the circuit s:pin, there will always exist five or more elements that can be removed from the circuit Sfrin, and from the remaining elements we can construct a new circuit Sn- 1 that realizes Fn_ 1 •

Pro o f o f Lemma 2 . Suppose we have a circuit Sfrin. Let us consider all the possible cases depending on the type of this circuit (Table 3).

1. Suppose that the variable Xn is applied at the inputs of some elements Ei, Ei, and Ek (Fig. 16) (if the variable Yn is applied at the inputs of three elements, the proof will be similar).

Since none of the elements of Ei. Ei, and Ek can be an output element of the circuit, and the circuit itself is minimal and does not contain cycles, there exists an element Ez to whose input we apply the output of at least one of the elements Ei, Ei, and Ek (for example, Ei). If Ei is a conjunctor or disjunctor, we shall write Xn = x (Ei), and if Ei is an inverter, we shall write xn = 0.

By virtue of Property 1 it is possible to remove the elements Ei, Ej, Ek, and Ez, thus obtaining a circuit that realizes the operator F n (x1, ••• , xn_1, const, y1, ... , y n-1, Yn>. Since in this circuit the variable Yn must be applied to the input of at least orte element of Ern, it is possible to remove for y n = x n = const this element of E m, thus obtaining a circuit Sn _ 1 that realizes F n _ 1•

PROOF OF MINIMALITY OF CffiCUITS

TABLE 3. Splitting of Proof of Lemma 2 into Cases

General case (the circuit realizing F 0 is minimal; we are allowed to apply the constants 0 and

to the circuit inputs)

l 1) Either of the

variables x0 and Yn is applied to

the inputs of three or more elements

I .~

I t

2) x 0 is applied to the input of only one element Ei

l

I J

3) Yn is applied to the input of only one element Ei

1

I t

4) Each of the variables x 0 and Yn is applied to

the inputs of pre·

cisely two elements

2.1) The

element Ei is an inverter

2.2) The element Ei is not an

inverter

4.1) The variable x1 is applied .to the in-

puts of som..e

elementsEi and Ej,

whereas Yn is applie to the in pu ts of Ek

and Ei

4.2) Tbe variable Xn is applied to the in ... puts of Ei and E~

whereas Yn is J

applied to the inputs

of Ej~ and Ek L_ _____ __,

4.3) Both variables! x 0 and Yn are applie

to the inputs of some elements E<:' and E?

1 1

I 1 J

2.1.1) The output 2.1. 2) The output of Ei is applied to of E: is applied to the inputs of more the input of only

than one element one element E?

I t

2.1.2.1) The variable Yn

is applied to the input of an inverter

Ei<

1 2. I. 2. 2) The ariable y 0 is

not applied irectly to the;

input of any inverter

J

I ,\

2.2.1.1) The

output of Ej

is applied to

the input of only one ele-

ment E~. The

variable Yn

is not applied tc

the inputs of any elements

other than Ej

Hence Lemma 2 is valid for Case 1.

l 2.2, I) Yn is applied to the input of an in•

verter E:-J

I I -~

2. 2. 1.2) There I exist two ete ..

ments E~ and Ej to whose inputs we apply the out..-

put of the ele-

ment Ej or the

variable y0

l 2. 2.2) Yn is not ap-plied directly to the input of any inverter

I J

2. 2.2. I) Yn is applied to the in.,

put of only one

element Ej

2, 2.2.2)

'Ibere exist two elements

Ej and E~ to whose inputs we apply Yn

93

2.1.1. Suppose that the output of the element Ei is applied to the inputs of the elements Ej and Eok (Fig. 17). Since the circuit is minimal and without cycles, and none of the elements Ej and Ek can be an output element of the circuit, it follows that there must exist an element Ez to whose input we apply the output of one of the elements Ej and Ek (for example, the output of the element Ej). Let us apply to the circuit input corresponding to the variable xn the constant

94

Fig. 17

Eo k

Fig. 19

N. P. RED'KIN

Fig. 18

Fig. 20

x(Ej). We then remove the elements Ei, Ej, Ek, and Ez (Property 1), and construct from the remaining elements a circuit S~ that realizes a function F~ of 2n- 1 variables:

F~ = YnX (Ej) V (X (Ej) V Yn)Fn-t•

InS~ there mustexist an element Ern to whose input we apply Yn. Let us write Yn = x(Ej). By virtue of Property 1 the element Ern can be removed, thus yielding a circuit Sn-t that realizes F n -t·

Hence Lemma 2 is valid for the case just considered.

2.1. 2.1. In this case there mustexist an element Ez (Fig. 18), other than Ej, to whose input we apply the output of Ek" (otherwise the output function of the circuit will not depend on Xn for some Yn = const). Since the circuit is minimal and without cycles, and none of the elements Ej and Ez can be the output of the circuit, there mustexist an element Ern (m ~ j, l) to whose input we apply the output of one of the elements Ej and E[ (for example, the output Ej). Let us apply the constant x(Ej) to the circuit inputs corresponding to the variables Xn and Yn. In this case constants will be also applied to the inputs of the elements Ei, Ej, Ek, E z, and Ern, and these elements can be removed (according to Property 1) in such a way that the thus-obtained circuit Sn-twill realize Fn-t•

Hence Lemma 2 is valid for this case.

2.1. 2.2. In this case there mustexist two elements Ek and Ez (other than Ej) to whose inputs we apply the variable Yn· Indeed, otherwise we would have to assume that Yn is applied to the input of Ef, or that only one element of Ek exists to whose input w~ apply Yn. It is not allowed to apply Yn to the input of Ej, since otherwise we obtain sgnn (yn - Xrt), which is impossible (for this same reason we cannot apply to the input of Ef any of the variables x1, ••• ,

Xn-t• y1, ••• , Yn-t• but only a function cp 1, see Fig. 19).

PROOF OF MINIMALITY OF CIRCUITS 95

Yu ___ L

-1 J Eil_;_

Fig. 21 Fig. 22

It is also easy to see that we cannot confine ourselves to applying y 11 to the input of only one element of Ek. In fact, we shall select from the circuit s(ilin a subcircuit S<p 1 that has one input and realizes the function q; 1. This subcircuit must contain the element of Ek to one of whose inputs we apply y11 (otherwise we have sgün (x1, ••• , x 11 _ 1, y1, .•. , y11 _ 1- x11 , which is impossible). Furthermore, if y11 is applied only to an input of E k• it is necessary to apply to the second input of this element a function q;2 that does not depend on x 11 and y11 , which is likewise impossible, since ityields Sfilin (x1, •.. , xn-1• Y1• •.. ,Yn-1-Yn).

Thus we must assume that the variable y 11 is applied to the inputs of the elements Ek and E[ (Fig. 20).

Since the circuit has no cycles and none of the outputs of the elements E'k and Ez can be the output of the entire circuit, and it is impossible to apply the output of E'k only to an input of Ez, or (conversely), the output of E[ only to an input of Ek (by virtue of the minimality of the circuit), it follows that there always exists an· element E m (m ~ j, k, l) to whose input we apply the output of at least one of the elements E'k and E[. Suppose, for example, that we apply the output of E'k to an input of Ern (as is shown in Fig. 20). If we now apply the constant x (Ek) to the circuit inputs corresponding to the variables x 11 and y11 , it is easy to verify the validity of Lemma 2 for the case under consideration.

2.2. 1.1. This case cannot occur.

Indeed, since cycles are not allowed, it follows that under the limitations mentioned above it is necessary to apply to the input of Ej or to the input of Ek a function q; 1 that does not depend on Xn and y 11 • Butthisleads to the obstruction of at least one of the variables x 11 and y 11

by the variables x1, ... , x 11 _ 1 and y1, ... , y 11 _ 1, which is not allowed.

2.2. 1.2. By virtue of the minimality of the circuit SWin, the output of the element Ej must be applied to the input of at least one element Ek. Moreover, in view of the limitations assumed for this case, there exists an element E z to one of whose inputs we apply either the variable Yu• or the output of the element Ej (Fig. 2l).t

Since the circuit under consideration is minimaland the output of the element E'k cannot be the output of the circuit, it is easy to show that there must exist an element E m (other than Ez) to whose input we apply the output of the element Ek.

Let us apply the constant x(EjJ to the circuit input corresponding to the variable y 11 • By virtue of Property 1 we shall remove the elements Ej, E k• Ez, and E m, thus obtaining a circuit Sri that realizes

t y signifies either y or y.

96 N. P. RED'KIN

Fig. 23 Fig. 24

In the newly obtained circuit S~ there exists at least one element to whose input we apply x 11

and that can be removed by virtue of Proparty 1 for x 11 = x (Et), thus yielding a circuit 811 _ 1 which realizes F 11_ 1• Hence Lemma 2 is valid in the case under consideration.

2.2. 2.1. This case cannot occur (see Case 2.2. 1.1).

2.2. 2.2. If Ej and Ek are identical elements, the proof of the lemmawill be almost evident (it suffices to write x 11 = Yn = x (Ej) = x (Et) and take into account that this circuit is minimal). Weshall therefore assume below that Ej is a disjunctor and Ek a conjunctor, and, moreover, that Yn is applied only to the inputs of Ej and Ek (if Yn is applied to the inputs of three or more elements, see Case 1). Suppose that Ej is a disjunctor (the case that E'f is a conjunctor can be analyzed in the same way).

Hence follows that:

1) It is impossible to apply the output of E{ (Fig. 22) to the inputs of both elements Ej and Et since the subcircuit realizing cp must contain at least one of the elements Ej and Ef, since otherwise cp will not depend on y 11 , which is impossible (see Case 2.2. 1.1).

2) The output of the element E'! cannot be the circuit output.

3) We cannot confine ourselves to applying the output of Ei1 to the input of only one of the elements Ej and Et (if, for example, the output of E'! is applied only to the input of Et, we obtain. S;pi11 (y11 - x11) (when the identical zero is applied to the circuit input corresponding to the variable y11 ).

It follows from 1-3 above that there mustexist an element Ez to whose input we apply the output of the element E'! • It is also easy to see that:

4) The output of Ej cannot be applied to the input of E'! (see 3)).

5) The output of Ej cannot be the output of the circuit.

6) We cannot confine ourselves to applying the output of the element Ej only to the input of the element Et (if the output of Ej is applied only to the input of Et , the circuit will not be minimal, which contradicts our original assumption).

It follows from 4-6 that the output of Ej must be applied to the input of an element, other than E'! and Et. Herewe can have two cases:

a) The output of Ej is applied to the input of an element Ern, other than Ez;

b) the output of Ej is applied to the input of Ez. In Case (a) Lemma 2 is valid, since by applying the identical unity to circuit inputs cor

responding to the variables x 11 and y11 , it is possible by virtue of Proparty 1 to remove five elements (E'!, Ej, Et, E1. Em) and obtain a new circuit 811 _ 1 that realizes F11 _ 1•


In Case (b) the output of the element E~ (a disjunctor or conjunctor) cannot be the circuit output, since otherwise when we apply the identical unity to the circuit inputs corresponding to the variables Xn and Yn, the output function will not depend on x1, ... , Xn-1 and y1, ... , Yn-1,

which is impossible. Furthermore, it is likewise not allowed to confine oneself to applying the output of E ~ only to the input of E~, the output of Ej only to the ~nput of E z, and the output of E~ only to the input of Ez (the application of the output of El only to the inputs of Ez and Ei and of the output of E'j only to the input of Ez cannot take place by virtue of the minimality of the circuit), since otherwise we obtain S)ilin (y n - Xn).

Hence in the Case (b) there mustexist an element Ern to whose input we apply the output of one of the elements E;, E'l and Ez. But in this case, when we apply the identical unity to the circuit inputs corresponding tothe variables Xn and y 11 , we can remove by virtue of Property 1 the elements E(, Ej, E~, EI, and Ern (to the inputs of the element Ez we apply "ones"; therefore the output of this element willlikewise have a "one"), and obtain a new circuit that realizes F n-1• Thus we have completed the proof of Lemma 2 for Case 2.2. 2.2.

3. The proof of Lemma 2 for the Case 3 is carried out in the same way as for Case 2.

4.1. Since the circuit under consideration is minimal and without cycles, and none of the elements Eh Ej, Ek, and Ez (Fig. 23) can be an output element of the circuit, there must exist an element Ern (m ~ i, j, k, l) to whose input we apply the output of at least one of the elements Ei, Ei, Ek, and Ez (for example, the output of Ed. For convincing ourselves of the validity of Lemma 2, it suffices to consider the case that a constant x (Ef') is applied to the circui t inputs corresponding to the variables x11 and y 11 if Ei is a conjunctor or disjunctor, and zero is applied if Ei is an inverter.

4.2. Let us assume that Ej is a disjunctor (Fig. 24; if Ej is a conjunctor, the proof will be similar). If Ei or Ek is an inverter, it is easy to prove Lemma 2 by writing x 11 = Yn = 1 or x 11 = y 11 = 0. Therefore weshall assume below that none of the elementsEi and Ek is an inverter.

The output of the element Ej must be applied to the input of an element Ez (l ~ i, k), since otherwise the circuit is not minimal. Next, there mustexist an element Ern (m ~ i, k, l) to whose input we apply the output of one of the elements Ei and E k (for example, Ei). Indeed, if this assumption does not hold, then (by virtue of the minimality of the circuit) the output of Ei must be applied only to the input of E'k (or conversely, the output of Ek only to the input of Ej') whereas the output of E'k (or Ef') must be applied only to the input of Ej' (Fig. 25). But in this case the element Ez must be a conjunctor (otherwise we have Sifil1 (y 11 - x11 )). Thus the element E k must be a disjunctor (otherwise we likewise have swin (y 11 - Xn)). But such a case is also impossible, since we likewise obtain sgün (y n.- xn). Hence we must assume the presence of the above-mentioned fifth element Ern (see Fig. 24). To the circuit inputs corresponding to the variables x 11 and y11 weshall apply the constant X (Ej') if Ei is a disjunctor or conjunctor, and the identical zero if Ei is an inverter. Then we can remove the elements Ei, Ej, Ek, Ez, and Ern (Property 1) and satisfy the conditions of Lemma 2.

4.3. By virtue of the minimality of the circuit under consideration there exist two elements Ek and Ez to whose inputs we apply the outputs of the elements Ei and Ej (Fig. 26). Hence we can have two cases.

a) There exists a fifth element Ern to whose input we apply the output of one of the elements Ek and Ez (for example, Ek). In this case weshall apply to the circuit inputs corresponding to the variables x11 and Yn. the constant x (E k) if E k is a disjunctor or conjunctor ~ and zero if Ek is an inverter. By virtue of Property 1 we can then remove five elements Ej', Ej, Ek, Ez, and Ern' and satisfy the conditions of Lemma 2.

98 N. P. RED'KIN

Xn Yn Xn Xn Yn Xn Yn

EJ~ Ei ,-0 q :.:i

Xn Yn Xn Yn

Eo L q~ ~q Fig. 25 Fig. 26 Fig. 27

Xn Yn Xn Yn

X y X y

a b

Fig. 28 Fig. 29 Fig. 30

b) The output of one of the elements Ek and Ez is applied only to the input of the other element, for example, as shown in Fig. 27.

But in this case the output of the element Ez cannot be the output of the circuit, since otherwise for X 11 = Yn = x (Ez) the output function will not depend on x1, ••• , x 11_ 1, and y 1, ••• ,

Y11 - 1, which is impossible.

Hence there exists a fifth element Ern (m ~ i, j, k, Z) to whose input we apply the output of Ei. But in this case, when we apply to the circuit inputs corresponding to the variables Xn

and Yn one of the constants (X (Ek) or 0), it is possible by virtue of Property 1 to remove five elements Ei, Ej, Ek, Ez, and Ern, thus obtaining a circuit that realizes Fn-1• This completes the examination of Case 4.3.

Wehave also completed the proof of Lemma 2.

§ 4. Minimal Realization of Coincidence Operator

A coincidence operator Rn is a (2n, 1)-operator such that

~ ~ { 1, Rn (x, Y) =

0,

if X=y,

if x=f=y.

PROOF OF MINIMALITY OF CIRCUITS

This operator can evidently be defined inductively as follows:

Rn (xll · · ·, Xn, Y11 · · ·, Yn) = ((Xn V Yn) V XnYn) Rn-1, R1 (x, y) = (x VY) V xy.

Theorem 3. For any n 2: 1 we have

L (Rn)= 5n-1.

99

Proof of Theorem 3. The upper bound follows directly from the inductive definition of the coincidence operator (Fig. 28).

The lower bound is a simple consequence of Theorem 1 and of the next Iemma.

Lemma 3. Let s;rin be a minimal circuit that realizes Rn, n 2: 2. If it is allowed to apply the constants 0 and 1 to the inputs of the circuit S/Pin, there will always exist five or more elements that can be removed from the circuit s~in, and from the remaining elements we can construct a new circuit Sn_ 1 that realizes Rn_ 1 •

Pr o o f o f Lemma 3 . Suppose we have a minimal circuit S ;rin that realizes Rn. In this circuit there evidently exists at least one two-input element E j_ such that to each of its inputs we apply either a variable belanging to x1, ... , Xn and y1, ••• , Yn• or the output of an inverter to whose input we apply one of the above-mentioned variables. Then the proof of Lemma 3 is continued separately for each of the cases listed in Table 4 (any of the given circuits can be referred to one of the cases listed in Table 4).

1.1. This case cannot occur.

Indeed, if a variable (for example, x 1) is applied only to an input of Ej (Fig. 29), whereas the output of Ej is applied only to an input of Ej, with the second input of Ei beingunder the action of z (z E {x2, ... ' Xn, x2, ... 'x;,, Yt. ... ' Yn, Yl, ... ' ~}) we obtain s~n (z- Xr)' which is not permissible.

1.2. Since the circuit und er consideration is minimal and the output of Ek cannot be the circuit output (Fig. 30; without loss of generality it can be assumed that x1 is applied to the input of Ej), it follows that there exists an element Ez to whose input we apply the output of Ek.

To the circuit input corresponding to the variable x1 we shall apply a constant 0 or 1 suchthat x1 = x (Ek). By virtue of Property 1 wt. ohall remove four elements Ef', Ej, Ek, Ez and construct a new circuit Sri that realizes R11 (const, x2, ••• , xn, y 1, y2, ••• , y 11). In this circuit S~ there must exist an element E m to whose input we apply the variable y 1• To the input of the circuit Sh corresponding to the variable y 1 we shall apply the same constant as was applied before to the input of the circuit s;rin corresponding to xr- Then (by virtue of Property 1) we can remove at least one more element (Ern) and construct a circuit Sn-1 that realizes Rn-i•

2.1. This case cannot occur (see Case 1.1).

2.2. Since the given circuit is minimaland without cycles, and none of the elements Ej', Ei, and Ek (Fig. 31) can be an output element of the circuit, there exists an element Ez (other than Ef', Ei, and Ek) to whose input we apply the output of one of the elements Ej', Ei' and Ek (for example, of Ei). But in this case, by applying the constant x (E[) to the circuit inputs corresponding to the variables x1 and y 1, it is possible to remove no less than five elements and construct a new circuit 8 11 _ 1 that realizes R 11 _ 1 (in the same way as was done, for example, for Case 1.2).

100 N. P. RED'KIN

T ABLE 4. Splitting of Proof of Lemma 3 into Cases

General case

(the circuit realizing the coincidence operator is minimal; to the

inputs of the circuit elements we are allcwed to apply the con~

stants 0 and 1)

To each of the inputs of an element Ef we apply either a variable

belanging to ~' ••• , x0 and y1 , •• o , y 0 , or the output of the

inverter to whose input we apply one of the just-mentioned vari

ables

1) To at least one of the

inputs of the element Ef we apply the output of an

2) To both inputs of Ei we

apply variables belanging to

"l_, ••• , Xnandy1 , ••• , Yn

inverter E7 J

(for example, X:t and some

other variable)

1. 1) The output

of Ej- is applied

only to the input

of E~ and the

variable applied

to the input of

Ej is applied to

the input of only

1.2) There exists

at least another

element Ek to

whose input we

apply either the

variable applied

to the input of E; or the output of Ej.

2,1) The

variable Xz is applied

to the input

2.2) There exist

at least another

two elements Ej

and Ek to whose

input we apply

the variable "1

2.3) The vari

able x" is ap

plied to the

inputs of pre

cisely two ele

ments Efand Ej

this element

2.3. 1) Ej is an inverter

2.3.2. 1) The variable y1 is applied I to the input of Ef _j

i 2,3,2,2.1) The vari

able y1 i s applied to

the input of Ej

2.3.2.2. 2) The vari-1

able y1 is applied to,

the input of EY

2.3.2.2.2.1) The variable Yi is applied to

the inputs of at least two elements

of E~

2.3.2) Ej is not an inverter

2.3.2,2) The variable y1 is not applied

to the input of Ei

2.3.2.2.1) The vari

able y1 is applied to

the input of Ej

2,3,2.2.2) The vari

able y1 is not applied

to the input of Ej

2 .3. 2. 2. 2. 2) The variable y1 is applied to

the input of only one element Ek


Fig. 31 Fig. 32

x, y,

Fig. 33 Fig. 34

2.3.1. Suppose that the output of E] is applied to the input of an element Ek (Fig. 32). By virtue of the minimality of the given circuit there exi.sts an element Ez (l ;r i) to whose input we apply the output of the element Ek. Then the proof can be continued in a similar way as in Case 1.2.

2.3. 2.1.L By virtue of the minimality of the given circuit there evidently exist elements Ek, E z, and Ern such that the output of Ei is applied to the input of Ek, the output of E j to the input of Ez, and the output of one of the elements Ek and Ez to the input of Ern (for example, Ek, as shown in Fig. 33).

To the circuit outputs corresponding to the variables x1 and y1 let us apply the constant x (Ek) if E k is a disjunctor or conjunctor, and the identical zero if E k is an inverter. In this case constants will be applied to the inputs of five elements (Ej, Ej, Ek, Ez, Ern), and these elements can be removed and a new circuit constructed that realizes R 11 _ 1•

2.3. 2.1.2. In this case there exists an element Ek (other than Ei and Ef) to whose input we apply the variable y 1 (otherwise we would have Slrin (x1 - y 1), which is impossible). Furthermore, by virtue of minimality of the circuit Sfilin there exists an element Ez other than Ei, Ej, E k to whose input we apply the output of the element Ei (Fig. 34).

None of the elements Ej, Ek, Ez can be the output element of the circuit, since otherwise when a constant (0 or 1) is applied to the circuit inputs corresponding to the variables x1 and y1,

the output function will not depend on the other variables (n 2: 2), which is impossible [for example, for x1 = y 1 = x {Ez ), if E z is a conjunctor or disjunctor, or for x1 = y 1 = O, if Ez is an inverter, we obtain qJz = const]; since the given circuit is minimaland without cycles, there evidently exists an element Ern (other than Ej, Ej, Ek, Ez) to whose input we apply the output of at least one of the elements Ej, Ek, Ek. But in this case, as is easy to see, when a constant (0 or 1) is applied to the circuit inputs corresponding to the variables x1 and y1, constants will be also applied to at least one input of each of the elements Ej, Ej, Ek, Ez, Ern, and these elements ca.n be removed from the circuit (Property 1) and a circuit constructed that realizes Rn-1•

102 N. P. RED'KIN

Fig. 35 Fig. 36

Fig. 37 Fig. 38

2.3. 2.2.1. In this case Lemma 3 is proved in the same way as in Case 2.3. 2.1.2.

2.3. 2.2.2.1. In this case the circuit has two elements Ek and Ez (Fig. 35) to whose inputs we apply the variable y 1• Next, it is easy to show that there exists an element E m (other than Ef, Ej, Ek, Ez) to whose input we apply the output of one of the elements Ef', Ej, Ek, Ez (for example, Ek). But in this case it is easy to convince oneself of the validity of Lemma 3 for this case [when we apply x (Ek) ü Ek is a disjunctor or conjunctor, and the identical zero if Ek is an inverter, to the circuit inputs corresponding to the variables x1 and y1, it is possible to remove five elements and obtain a circuit that realizes Rn _1].

2.3. 2.2.2.2. Fig. 36 (zE{x2, .•. , Xn, Y2• · · ., Yn}).

Let us assume that Ei and Ej are the same elements. To the circuit input corresponding to x1 weshall apply the constant x (Ei) = x (Ej). In this case it is possible to remove no less than four elements (if the circuit has at least two elements E p and E q to whose inputs we apply the outputs of Ej_ and Ej, it is possible to remove Ei, Ef, Ep, Eq, whereas if the outputs of the elements Ej' and Ei are applied to the inputs of only one element Eu it is possible to remove E'f, Ej, Er and yet another element Es to whose input we apply the output of Er; such an element E 5 evidently exists under our assumptions), and obtain a circuit that realizes Rn <x (Ei), x2, ••• ,

Xn, y1, ••• , Yn>· Furthermore, when x (Ej) is applied to the input of the newly obtained circuit corresponding to the variable y 1, it is possible to remove yet another element and obtain a circuit Sn-1 that realizes Rn_1• Thus Lemma 3 is valid under our assumptions.

It remains to consider the case that the elementsEi and Ej are distinct. Let us assume that Ek is a two-input element, for example, a disjunctor (if Ek is an inverter, the reasoning will be the same). Since Ej' and Ej are distinct elements, one of them (for example, Ej) will be a disjunctor (Fig. 37).

Under the above assumptions the outp~t of Ei! cannot be applied only to the input of Ej, whereas the output of Ej cannot be applied at all to the input of E';! (otherwise we have in both cases s~in (x1- y1)). Therefore only the following two cases are possible:

a) There exist two elements E z and E m (other than Ef, E'l, E';!) to whose inputs we apply the outputs of the elements E'l and E';!.

b) The outputs of the elements E'l and Ei! are applied to the inputs of only one element Ez.

PROOF OF MINIMALITY OF CIRCUITS

If the identical unity is applied to the circuit inputs corresponding to the variables x1 and y1, it is possible (by virtue of Property 1) to remove five elements in Case (a) (the ele-

103

ments Ef, E'/, E~, Ez, Em), as well as in Case (b) (the elements Ef, E'l, E~, E1 and yet another element at least, towhose input we apply the output of the element Ez) and obtain a circuit that realizes R11 - 1•

If the element Ei is a disjunctor (Fig. 38), the reasoningwill be as follows. The output of E';! cannot be applied only to the input of Ef (otherwise we have sgün (x1 -- y1)); hence there exists an element Ez to whose input we apply the output of E~. Furthermore, the output of Et cannot be applied to the input of E~ (since S)pin (x1 - y1)) and by virtue of the minimality of the circuit we cannot confine ourselves to applying the output of Ei only to the input of Ef. Hence we can have two cases:

c) The output of E':/ is applied to the input of an element Ern other than Ef, E~, E1;

d) the output of E} is applied to the input of E z • N ow it is easy to see that Lemma 3 is valid for Case (c) as well as Case (d).

Thus we have completed the proof of Lemma 3.

Literature Cited

1. 0. B. Lupanov, 11Synthesis of certain classes of control systems,11 Problemy Kibernetiki, Vol. 10, Fizmatgiz, Moscow (1963), pp. 63-97.

2. 0. B. Lupanov, 11 A method of synthesis of control systems - the principle of local coding, 11

Problemy Kibernetiki, Vol. 14, Nauka, Moscow (1965), pp. 31-110. 3. E. P. Soprunenko, "Minimal realization of functions by circuits using functional ele

ments,11 Vol. 15, Problemy Kibernetiki, Nauka, Moscow (1965), pp. 117-134. 4. S. V. Yablonskii, 11 Functional constructions in k-valued logic," Proc. Steklov Mathematical

Institute of the Academy of Seiences of the USSR, 51:5-142 (1958). 4. C. Cardot, "Some results concerning the use of Boolean algebra in the synthesis of relay

contact circuits," Annales des Telecommunications, 7(2):75-84 (1952).

FULL TEST FOR NONREPETITIVE SWITCHING CIRCUITSt

Kh. A. Madatyan

Moscow

One of the main subjects of cybernetics is investigation of the reliability of control systems [11]. To test the performance of a control system and to locate failures occurring in it is one of the most difficult and important problems. First investigations in this direction were made by I. A. Chegis and S. V. Yablonskii [9] who proposed an algorithm for testing electric circuits which with a relatively small sequence of instructions (tests) makes it possible to determine not only whether the system operates properly but also to find the nature and location of failures in the system to within electrically distinguishable components.

However, as noted by the authors, the large amount of work involved makes it impracticable even in the most simple cases to use the general algorithm for designing minimal tests and for evaluating their complexity. It is thus of considerable importance to consider the design of minimal tests for specific classes of circuits. Early results in this field can be found in [9] (Chapter 2). The tests discussed there were designed for circuits implementing symmetrical and linear functions, for comparators, and for a binary adder.

Further results were obtained by V. V. Glagolev [2] (tests for block switching circuit), I. V. Kogan [4] (diagnostic tests for nonrepretitive switching circuits), and V. V. Vaksov [1] (single diagnostic tests for nonrepetitive circuits). These works were concerned mainly with the design of minimal tests and with evaluating their complexity.

A comprehensive discussion of full diagnostic tests must also consider the construction of algorithms for carrying out the test procedure.

In this article we consider a class of nonrepetitive circuits. A full diagnostic test is designed for such circuits and its length is evaluated; also, a minimal test is constructed for nonrepetitive I1 circuits. A simple algorithm for carrying out the test is formulated for this class of circuits.

Let us now define the basic concepts dealt with in this article.

Let the switching circuit 21. implement the logical algebra function f (x1, x2, ••• , x 11 ).

Assurne that the only failures that can occur in the circuit 21. are either short- or open-circuit failures. Thus (if a failure takes place), the circuit 21. turns into the circuit 21.'. Let us denote by f' (x1, x2, ••• , x 11) the conduction of the circuit 21.'. The function f' (x1, x2, ••• , x 11 ) is called

t Original article submitted February 15, 1968; revision submitted February 14, 1969.

105

106 KHoAoMADATYAN

the failure function o The set of all failure functions is partitioned into the classes F 0, F 1, .. o' Fm suchthat f (x1, x2, .. 0, x ) belongs to F 0, and the functions f' and f" belong to one and the same class if and only if

f' (xt, Xz, ... , Xn) = f" (xt, Xz 1 ••• , Xn) ·

To distinguish between the above-indicated classes one must carry out a certain experiment that is characterized by the set of sequences applied to the circuit inputso A general definition of a test is given in (9]o Certain specific types of tests are defined belowo

Definition 1. A totality of sequences is called a checktest (Tc) with respect to a given list of failures iffromtheconductionofthecircuitwiththesesequences it is possible to determine whether the circuit is in good working ordero Obviously, Tc

m allows the class F 0 tobe distinguished from u F1•

i=!

Definition 2o A totality of sequences is called a diagnostic test (Td) with respect to a given list of failures iffromtheconductionofthecircuitwiththese sequences one can determine the nature of the failureo

Obviously, the test T d makes it possible to distinguish between the classes F 0, F 1, .. o, Fm o

Definition 3 o The number of sequences comprising a test is called the 1 eng t h o f the tes t.

From the point of view of the possible failures we shall consider single and full testso

A s in g 1 e te s t is a test capable of detecting failures when it is known a priori that the failure can occur in any but only one contacto

A f u 11 t es t is a test capable of detecting failures when the possible failures are ei ther short or open-circuits in any contact (or several contacts simultaneously)o

§ 1. Full Test for Switching Circuits

Let t1 <x1, xo, ... , xn> be the shortest test applicable to all circuit realizations of the function f (x1, x2, .. o, Xn) o Further, let t (n) = max t1 (x1 , x 2 , •.• , xnh where the maximum is taken over all

I logical algebra functions depending on n argumentso In (9], S, V o Yablonskii asked what is the asymptotics of the function t(n)? We will show that

t(n) = 2n o

Definition 4 o A test that detects a short- (open-) circuit only in any contact (or contacts) is called a f u 11 s h o r t- ( o p e n-) c i r c u i t t es t.

Let Wf be the set of vertices (a1, a 2, ... , an) of an n-dimensional unit cube such that f (a1, a 2, ... , an) = 1. Let H,.o (f) (Bko (!)) be a subset of the set W f (W j) whose elements do not overlap with the intervals of an abridged disjunctive normal form of a dimension not less than k0, and let t} (tj) be the smallest full open- (short-) circuit test.

o I H ko (/) I ( s I Bko (!)! } t Lemma 1. t1> 2,.0 it> 2,.0 •

t 1 H,.0 (/) 1 is the number of elements in the set Hko (!).

TEST FOR NONREPETITIVE SWITCHING CffiCUITS 107

Pro o f. To any chain s of the circuit 21 corresponds a conjunction of variables, and to eachconjunctioncorresponds a certain interval in the n-dimensional unit cube.

Let an arbitrary circuit 21 realize the function f, and let s 1, s 2, ... , sz 1 be chains in the circuit ~1, covering the points of the set H k" (f). Clearly, the chain s i (i = 1, 2, ... , Z1) realizes a k-dhnensional interval (k :::; k0). In the circuit ~ let us now open all contacts not belanging to the chain si. The resulting defective circuit realizes a function equal to unity only with those sequences which realize the chain si and, consequently, is distinct from f = 0 only with these sequences. Let us do the same for all i (i = 1, 2, ... , Z1). As a result we obtain apart of the

failure function table. The length of the test for such failure functions is not less than

and thus certainly tP-;;, 1 H ko (f) 1

I ""' zk11

I H ko (/)I

To any dead-end cut [7] in the circuit ~ corresponds a disjunction of variables and to each disjunction corresponds a certain interval in the n-dimensional unit cube.

Let w11 w2, .•• , w12 be tlle set of dead-end cuts in the circuit \'ll, that cover the points of the set Bko (/).

Let us close all contacts not belanging to the cut wi. The resulting defective circuit realizes a function equal to zero only with sequences that realize the cut w i (i = 1, 2, ... , Z2).

Reasoning similarly we have

This proves the lemma.

Theorem 1. t(n) = 211 , and the fraction of almost all functions of the algebra of logic is

zn t (f) > ( i - Ö) log2 n log2 log2 n (O<ö<i).

Pr oof, tj + tj = 211 •

Fora parity counter H0(j) = B0(j) = 211 - 1, and according to Lemma 1, t f::::

It has been shown in [3] that for any e > 0 and assuming K0 = [log2 ((1 + e) log2n • log2 log2 n)], we have for almost all functions of the algebra of logic

In virtue of duality, the same basic premises show that for almost alllogic algebra func-tions, 1 Bk, (f) 1 ~ 2n-!. Hence we can conclude that for almost all functions

Since k0 = [log2 ((1 + e)log2n • log2 log2 n)], and considering Lemma 1, we obtain the second statement of the theorem, i.e., that for almost all function of the algebra of logic

Where ö = e+o (i) 1+e ·

108 KH.A.MADATYAN

§ 2. Full Test for Nonrepetitive Switching Circuits

Let T~(n), Tc (n), TJ(n), and T d (n) denote the Shannon functions of respectively a single check test, a full checktest, a single diagnostic test, and a full diagnostic test for nonrepetitive switching circuits.

I. V. Kogan [4] proved that Tc (f) ::::; n + 1, where Tc (f 1) = n + 1 for f 1 = x 1 • x2 ... X 11 •

V. V. Vaksov has shown that TJ(n) = n + 1. It is easy to observe that T~(n) = n + 1.

The number of edges in a dead-end cutwill be called the width of the cut [7]. The wid th of the circuit ~ is said to be the maximum number of edges in the dead-end cut of this circuit. The length of the Iongest chain of a circuit is called the 1 eng th of this circuit.

Let the nonrepetitive circuit have a length l and a width b.

In a certain sense we shall improve on the results obtained by I. V. Kogan; thus:

Theorem 2. Tc (f)<.b+ l.

First, let us formulate certain well-known facts and prove one lemma.

Consider a two-terminal net with terminals A and B. Let each vertex ofthe net be assigned a certain number. A chain between the terminals is said tobe monotonic if the numbers of the vertices of this chain increase in moving from terminal A to terminal B.

Lemma 2. The vertices of a strongly connected net can be numerated with the numbers 0, 1, 2, ... , l (where l is the length of the net) so that a monotonic chain passes through every edge.

A proof of this lemma is given in [7].

A segment of any arbitrary chain ö from the vertex c to vertex b is denoted ö (c, d).

Let c and d be two adjacent vertices of contact of two monotonic chains o1 and o2, and y be an edge through which only one of these chains passes in the interval (c, d) (Fig. 1). Denote Öt(C, d) by (Öt, '}',<52) and o2(c, d) by (<52,'}', Ot>·

Lemma 3. Let the vertices of a strongly connected net be numerated sothat a monotonic chain passes through every edge, and let the width of the net beb. One can then indicate b monotonic chains such that at least one such chain passes through every edge of the net.

Pr oof. Let S(b) be a two-terminal strongly connected net of width b. Let us draw b monotonic chains a 1, a 2, ... , ab which altogether cover a maximum number of edges of the net S(b). Weshallshow that these chains cover all edges of the net S(b). Let us assume that the opposite is true, i.e., no matter how we choose the chains a 1, ... , ab there exists one edge a such that none of the chosen chains passes through it. We will show that the width of such a net is greater than b. Through a let us draw a monotonic chain a.

The following statement is then true:

For any monotonic chain ö that does not pass through a there exists in the segment (ö, a, a) at least one edge aö through which passes at most one chain of the set { a1, a2, ... , ab}• Let us assume the contrary, i.e., that there exists a monotonic chain <5 1 in which at least two chains from the set { a 1, a 2, ... , ab} pass through every edge of the segment (o1, a, a).

Two cases are possible.

1. There exists a chain ai in which the segment (ai, a, a) coincides with the segment (ö ', a, a).

TEST FüR NONREPETITIVE SWITCHING CIRCUITS 109

6 Ao-----

~-~~~-~ -----• B . c d

Fig. 1 Fig. 2

2. There is no chain ai such that a i covers the entire segment ( o 1, a, a).

If the second case is true, then on the segment ( o 1, a, a) let us select the vertices d1 and d2 (Fig. 2) so that the chain ai1 passes through (c, d1) and the chain a~ passes through (d1, d1). In place of the chains ai1 and a i2 let us take the chains a I1 and a ~, where

ai1 : Gi1 (A, d1)-Gi2 (d!, B),

ai2 : Gi 2 (A, d1)- Gi1 (d1, B)

and at covers the segment o1 (c, d2), greater than the segment covered by the chain ai1 • Repeating this argument, we arrive at the first case, i.e., the chain ai covers the segment ( o1, a, a). Instead of the chain a let us take the chain

ai: a; (A, c) -a (c, d) -a; (d, B).

The chains a 1, a 2, ••• , a{, ... , ab cover then also the edge a, which is impossible.

The statement is thus true.

Obviously, the set of edges {a6} U a forms a cut of which not a single one of the edges a, acrr, acr2, ... , acr3 can be rejected. Hence follows that the width of the net exceeds b, which

is impossible.


In the nonrepetitive circuit let us isolate a certain dead-end cut (chain). To all variables in the cut (chain) let us assign the value 0 (1), the value 1 (0) being assigned to all other variables. Such a sequence of values of the variables is denoted by w(s). In the following no distinction will be made between the cut (chain) and the sequence which realizes this cut (chain). The set of sequences w is denoted by Q, and the set of sequences s by S.

Achainissaidtobea unit chain with respect to a certain cut ifitcrosses it exactly once.

Theorem 3. The subsets of the set QUS form a full check test for the circuit ~, if through every contact of the circuit passes a cut and a unit chain with respect to this cut.

A proof of this theorem is given [4].

Now let us prove Theorem 2.

Takel + 1 parallel planes L1, L2, ••• , Ll+1 arranged in order of their numeration. On the plane Li let us place all vertices of the netto which the number i - 1 (i = 1, 2, ••• , l + 1) has been assigned. From Lemma 2 follows that two ends of one edge cannot lie in one and the same plane. Consequently, any net of length l can be geometrically realized so that any monotonic chain crosses the plane Li (i = 1, 2, ••. , l + 1) exactly once.

The set of all edges that cross the portion of space between the planes Li and Li +i is denoted by wi. Evidently, the sets of edges wi form dead-end cuts. Let us choose all dead-end cuts

110

Fig. 3

KH.A.MADATYAN

wi, w2, ••• , wz. According to Lemma 3, in a net of width b we can select b monotonic chains so that at least one of these nets passes through every edge of the net. These chains are unit chains with respect to all chosen cuts. According to Theorem 3, the selected dead-end cuts and the chains form a full checktest for the given circuit.

Consequently, a minimal full check test does not exceed the sum of the length and width of the given circuit.

This proves Theorem 2.

This theorem actually improves the results obtained by I. V. Kogan. lf the width and length of a net are of the order rn, the check test for this net is less than the cdD. and not n + 1, where c is some CQnstant.

In a nonrepetitive circuit let us take the dead-end cut w and a unit chain s with respect to this cut. Let us denote by Ws (sw) a sequence coincidingwith the sequence w (s) except that variable through which the chain s (cut w) intersects the cut w (chain s).

E xample. In the circuit shown in Fig. 3 let us take the cut w = (001101) and s = (101000); then Ws = (101101) and Sw = (001000).

The net of sequences w5 (sw) will be denoted by !Js (SJ.

Lemma 4. The set of sequences QUQ8 forms a full diagnostic test for nonrepetitive circuits.

Pr oof. Let 21' and 21" be distinct defective states of the circuit 21, that realize the functions f 1 and f" respectively.

Two cases are possible:

1. There is a variable xi suchthat one of the functions (either f 1 or f") depends (essentially) on the variable xi while the other is independent of xi.

2. The functions f 1 and f" depend on the same variables.

In the first case let xi be a variable suchthat f 1 depends essentially on xi and f" is independent of xi.

In virtue of the fact that f 1 depends essentially on xi, we can make in the circuit 21' a cut w 1 which pass es through the contact xi ( f 1 (w 1)) = 0. t Let us consider the cut w in cir-cuit W, corresponding to the cut w1 in the circuit W. Thus, j 1 (w) = 0 since both wand w1

differ only in those variables on which f 1 does not depend essentially. lf f" (w) = 1, the lemma is true since w E Q. lf, however, f" (w) = 0, let us consider the sequence ~. which coincides with the sequence w except in the variable xi; then, f' (~) = 1, and j" (~) = 0 and ~ E Qs.

In the second case, as f' is distinct from f" we can find a cut w' suchthat j 1 (w') ~ f" (w1), and, consequently, f' (w) ~ f" (w) is satisfied also on the cut w of the circuit W, corresponding to the cut w'.


The following lemmas are proved in exactly the same way.

Lemma 5. The set of sequences SUS!:! forms a full diagnostic test for nonrepetitive circuits.

t w' is a sequence that realizes the cut w1 •

TEST FOR NONREPETITIVE SWITCHING CIRCUITS 111

Fig. 4 Fig. 5

Lemma 6. The set of sequences QUS forms a full diagnostic test for nonrepetitive circuits.

N o t e . As will become evident below, tests constructed according to Lemmas 4, 5, and 6 are minimal tests for some circuits.

Let A(n) be the maximum number of chains in a two-terminal net with n edges.

Lemma 7. A(n)=3 3 •

Pro o f. For every two-terminal net (A, B) with n edges one can find a two-terminal TI~ 1, 1: 2, ... , 1:z) net [8] with n edges (Fig. 4) in which the number of chains is not less than the number of chains in the gi ven net.

In fact, let us denote the edges incidental to one of the terminals, say A, by the numbers 1, 2, 3, ... , k.

Let a i (i = 1, 2, ... , k) be the number of chains passing through the i-th edge.

Let also a = max ai and let a be reached at the j-th edge. Let us disconnect the i-th edge (i ~ j) from its corresponding internal vertex and attach it to the internal vertex c to which the edge j is incidental. If the net resulting from this operation has a branch, we identify the internal vertices of the edge s i and j. Let us carry out this operation for all i (i ~ j). Obviously, the number of chains in the new net is not less than the number of chains in the original net. Let us now take the two-terminal subnet (C, B) and apply to it the same operation. Let us continue this process till the result is a TI~ 1, 1: 2, ... , 1: z) net. Since A(n) is attained on TI~ 1,

1: 2, ... , 1:z) nets, it suffices if we consider only these nets.

Let ui be the number of edges in each subnet 1:i of the TI~ 1, 1: 2, ... , 1: z) net. Then, the number of chains in this net, A (TI), is equal to u 1 • u2 • u3 ... u 1, where u1 + u2 + ... + u z = n. It is clear that max A(TI) = 311/ 3,


Lemma 8.

The proof of this lemma follows immediately from Lemmas 5 and 7 as to every chain s correspond not more than n - 1 sequences of the type Sn .

Let B(r) be the number of dead-end cuts in a two-terminal net with r vertices.

Lemma 9. B(r)<.2r-2.

Pro o f. Every dead-end cut partitions the internal vertices of a net into two parts.

Thus, B(r)<;:C~-2+C;_2+ ...

Theorem 4. For any e > 0 and sufficiently large n, for almost a 11 non r e p e t i t i v e c i r c u i t s t hat r e a li z e th e f u n ct i o n f ( x 1, x 2 , ••• , X 11 )

112 KH.A.MADATYAN

TABLE 1

Yl Y2 Y3 Y4 X I I pi I I pi I I pi I I pi Fo 2 Fo a Fo 1 Fo • 2 i=1, 2, ... , s 3 i=1, 2, ... , s 1 i=1, 2, .,., s 4 i=1, 2, ... , s

{ ~ ~ ~ ~ :i 1 1 I 0 0 At Ii

1 I t; 0 0

1 0 0 1 a; 1 0 0

{ ~ ~ ~ ~ :i 1 1 0 1 A2 Ii 1 0 1

1 1 0 1 a; 1 1 0 1

{ ~ ~ ~ ~ :i I 1 1 1 0 Aa

11 1 /; 1 ' 0

1 0 1 1 a; 1 1 0

{ ~ ~ ~ ~ :i 0 I~ 1 1 A4 0 0 Ii I;

0 1 1 0 a; 0 10 1 1

{ ~ ~ ~ ~ :i 0 1 1 1 A5 0 1 Ii 1

0 1 1 1 a; 0 1 1 1

A6 { ~ ~ ~ ~ :; 1 ol 1 1 1

öl 0 1 I;

1 1 1 0 a; 1 1 1

Pr oof. Let rn be the number of vertices in a two-terminal net with n edges. In [6] it is shown that for almost all nets r ~ (2n/ln n) (1 + e ), where e - 0 when n- oo. According to

Lemma 9, the number of dead-end cuts in such nets 9 B (rn) <:;: 2'n- 2 and according to Lemma 4, · ~ (i+e)

.T d (!) <:;: 2"n- 2 (n + 1) = 2Inn , where e - 0 when n- oo.

The theorem is thus proved.

Let the nonrepetitive circuit 21 realize the function f (x1, x2, ••• , Xn) and let Td (21) = t. Let us construct a new nonrepetitive circuit ß (Fig. 5) that realizes the function F(y1, y2, y3, y4, Xt, x2, ••• , Xn) •

Lemma 10. T~(~)>2t+2.

Pr oof. Let us compile a fractional table of failure functions for the circuit ß and show that for these failures the length of the test is not less than 2t + 2.

Let 2Io, 2It. ... , 2Is be all the possible defective states of the circuit 2I (it is assumed that the circuit contacts can only open as a result of action of the failure source) that realize the functions f 0, f 1, ••• , / 8 (j 0 = f) respectively.

Consider the fol~owing defective states ß~ (j = 1, 2, 3, 4; i = 1, 2, ••• , s) of the circuit ß. The defective states ßj occur when the contact yi is open and the circuit 2I is in the defective state 2!;.

The failure

~~ has the conduction t F 1 = YzYafi V YzY4,

~~ has the conduction F~ = YtYdi V YtYa,

TEST FüR NONREPETITIVE SWITCHING CIRCUITS

111 has the conduction F1 = Y1Yd; V YzY4,

11~ has the conduction F~ = y2y3f; V Y1Y3,

where i == 0, 1, 2, ... , s.

Let us compile a table of failure functions for these defective states (Table 1). In the table weshall consider only those sequences in which the defective states ~ i (j == 1, 2, 3, 4; i == 0, 1, 2, ••. , s) are distinct. 1

Sequences of this kind are:

Yt Yz Y3 Y• X

Al 1 0 0 1 a;

Az 1 1 0 1 a;

A3 1 0 1 1 a;

A. 0 1 1 0 a;

A5 0 1 1 1 a;

A6 1 1 1 0 a;,

113

where ä i (i == 1, 2, .•• , l) are all the possible sequences in which f (a i) == 1. The minimal set of sequences that distinguishes between all defective states ~~ (i == 0, 1, 2, ••. , s) and between

the states ~~ is denoted by T 2, 3, and those concerning ~~ and ~i by T 1, 4• As seen in Table 1,

so that

Tz,3cA1UAzUA3 T1, 4 c A 4 U A 5 U As

([Tz,3[>t),

([Tt,,[>t),

T2,3 n T1, 4 = 0. (1)

If in the set of sequences T 2 3 there exists at least one sequence a of the set A2 or A3, I T 2 31 2: ' ~ ~ '

t + 1. In fact, let us assume that I T 2,3 1 < t + 1 and a E Tz. 3 n (Az U A3) (for example, a E Az);

then there are two defective states ~~ and ~ ~ which differ only in the sequence Oi of the set T2,3• In this case the defective states ~~ and ~~ do not differ in the sequences of the set T2 ,3

which is impossible.

Similarly, if in the set of sequences T 1, 4 there is at least one sequence of the set A5 or A6,

then I T 1,4 I 2: t + 1.

The following cases are possible:

I. T2,3 c Ah T1, 4 cA4t then T~(~)>2t,

I I. T2 , 3cA1, T,," n (A5 U A6) =I= 0, then T d (~) > 2t + 1.

III. T2,3 n (AzUAa)=l=-0, T~, 4 c A4, then T~(~) > 2t+- 1,

IV. Tz. a n (A2 U Aa) =I= 0, T1,, n (A5 U Ae) =I= 0, then T~(~) > 2t + 2.

It is sufficient to consider the cases I and II, since II and III are not essentially different, and the lemma is true for case IV.

I. The defective states ~B and ~~ differ only in sequences belanging to the set A5 U Aa, a ~~

while ~~ and ~g differ in sequences of A2 U A3, that do not belong to T2, 3 U T~. •· Thus, in order to distinguish between the defective states ~r (j == 1, 2, 3, 4) we need at least two more sequences. Thus, T~(~) 2: 2t + 2.

114

Fig. 6

KH.A.MADATYAN

li. The defective states ~~ and ~~ differ only in sequences of the set A2 U Aa, that are not included in the test T 2, a U T 1, 4· Thus, T d (~) 2:

2t + 2. The lemma is proved completely.

In the bridge circuit M1 (Fig. 6) let us replace the contact x by the subcircuit Mt; we obtain the circuit M2, etc. The circuit Mk results from replacing the contact x in the circuit Mt by the circuit M k -t·

n+3

Lemma 11. Td(Mk)=3·2-4--4, where n = 4k + 1.

Pr oof. According to Lemma 10, Tci(Mk) 2: 2Td(Mk-t) + 2, whence Td(Mk) 2: 3·2k- 2.

In virtue of duality: T~ (Mk) 2: 3 o 2k -2. Whence

T d (Mk) >Td_(Mk) + Td (Mk) > 3·2k+1-4. (2)

In the circuit Mkl Q I =I SI = 3 · 2k- 2, so that from Lemma 6 follows that T k+! 4 d (Mk)-<3·2 - .

n+3

From (2) and (3) we have T d(Mk) = 3·2''+1-4=3·2-4--4.

(3)

Rem a r k. The minimal test of circuit M k coincides with the test designed in accordance with Lemma 6 (see rtote on p. 111). From Lemmas 7 and 11 follows immediately

n+3 n

Theorem 5. 3·2 -.--4-<T d (n) <;:n3 3 .

§ 3. Full Test for Nonrepetitive II Circuits

Let T~(n) be the Shannon function of a full diagnostic test for nonrepetitive II circuits.

TI 3 Theorem 6. Td (n)= 2 n.

To prove this theorem we must first consider several lemmas.

Let Wt and W2 be two nonrepetitive circuits which realize the functions f 1 (x1, x2, ... , xn) and f 2(y1, y2, ... , Ym), and let the sets of sequences tt and ~ be their full diagnostic tests respectively. Joining these circuits in series (parallel) we obtain the circuit W, which realizes the function

f=ft(x1, · · ., Xn)·f2(Yi> · · ., Ym) lf=ft(xi> · · ., Xn) V f2(Y~o · · ., Ym)].

L e m m a 1 2 o Tf (!)-< I t 1 I + I t2l·

Pro o f • Consider the case when W 1 and W2 are connected in series. To every sequence from the set tt we add m ones on the right and to every sequence from t:! we add n ones on the left. The result is a set of sequences t = t~ U t~ of length n + m. Clearly, I t I ::5 I t11 + I t:! 1. We shall now prove that the set of sequences t is a full diagnostic test for the circuit ~1 • To do this we take two different failures of the circuit

and show that they are distinct in the sequences t. Since f' ~ f", either fi ~ fY or 1; ~ f~. Assurne that f~ differs from f~ in the sequence a from t1, i.e., f~(a) ;r f~(a); then j'(a') ;r !" (a') (where a' is a sequence obtained from a by the above method) since n and tr cannot be identically equal to zero.

TEST FOR NONREPETITIVE SWITCHING CIRCUITS 115

xl,~~2~' ·~· :_o_"j~~~~~

p 1

A I I

-----1 ---- (P12 I

!'" I [J

I I

c

Fig. 7

Consider now the case when 2!1 and 2Iz are connected in parallel. To every sequence belonging to the set t1 we add m zeros on the right and to every sequence from t2, n zeros on the left.

The proof that the set of sequences t = t~ U t; is a full diagnostic test for 2I is similar to that above.

This proves the Iemma.

Corollary. The connection of a single contact in series (parallel) to a nonrepetitive circuit increases the length of its full diagnostic test by not more than unity.

Lemma 13. T~ (n)<-}n, n> 1.

Pr o o f . The Iemma is proved by applying the method of mathematical induction to the number of contacts. The statement is clearly true for n = 2.

Let us assume that the statement is true for all k :s n and prove that it is also true for n + 1.

A TI circuit with n + 1 edges is a parallel or series connection of two n circuits with l and s edges: l + s = n + 1. According to Lemma 12,

TIT ( I 1) 3 l 3 ~ 3 ( 1) d n 1 <z- +z-s-- 2 n-t- .

This proves the Iemma.

Let TJ•rr (f) (T d'rr (f)) be a full short- (open-) circuit test of a nonrepetitive TI circuit 1: that realizes the function f. Let f = fdx11 x3, ... , :rn) V !2 (y1, Y2, ... , Ym),

" If !1 = ;~ 1 x;, then TJ•rr (f) = I t2 1, since there exists only one failure function for f i which is

identically equal to 1.

n m Lemma 14. If f=!Nfz, where k=I=V :r;, / 2 =1= V y;, then

i~! i~!

Td 11 (!)>I t1i +I lz\.

Pr o o f. Assurne that the set of sequences t is a test t for the function f. This set should contain tests for both f 1 and f 2·

Let us take the set of sequences 1) = P 1 U P 2 from t, where the set P 1 forms a dead-end testminimal with respect tot for the function f 1 and P 2 is a similar test for the function f 2•

Assurne that the sets of sequences Pi and P2 intersect on the sequences forming the set P 12 •

Let us first arrange the set of sequences Pi so that all sequences belonging to Pi2 are at the end, next the remairring sequences of P 2, and at last the set of sequences C, where C is formed of all sequences that to not belong to the set P (Fig. 7). We will prove that the number of rows

t A short-circuit test is meant here.

116 KH.A.MADATYAN

in the matrix C is not less than I P 12 1 • Without loss of generality we can presuppose that the number of I A' I of nonzero rows of the matrix A is not less than the number of nonzero rows of the matrix B. Let us show that the matrices B and C should contain I P12 1 + I A' I nonzero rows. Let us take the first sequence a ofthe set Pi to which a nonzero sequence from the matrix A has been added. Since the set Pi forms a dead-end test for the function f i• there are two failure functions ~~ and ff that düfer in the sequence a E PI only. Let in this sequence the variable Yi = 1. Let us take in the circuit W2 , a chain that passes through the contact Yi and close all contacts (exceptyd along this chain. Theresultisafailurefunctionj~ofthecircuit ~I 2• t; 'jE 1. The failure functions f' = t; V ~~ and r = ~~ V ~~ do not differ in the sequences P 1, so that the matrix B or C should have a sequence ß (ß ;e 0) such that f 1 (ß) ;e f" (ß). The same procedure is applied also to any two failure functions of the circuit I: that düfer in the sequence a only.

An analogous reasoning is carried out for any arbitrary sequences P 12 U A' of the type a; as a result we obtain sequences of the type ß in the matrices B and C. Let us show that s 2::

I P i21 + I A' I· Assurne that s < I P i21 + I A' I ; from the set of sequences Pi let us eliminate the sequences P 12 U A;, considered above and replace them with sequences of the type ß. The result is a test for the circuit ~ 11 whose length is less than I P il ; this is impossible as the set of sequences P 1 is aminimal test with respect tot for the circuit ~ 1 • However, since the number of nonzero rows in the matrix B is less than I A' I, the number of rows in matrix C is not less than I P 12 1.


Obviously, if f = !N !2, then T dJI( f i) 2:: Tc!•rr(j 1 ) + Td•II(j 2 ) and Tr(j) 2:: T~·II( f i) + T J' II ( f 1) + T~· II ( f2) + T a·II ( f2). In virtue of duality from f = f i • f2 follows that T ~( f) 2::

Td•II ( f i) + TJ·II ( f i) + Td,II (f2) + Td,II (f 2) •

Thus, a full test constructed in accordance with Lemma 12 is minimal provided ti and t2 are minimal tests for f 1 and f 2 respectively. Hence and from Lemma 13 follows the proof of Theorem 6.

Rem a r k. From Lemmas 12 and 14 follows that there exists a sufficiently simple method for constructing minimal full diagnostic tests for nonrepetitive II circuits. This method does not presuppose the construction of a failure-function table. The question which now arises is how to apply this test to the localization of the failure, i.e., how to find which particular contacts are defective to within electrically distinguishable components.

Each sequence of a minimal test checks the failure of contact xi if the value of the correct function differs from the value of the incorrect function generated by the failure of the con-tact xi. To every sequence corresponds a set of single failures which it checks. In front of each sequence let us write out the contacts checked out by the given sequence.

Ex ampl e. Let us design a minimal test for 1he circuit shown in Fig. Sa. For this purpose let us first consider its individual subcircuits.

The subcircuits shown in Fig. 8, b, c, and d have the following sequences as full tests:

1 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0 1 1

TEST FüR NONREPETITIVE SWITCHING CIRCUITS 117

x, ~~x,~~x, Xs x, x2

Xa x6

Xa

a b c d d

Fig. 8

Fig. 9

Subcircuits b) and c) are connected in series. Consequently, in accordance with Lemma 12 the following set of sequences is a full test for the subcircuit e) in Fig. 8:

Xt Xz X3 X4

1 0 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 1 1 0 0

Now, taking into account that the subcircuits d) and e) are connected in parallel, the minimal full test of the complete circuit is

x 1 x2 X3 x, X5 Xa

(xt) 0 1 0 1 1 0 0 (xz) o 0 1 1 1 0 0

(xt)5 (xz)5 0 0 1 1 0 0

(x3) 0 1 1 1 0 0 0 (x4)o 1 1 0 1 0 0 (x3)S (x,)s 1 1 0 0 0 0 (xa)S 0 0 0 0 1 0 (x5)s 0 0 0 0 0 1

(x5) 0 (x6) 0 0 0 0 0 1 1

The failure checked out by the given sequence is shown at the left. The opening and closing of contact is denoted by (x)0 and (x)S respectively.

Rem a r k. A minimal test for l1 a;) circuits designed in accordance with Lemma 12 coincides with the test constructed according to Lemma 4 (5) (see note on p. 111).

Let the nonrepetitive l1 circuit that realizes the function f (x1, x2, ••• , Xn) have a length l and width b. Let T1,rr (j) and T~(j) be respectively a single and a full checktest for the nonrepetitive circuit which realizes the function f.

Theorem 7. T~'rr(j) = T~(f) = b + l.

118 KH.A.MADATYAN

TABLE 2

10 11 12

I I I I I 16 1 8 I I I

n=7 I I I I I I I I I I I I 11281 3181 68 1 10

Proof. The upper bound follows from Theorem 2. Considering that T~,rr(j) s T~(j), it is sufficient to prove that

T~· rr (!) > b+ l. (4)

Since the width of the circuit is b, there exists a dead-end cut with b contacts. An open-circuit in each of the contacts entering into this dead-end cut can be detected only ü the unit chain passes through one of contacts in the cut. Thus, in order to detect the opening of all contacts we must have at least b sequences. Hence

r~· o. rr (f) > b.

In virtue of duality:

T~ s. rr (f) > l, From (5) and (6) follows (4). The theorem is thus proved.

Corollary. b+l<T~·rr (f)<n+i.

From Lemmas 13 and 14 follows that:

n+i<T~(f)<[~ nJ.

The length of the full diagnostictestforthe circuit in Fig. 9 is exactly 3n/2.

(5)

(6)

The subset of II nets whose minimal diagnostic test has a length n + k is denoted by sn,k .

It can be easily observed that for any k (k = 1, 2, ••• , [n/ 2]) the subset s n,k is nonempty.

Let Iln,k be the number of sn,k nets.

The subset sn.t results from a parallel or series connection "of one edge to nets with n- 1 edges which, in turn, are obtained by parallel or series connection of one edge to nets with n- 2 edges, etc., ~,1 = 2n-t. In fact, Iln,t = 2IIn-t,1 = 22IIn-2, 1 = .•. = 2n-1rr 1, 1 = 2n-1• From Lemmas 12 and 14 follows that for such nets

(7)

TEST FOR NONREPETITIVE SWITCHING CffiCUITS

and from Theorem 7 follows that

T~· rr (f) =n+ 1,

since the sum of the length and width of such circuits is n + 1. From (7) and (8) follows:

Theorem 8. If fEsn,~> then T1'rr(/) = T~(j) = T~,rr(j) = T~(/) n + 1.

119

(8)

Table 2 indicates the length of the full diagnostic test and the munber of circuits having a test of this length for small n. Let H(j) be the number of distinct failure functions of a nonrepetitive II circuit ~. that realizes the function f (x1, x2, ••• , x11 ).

Theorem 9. H(j) = 2 11 •

The proof is obtained by mathematical induction applied to the number of contacts.

The statement is obviously true for n = 1.

Assurne that the statement is true for all k < n and prove that it is also true for n.

The II circuit ~ with n contacts is a parallel or series connection of two II circuits (~~ and ~z) with k1 and k2 contacts, where k1 + k2 = n. Let f = ft · j 2; then to each defective state of the circuit 'tll~> whose conduction is not identically equal to zero correspond 2~<2 defective states of the circuit ~ 2 with a conduction not identically equal to zero (including also the circuit ~~)· To the correct state of ~~ correspond 2~<2 failure functions of the circuit ~h

Consequently,

The theorem is thus proved.

In conclusion the author expresses his gratitude to s. V. Yablonskii under whose guidance this work was conducted.

Literature Cited

1. V. V. Vaksov, "On tests for nonrepetitive switching circuits," Avtomat. i Telemekh., 26(3):521-524 (1965).

2. V. V. Glagolev, "Design of tests for block circuits," Dokl. Akad. Nauk SSSR, 144:6 (1962). 3. V. V. Glagolev, 11Some bounds for disjunctive normal forms of functions of the algebra

of logic," in: Systems Theory Research, Vol. 19, Consultants Bureau, New York (1970), p. 74.

4. I. V. Kogan, "On tests for nonrepetitive switching circuits," Problemy Kibernetiki, Vol. 12, Fizmatgiz, Moscow (1964), pp. 39-44.

5. A. V. Kuznetsov, "On nonrepetitive switching circuits and nonrepetitive Superposition of logical-algebra functions," Trudy MIAN SSSR, Vol. 60, Moscow (1958), pp. 186-225.

6. 0. B. Lupanov, "On asymptotic estimates of graphs and nets with edges," in: Problemy Kibernetiki, No. 4, Fizmatgiz, Moscow (1960), pp. 5-21.

7. Kh. A. Madatyan, 11 Synthesis of switching circuit of limited width, 11 Problemy Kibernetiki, Vol. 14, N auka, Moscow (1965), pp. 301-307.

8. B. A. Trakhtenbrot, 11 0n the theory of nonrepetitive switching circuit," Trudy MIAN SSSR, Vol. 60, Moscow (1958), pp. 226-269.

9. I. A. Chegis and S. V. Yablonskii, "Logical methods for testing electrical circuits," Trudy MIAN SSSR, Vol. 60, Moscow (1958), pp. 270-362.

10. S. V. Yablonskii, "Functional designs in k-valued logic," Trudy MIAN SSSR, Vol. 60, Moscow (1958), pp. 5-142.

120 KH. A. MADATYAN

11. S. V. Yablonskii, "Basic concepts of cybernetics," Problemy Kibernetiki, Vol. 2, Fizmatgiz, Moscow (1959), pp. 7-38.

12. J. Riordan and C. E. Shannon, "The number oftwo-terminal series-parallel network.s," J. Math. and Phys., 21(2):83-83 (1942).

ON FINITE MODEL SCHEMES HA VING DISCRETE FUNCTIONING t

Yu. A. Vinogradov

Moscow

Schemes with discrete functioning are ordinarily presented in the form of graphs whose directed edges are lines for transmitting information while the nodes are functional elements. Functions of memoryless schemes can be obtained as Superpositions of functions of operating (functioning) elements.

Combined with the fact that such schemes are often very complex is the fact that calculation of their functioning is negotiated by a comparatively simple mathematical model of the functional elements. Serving as such models are the usual models of the algebra of logic. Obviously, faithful representation of the functioning of the scheme as a whole can be obtained only with weil fitting functions of the elements and their models. The necessary good fit is ordinarily obtained by constructing special elements whose characteristics admit a good binary interpretation. However, there are cases of certain unnatural matchings of characteristics of elements to binary models in which such matchings are found not to conduce to the objective. In this case the binary model must be provided with certain restrictions within which the model has meaning. The elaboration of these, of course, is not characteristically a binary affair.

There is in principle another way of approximating functions of elements and their models, namely, the construction of more refined discrete models for the scheme elements. Suchmodels may be functions of k-valued logic.

The possibility in principle of using many-valuedlogic in synthesis of discretely functioning electrical schemes was proved by S. V. Yablonskii [4], andin 1956-57 a group of diploma students at Gorky University built such a model for various electronic computer schemes [1, 2, 3]. The present work is a continuation of this and deals with certain general questions of construction of finite models of schemes realized by functional elements.

§ 1. Finite-Valued Models

It is clear that a specific functional element f with n inputs and one output can be posed in the form of a function f (x1, x2, ••• , x11 ) = x n+i expressing the dependence of the signal at the element's output on its inputs. Such functions of elements we call natural and shall, in the sequel, study not only elements but also their natural functions. We shall construct finite models

t Original article submitted September 10, 1968.

121

122

TABLE 1

X II F 1 (X) I F2 (X) I Fa (X)

0 4 4 1 1 4 0 2 2 3 1 3 3 2 2 I 4 4 1 3 I 0

I

f(J:}

g5

gq.

g3

g2

g,

g, !h g3 gq. g5

Fig. 1

.z

YU. A. VINOGRADOV

for schemes whose elements are realizations of natural functions defined on the segment A = [a, b] for each variable (in space An), taking values, also, from A.

Consider a partition H of the set A into points of division a = g 0, g1, ... , gk_1, gk = b, k > 1, and intervals (g0, g1), (g1, g2), ... , (gk_1, gk). Supposethedivisionpointsarearranged in increasing order. Denote the set of points {gi} by I; the set of intervals (gi, gi+1) by G. Weshall use the terminology, H-partition, I-points, and G-intervals.

Definition. The function f (x1, x2, ... , Xn) = Xn+1 has k-valued model (k-valued description) if there exists an H-partition suchthat

Xn+i = f (xl, Xz, ... , Xn) E (gz, gz+!) E G, if Xi E (gs;, g•;+i) E G. (1)

Substituting for the variable xi respectively the discrete variable Xi, taking the value l if x; E (gz, gz+l), we transform a function f (x1, x2, ... , xn) defined on A and taking values from A into a finitely many-valuedfunction F(X1, X2, ... , Xn) defined on the set { 0, 1, ... , k- 1} and taking values from this same set. We call the function F an I-model of the function f. In this work, we confine our attention to models of continuous single-valued functions of one variable.

§ 2. k-Valued Models of Functions of One Variable

The continuous function f (x) has a finite-valuedmodel if there exists an H-partition such that, for any (g;, g;+1) E G there exists a (gz, gz+ 1) E G and

(1')

Condition (1') is fulfilled if none of the sets f[(gi, gi+i)], i = 0, 1, ... , k -1 include values from I.

Ex a m p 1 e 1. Consider some functions (not confined to continuous and single valued, with the aim of demonstrating the modelling possibility) prescribed by the graphs in Fig. 1, along with their models under the given H-partition. Nurober the G-intervals. Condition (1') is fulfilled for functions h• f2, and j 3• These functions will correspond respectively to the model functions from P5 (Table 1). As for the function f 4, we cannot, for the given H-partition, place it in correspondence with a k-valued model function, since Condition (11) is not fulfilled for / 4• Actually / 4[(g1, g2)] includes the I-points g2 and g3•

We note that the H-partition considered in the example is suitable for some functions, and the grating formed by the prescribed I-points is common for these functions. The existence of a common grating for the system of elements forming a single scheme is necessary for the construction of a finite model ofthat scheme.

It is evident from the example just introduced that taking the definition of a finite model permits construction of models of functions of various forms. However, by no means can allfunctions have finite models. Below we consider necessary and sufficient conditions for the existence offinite models as well as methods of obtaining such models.

Lemma 1. If f (x) has an I-model and gEI, then /-1 (g)El.

FINITE MODEL SCHEMES HAVING DISCRETE FUNCTIONING 123

c d J:

Fig. 2 Fig. 3

Pro o f. In the contrary case, we can find x 'fJ, such that f (x) == g, which contradicts Condition (11).

Co r o 11 a r y . If g Ef, t h e n f - 1 ( g) i s finite .

Introduce the notation

00

t;t (g) = f-1 (f-1 ( ... u-1 (g)) ... )) '-.-'

i times

and call the sum U /i1 (g)=f*(g) the track of g. We remark that the tracks of two different i=l

points either do not intersect or else one contains the other. The set consisting of a, g, and b we denote by E g.

Theorem 1. The continuous function f (x) has a k-valued model if there exists gE(a, b) such that f* (Eg) is finite.

Pro of. Ne c es s i ty. Let f (x) have an I-model for a finite set I. Then there exists gEf n (a, b), and tracks f * (a), f * (g), and f * (b) arefinite (Corollary of Lemma 1).

Sufficiency. Letj* (Eg) be finite. Then, in [a, b] there isafinite nonempty set of points Eg Uf* (Eg) c.c j_ We show that i can be a set in which an I-model of the given function is constructed. The condition of existence of a k-valued model for f (x) falls to be fulfilled only if there exists a value g s E i whose pre-image does not belang to the set i; this is impossible, since i contains all its pre-images.

However, this theorem does not give an effective method of finding a set of I-points. More effective methods will be considered below.

We prove two preliminary lemmas.

Denote by r 1 and r 2 the graphs of the functions y == f 1 (x) and x == f 2 (y) defined on the segments [c, d] and [p, e], respectively.

Lemma 2. If

/!([c, d]) 2 [p, e],

fz ([p, e]) 2 [c, d],

andin the closed reetangle [c, d] x [p, e] there are precisely two (distinct) points of intersection of the graphs r 1 and r 2 : M(c, e) and N(d, p), then for any point g1E(p. e) there can be found a point gzE(p, e),

gzEf;1 (f~1 (g 1 )), such that

124 YU. A. VINOGRADOV

Pr oof. It follows from continuity of f 1 (x) that for any g1 E (p, e) there exists mE (c, d), suchthat m E /~1 (g1) • It is easy to see that g1 ~ g2 since otherwise the point (m, g1) would be a point of intersection ofthe graphs r 1 and r 2• (Shown in Fig. 2 is one of four varieties of possible mutual placement of the graphs r 1 and r 2 and the points M and N.)

Corollary. Since g1 from (p, e) generates g2 on (p, g1) (or on (g1, e)) it follows that unlimited repetition of this procedure would lead to the formation of an infinite set of points g 1, g2, ... , g5 , ... , located on the interval (p, e).

Lemma 3. If functions j 1 (x) and f2(y) are defined respectively on Land Q, where L and Q (or Q and L) are either

1) semiintervals or

2) semiinterval and interval or

3) segmen ts, and if

then the graphs r 1 and r 2 have a common point.

Pr oof. We prove only Case 1). Let L = [c, d), Q = (p, e], j 1(c) = h, and let x 0 be a point close to c at which j 1 (x0) = e. We denote by rt the part of the graph r 1 included between the points (c, h) and (x0, e), while by r1 we denote the remaining part of the graph r 1 along with the point (x0, e) adjoined to i t (Fig. 3).

Consider the values of the function f2(e):

1) f2 (e) = x 0,

2) ! 2 (e) < x 0, and

3) f2(e) > x 0•

In Case 1), r 1 and r 2 have a common point (x0, e). In Case 2) the point (f2(e), e) is separated from all points of the reetangle [x0, d] x (p, e) by the arc rt, and r 2 must intersect the arc rt. In Case 3) the same point is separated by the arc r 1 from all points of the reetangle (c, Xo] X (p, e], and r 2 must intersect the arc r t•

Rem a r k 1 . It is easy to see that Lemma 3 is also true in case

/ 1 ([c, d)) = (p', e] = (p, e], / 2 ((p, e]) = [c, d') 2 [c, d).

Remark 2. If Land Q (or Q and L) are intervals, then the lemma is false; ifthey are segment and interval or segment and semiinterval, then its hypothesis is inconsistent.

Denote by rf the graph of the function f (x). Plotting for each point B (x;, y;) E r 1 the corresponding point B' (yp xi) we get r f-1, the graphof the inverse function j-1(x). Let \R = r 1 n ff-1. We note that the set of coordinates of points from ffi coincides with the set of solutions of the equation f (x) = j-1(x) since to each point B (x;, y;) E \R there corresponds the point B' (y;, x;) E \R, and the abscissas of points in \R are solutions of the equation f (x) = j-1(x). Denote the set of solutions of the equation f (x) = j-1(x) by s. Since f (x) and j-1(x) are continuous functions, it follows that ffi and S are closed linearly ordered sets (finite, countable, or continual).

2.1. Models of Functions for Which f (A) = A

Lemma 4. If j(A) = A and gE_S, then f* (g) is infinite.

FINITE MODEL SCHEMES HAVING DISCRETE FUNCTIONING 125

y

fkdlt'(g,;J p r---+------+--"·

Fig. 4

Pro o f. Suppose S = A ""- S. Then S is either empty or consists of intervals and (or) semiintervals (abutting on the ends of A).

The points B(c, e) and C (d, p) E ffi are said tobe n e i g h b o r s in lH, if the interval (B, C) (on the contour r f- 1) contains no points of ffi (Fig. 4). Denote a' and b 1 the least and the greatest values from S. We note that ffi c:: !a'. b'] x [a', b'].

I. We show that j* (g1) is infinite if g1 belongs to an interval inS. Suppose points B(c, e) and C(d, p) are neighbors in ffi . Then (p, e) ES, and the hypothesis of Lemma 2 is fulfilled for the points B and C and the closed reetangle [c, d] x [p, e]. Here ] 1 = h = f and for any gl E (p, e), f* (gl) is infinite.

II. We show that j* (g) is infinite if g belongs to a semiinterval inS. We consider the following cases.

1. If XE (b', b], then f (x) ;r b. Otherwise, in the square (b', b] X (b', b] the graph r f would have a point in common with the diagonal y = x. Such a point would belong to \R, but by hypothesis, all points from \R belong to [a', b'] x [a', b'].

Analogously j(x) ;r a if xE[a, a').

2. If x 1 E[a, a') and x2 E(b', b], then it is impossible that j(x1) =band ]1)1..2) = a can both be fulfilled. Otherwise

I ((b', bl) = (a, a") ;::2 [a, a'),

f ([a, a')) = (b", b'] :=:2 (b', b]

and the reetangle [a, a') x (b', b] contains a point of \R (Lemma 3), but by hypothesis all points of ffi belong to [a', b'] x [a 1, b'].

Thus we have either 3) j ([a', b']) = A or 4) f 1)1..1) = b, x1 E [a, a') and f (x2) = a, Xz E (a', b'],

or 5) j(x1) =b, x1 E[a', b'J and ]1)1..2) = a, x2 E(b', b]. We consider these cases.

3) f ([a', b']) = A. Suppose g E [a, a') U (b', b]. Then there is a point P (m, g), m Ef-1 (g) n [a', b'], suchthat PE \R, since Pis not contained in [a', b'] x [a', b']. But then m belongs to an interval from Sand the track f* (g) is infinite since j* (m) is infinite.

4) f(x 1)=lJ., x1 E[a, a') and f(x2)=a, x2 E[a', b']. Suppose gE[a, a'). Butthen, as shown in 3, the track f* (g) is infinite. Suppose g E ( b', b ]. But since f ([a, a')) = (b', b), there exists m E f-1 (g) n [a, a'), and the track j * (g) is infinite by reason of the fact that j* (m) is infinite.

5) f (x1) = b, x 1 E [a', b'] and I (x2) = a, x2 E (b', b]. The proof of infinitude of the track j* (g)

for gE [a, a') U (b', b] is analogaus to the proof in Point 4.

Accordingly, the track j* (g) is infinite if g (S, as was tobe shown.

Let j (x) be an arbitrary continuous function. Construct for it a set 3 consisting of those and only of those g for which the following conditionst are fulfilled: 1) g belongs toS,

2) j-1 (g) = 1 and 3) t 1 (f 1 (g)) = 1.

v-1(g) is the cardinality of the set j-1(g).

126 YU o Ao VINOGRADOV

TABLE 2

"§ 113g, HS· (a,b):f-l(g)=g I Vg,HS· (a,b):j-l(g)ofr-g

I Arbitrary finite, k-valued- Arbitrary odd k-valuedness

<No ness

I 2<k<3-1 3--(:k<;:"§-1

No II Arbitrary finite and denu-1 Arbitrary odd and denu-

merable valuedness merable valuedness

Arbitrary finite, denumer- Arbitrary odd, denumer-c able, and continual able, and continual

valuedness valuedness

Theorem 2o If the continuous function f (x) defined on A takes all va1ues in A, then the track f* (g), gEA, is finite if and on1y if gE'J.

Pro o f. Ne c es s i ty of the condition g ES follows from Lemma 4o We prove necessity

of the conditions f-1 (g) = 1 and f-1 (f 1 (g)) = 1.

a) r (g) > 0, since f (A) == Ao

b) We show that t 1 (g) < 2. Let f-1(g) == {gm, gn, .. o} o Then there exist points M(g m' g) and N (gn, g) E ft· But the graph r f _1 has a unique point with ordinate g, and either M or NE \R. Then either gm or gn ES and either f * (g m> or f* (gn) is infinite o Consequently the track f* (g) too is infinite since f* (g) 2 f* (gm) U f* (gn)·

Necessity of f 1 W1 (g)) = 1 is proved analogously, putting f- 1(g) = g' o

Sufficiencyo Let gES, f-1 (g)=grz, and t 1 (ga)=gß. Then there existpoints B(grz, g),

C (gß, ga) and B' (g, grz), C' (grz, gß) E ffi. But since r (ga) = 1, it follows that gß = g (C coincides with B' and C' coincides with B)o Consequent1y, f* (g) == {gc:o g}, as wastobe proved.

Co r o 11 a r y 1 o A continuous function f (x) defined in A == [a, b] and taking all values of A has a finite mode1 if and on1y if there exists a g E (a, b) such that {a, g, b} E ;). In this case f(x) will have a model with I== { a, g, J-1(g), b}o

Co r o 11 a r y 2 o If f (x) has a finite model, then at least one of its modelled functions is contained either in P2 or in P 3o

Note o A k-va1ued mode1 of the function f (x) defined in A and taking all values in A can only be a strictly monotone k-valued function, i.eo, either F1(X) =X or F2(X) == k- 1- X, where Fand X take values from { 0, 1, .. 0, k- 1} o

Thus for the function f (x) taking all values from A, the set of I-points is formed only of values 1ying in the set J. It is obvious that the number of finite-valued models of a given function depends on the cardinality of 'J and is equal to the number of admissib1e subsets of 'J (an admissib1e subset is a subset including in itself a and b and a1ong with each g the preimage j-1(g))o The set 'J can be finite, infinite, or continualo

The va1uedness of the mode1 is tied to the cardinality of 3 . The possible cases ar.e exhibited in Tab1e 2 o

Ex a m p 1 e 2 o We construct a finite model for the function f (x) given by the graph r f of Figo 5o

FINITE MODEL SCHEMES HAVING DISCRETE FUNCTIONING

Fi.g. 5

y

!lz f-----+----+-+--l cr----+~~--4-r-~

IJ cg2 b JJ

Fig. 6

127

The mirror image of the graph r f relative to the diagonal y = x gives the graph of the inverse function r f-i and the intersection r 1 n ri-1 in the set m = { r 1, r 2, ••• , r 11}. The projection of the set lH on the x axis forms the set S and the values a, g1, g2, and b from S form the set Putting I=~ and numbering the G-intervals we get the model of largest valuedness F(X) = 2- X, XE { 0, 1, 2,}. Since for any g from :1· n (a, b), j-1(g) ~ g, and ~ = 4, it follows that this model will be the only possible model of the function f (x).

2.2. Models of Functions for Which f (A) C A. We consider conditions under which continuous functions not taki.ng all values from A have finite models.

For such functions the sequence A, f (A), f (j (A)), •.. forms a system of segments, nested in one another. Their intersection is either a segment (and we denote i t by [u, v]) or a point. The segment [u, v] is either the intersection of a finite system of nested segments or eise such a finite system does not exist.

Consider semiintervals from A ', [u, v]. Denote A ""f (A), j (A) "'- f (f (A)), ... by A1,

A2, •••• We note that if g E A;, then F 1 (g) E A;_1, and the complete preimage of each g E A 1 is empty. We select from each Ai a subset Ai suchthat gEAi if its complete preimage j-1(g)

is finite and is contained in A\-1 if A\_1 is not empty.

Any values lying in the sum U Ai=2I', have a finite track since f'f'1(g) = 0 for any gEA;.

Consider the segment [u, v ]. Here f ( [u, v]) = [u, v ]. We define the set ~uv for the function f (x) and the segment [u. v] in the same way that in 2.1 the set ~ was defined for the function f (x) and the segment [a, b ].

It is readily seen that a finite track has only those values g E [u, v], for which /-1 (g) E ~uv U 2I' Since to serve as an I-point in the interval (a, b) there can be taken any value from the whole nonempty set '21', it follows from Theorem 1 that a necessary and sufficient condition for the existence of a finite model of a function not taki.ng all values from A is finiteness of the tracks j* (a) and j* (b). We consider the followihg cases.

1. a, bEI(A). Here f *(a) and f *(b) are empty and such functions always.have finite-valued models.

2. bEI (A). Here j* (b) = 0, u = a, and the track f * (a) is finite when and only when

a) j-1(a) = {a, Ca} or

b) j- 1(a) = { v, Ca}, and j-1(v) = {a, Ca}, where Ca and CvE2I'.

3. a EI (A). The condition is analogous to Conditions a) and b) in Point 2.

128 YU. A. VINOGRADOV

Example 3. Construct a finite-valuedmodel for the function cp(x), given by the graph in Fig. 6.

Here b € cp(A), cp-1(a) = a, andhencethe function cp(x) has a model (Case 2 of the ones considered) inwhich ~, =A1 = (c, b], [u, v] = [a, c], S'ac= {a, g1}, cp- 1(g1) = {g1, g2}, g2 E~'. The I-points of the model are chosen from the set {a, g1, (c, b]}. The set I n(c, b) is arbitrarily formed by excluding the values g2 which necessarily fall in I if g1 E I.

Taking for instance I = { a, g1, g2, b} and numbering the G intervals, we get the model

Literature Cited

{ X, X=!= 2, <D(X)= 0, X=2.

1. G. A. Andreev, "Application of k-valued logic to schematic synthesis of mathematical machines" (diploma thesis), RFF GGU (1956).

2. T. I. Kir 1yanova, "Synthesis of electronic computers and control schemes by means of valued logic" (diploma thesis), RFF GGU (1956).

3. N. A. Loginova, "Synthesis of digital computers and control schemes by means of manyvalued logic" (diploma thesis), RFF GGU (1957).

4. S. V. Yablonskii, Proceedings of the All-Union Mathematical Society, Vol. 3, Akad. Nauk SSSR (1956), pp. 425-431.

ON A CER TAIN G ENERALIZA TION OF FINITE AUTOMAT A, WHICH FORMS A HIERARCHY ANALOGOUS TO THE GRZEGORCZYK CLASSIFICATION OF PRIMITIVEL Y

RECURSIVE FUNCTIONS t

V. A. Kozmidiadi

Moscow

This paper considers a certain generalization of the notion of a finite automaton. Asequence of expanding classes of n-automata (n = 0, 1, 2, ••• ) is formed. Each of the classes is formed by closure via a composition in the class of primitive n-automata. Under these conditions a primitive n-automaton operates similarly to a conventional finite automaton: it has an initial state and is stipulated by a certain function of transitions that determine the new state as a function of the previous state and the next input level. However, the states of the automaton are words in the input alphabet; the output word is formed as a sequence of states that are passed through by the automaton due to the action of the input word. The function of transitions for a primitive automaton of the (n + 1)-st rank is stipulated by means of an automaton of the n-th rank.

Chapter I gives the definition of an automaton of the n-th rank (n = 0, 1, 2, ... ) and presents a nurober of examples of such automata.

Chapter II proves combination theorems for n-automata: extension, branching, union, repetition, and configuration, Moreover, a theorem of rank elevation is proved which states that any transformation which is performable on an n-automaton may also be performed on an (n + 1)-automaton.

In Chapter III an example is constructed of the wocd numbering carried out in the alphabet A by 1-automata. On the basis of this numbering it is proved that any primitively recursive function is calculable on the appropriate n-automaton; on the other hand, any n-automaton is equivalent in adefinite sense to a certain primitively recursive function.

Chapter IV compares classes of n-automata and classes of primitively recursive functions from the A. Grzegorczyk classification. It is established that the class of functions calculated on n-automata (n 2: 1) coincides with the (n + 2) class of Grzegorczyk.

t Original article submitted October 26, 1967.

129

130 V. A. KOZMIDIADI

CHAPTER I

The Notion of an n-Automaton

In the present chapter a basic notion of the paper is defined - the notion of an n-automaton for n = 0, 1, 2, .... Any n-automaton is fully equivalent, relative to the alphabet A (A is the alphabet of an n-automaton), to a certain normal algorithm t [5]. The reverse, however, is not true. Moreover, n-automata in a certain sense occupy the same position among all dictionary calculable functions as primitively recursive functions do among all partially recursive functions.

In the present chapter we shall consider a series of examples of n-automatfl, besides defining an n-automaton. Many of these examples will be used in the later exposition.

§ 1. Definition of an n-Automaton

1. The definition of an n-automaton is given inductively (by induction with respect to n).

A primitive 0-automaton iJ,<o> in the alphabet Ais stipulated as follows. The word S0 in the alphabet A "'- {*} and the finite collection of Substitution formulas of the form ~ S- T, where ~ E AU{*}, S, TQA ',, {*}. are indicated. The word S0 is called th e in i ti al state of the primitive 0-automaton ~< 0 >. The primitive 0-automaton ~~< 0 > prescribes sequential transformation of the word P in the alphabet A according to the following rule.

Assurne P o ~ 1~2 ••• ~h (k > 0, ~ 1 , ••. , ~k are letters of the alphabet A). Let us form the word P*. Let us assume, moreover, that ~Mt o *· Let us form the word ~ 1S 0 • In the collection of substitution formulas let us find the formula ~ 1s0 - S1 (if, of course, such a formula exists there); then we use the word ~ 2s1 to find the formula ~ 2s1 - S2, etc. A string of words S1, S2, ... , Sk+1 is formed. If such astring cannot be formed (this is the case when for a certain i (0 :S i ::::; k) the collection of Substitution formulas does not contain formulas of the type ~ i+tSi- Si+ 1), we assume that the result of the transformation of the original word by the primitive 0-automaton ~<O> is not defined. However, if such a string may be formed, then we assume that the result of applying the primitive 0-automaton 2[<0> to the word P is the word s1s2 ... Sk+1; in other words, the primitive 0-automaton 2!< 0> transforms (manipulates) the original word P into the word 8182 ••• 8k+1• Any word 8 in the alphabet A "'- {*} is called a s t a t e of 2l' 0>.

Assurne that ~!< 0 > isaprimitive 0-automaton in the alphabet A; then ~<c, is called intri n s i c if * ~ A , and non in tri n s i c if * E A.

Assuming that the notion of composition of algorithms is known, we give the following definition of the notion of a 0-automaton.

A primitive 0-automaton in the alphabet A is a 0- a u to ma ton in the alphabe t A.

Assurne 2[<0> is a primitive 0-automaton in the alphabet A, while Q3<0> is a 0-automaton in the same alphabet. Then the algorithm that is a composition of 2[< 0> and Q<.< 0>, (i.e., the algorithm prescribing first the application of the primitive 0-automaton ~< 0>, to arbitrary initial data, and then the application of the 0-automaton Q)<O>, to the result of its work) is a 0-automaton in the alphabet A. This 0-automaton is intrinsic if the 0-automata 2[< 0), Q3<0> are intrinsic, and it is nonintrinsic if ~<O> or Q3< 0> are nonintrinsic.

t In this paper we shall strive to adhere to the notation in [5]. Moreover, the plan of the paper itself in many ways coincides with the plan of the monograph [5]: a nurober of terms, etc., are taken from there.

FINITE AUTOMATA AND PRIMITIVELY RECURSIVE FUNCTIONS 131

I 1 ... 1 I I . . ,

Input tape

A primitive (n + 1)-automaton ~<nH> is stipulated in the alphabet A as follows. The word S0 is indicated in the alphabet A along with the nonintrinsic n-automaton ~<n> in the alphabet AU {•}. The word S0 is called th e in itial state of the primitive (n + 1)automaton ~<n+u.

l!t<n+l> prescribes sequential transformation of the word P in the alphabet A thus:

Fig. 1 Assurne Po ~1~2 ..• ~h(k>O, ~1, ~z, ... , ~k

are letters of the alphabet A). Let us form the word P *. Let us place ~M1 o *· We construct

the word ~iSo and apply the n-automaton 2f<m to it; as a result we obtain the word Si; then we apply ~~ <n> to the word ~ 2Si, etc.

A string of words Si, S2, ••• is formed. If such a string cannot be formed (this is the case when for a certain i (0 ::o; i ::o; k) the results of transforming the word ~ i+is by the n-automaton ~<n> is not defined), we assume that the result of transforming the original word P by the primitive (n + 1)-automaton ~<n+u is not defined.

If such a string may be formed, then we assume that the result of applying the primitive (n + 1)-automaton ~<nm to the word P is the word Sis2 ••• Sk+i; in other words, ~<n+J> trans

forms (manipulates) the word P into the word SiS2 ••• Sk+i·

AnywordSinthealphabet A",{•} iscalleda state of the automaton ~<n+v.

The primitive (n + 1)-automaton ~ o>+u is called in tri n s i c if * ~ A, and non intrinsic if *EA.

Let us continue the definition of an (n + 1)-automaton. A primitive (n + 1)-automaton in the alphabet Ais an (n + 1)- a u to m a t on in the alphabet A.

Assurne ~<n+v isaprimitive (n + 1)-automaton in the alphabet A, and that )E<k> (k<n+1) is a k-automaton in the same alphabet. Then the algorithm which is a composition of ~<n+l>

and g)<k>, is an (n + 1) -automaton in the alphabet A; this (n + 1) -automaton is in tri n s i c if the (n + 1)- and k-automata ~<n+u and Q3<k> are intrinsic, and it is nonintrinsic if ~'n+" or m<h> are nonintrinsic.

2. The primitive n-automaton ~<n> functions in a certain sense in almost the same way as a conventional finite automaton (see [4]). In order to understand the functioning of an nautomaton more easily, it is convenient to represent it as a machine consisting of a readout head and a manipulation block ~ <n-n (Fig. 1); we shall not enter into the arrangement of the manipulation block at present. The original word P is fed letter-by-letter into the ~<n-o block which forms a new state from the preceding state and the next Ietter.

3. Taking account of the external similarity between the functioning of finite automata and n-automata, it would be desirable to provide the possibility of discussing primitive nautomata in the same terms in which conventional finite automata are discussed. We shall sometimes use such a terminology. Let us consider the primitive n-automaton ~<n> in the alphabet A and the original word P:

p 0 ~1~2 ..• ~i~i+l ... ~k,

where k::::: 0; ~ i• ~ 2, ••• , h are letters of the alphabet A. Assurne


is the string of words described in subsection 1. In this case we say that upon receiving the ne.xt, (i + 1)-st, letter of the original word the primitive n-automaton w<n> is in the state Si; due to the effect of the input letter ~ i+i it transits from the state Si to the state Si+t· Hereafter n 2{ 0»: S f-- ,T n means that the primitive n-automaton 9l<m transits from the state S to the state T due to the action of the letter ~.

"2! 00 : S 1--~T(T)" meansthat the n-automaton w<n> transits from the state S to the state T and prints out the word T due to the action of the letter ~ •

Instead of

and instead of

we shall usually write the abbreviated relation

and, correspondingly,

Assurne P is a word in the alphabet A, the condition P o ~~~2 ••• ~k, k > 0 being valid. Assurne that upon receiving the first letter ~ 1 of the word P the n-automaton wm> is in the state Si, and assume

In this case we shall say that the n-automaton w<m transits from the state Si to the state Si + k due to the action of the word P and shall denote this event thus:

or weshall say that the n-automaton w<n> transits from the state Si to the state Si+kdue to the action of the word P; this is denoted by

by definition we place

and

4. Let us introduce certain additional simplifications of the terminology, and let us likewise give certain definitions.

Assurne there is an n-automaton W<n>. Weshall call the number n the rank of W<n>.

We shall often speak loosely, saying "n-automaton" or simply "automaton" instead of "primitive n-automaton."


Fig. 2

Hereafter we shall denote automata and collections of Substitution formulas by capital Gothic letters (sometimes with indices). Under these conditions a superscript in parentheses will always denote the rank of the corresponding automaton. It is allowable, in denoting a certain n-automaton, to drop the rank in its designation if for some reason we do not wish to indicate it.

We shalllikewise call 0-automata finite automata, while n-automatahaving n> 0 shall be called in f i n i t e a u t o m a t a .

5. For the c ase in which the n-automaton 21<"> transforms a certain word P into a certain word Q weshall say that the result of the transformathn of the original word by the n- a utom aton 21un i s defined, w h ile the n- au to ma ton 21<n> is applicable to the word P.

If the n-automaton 1ll()l) is applicable to the word P, then it manipulates it into a certain completely defined word. We shall denote this word by 21(1'> (P).

Hereafter we shall sometimes make use of the symbol !. By placing this symbol in front of a certain expression, we shall thereby assert that this expression has meaning.

6. An n-automaton over the alphabet A is called an n-automaton in any expansion of the alphabet A (i.e., in an alphabet B which is suchthat A <: B). By analogy with the theory of algorithms [5], the notion of equivalence and complete equivalence of automata relative to the alphabet A is introduced.

Later on we shall sometimes make use of the symbol of conditional equality ~. By placing this symbol between two expressions we thereby state that these expressions denote the same thing if just one of them has meaning. Here we shall write the additional conditions imposed on the composite part of the expression considered in parentheses if necessary.

7. Let us consider a certain primitive automaton 21 in the alphabet A. According to the definition given in subsection 1, any word in the alphabet A""' {*} is a state of the automaton 21.

The state S of the automaton 21 is called accessible if there exists a word P in the alphabet A such that

where S0 is the initial state of the automaton 21. From the definition it is evident that the initial state of any automaton is accessible, since

It is not difficult to see that the nurober of states for which such a word P exists (i.e.~

accessible states) is finite for a primitive 0-automaton. Actually, the initial state S0 and, perhaps, certain of those states which are encountered in the right sides of the ~ubstitution formulas from the collection ~, are accessible for a 0-automaton. However, since the collection of substitution formulas of a primitive 0-automaton is finite, it follows that the nurober of accessible states is also finite. This fact explains why primitive 0-automata are also called finite automata.

8. We shall use the convenient representation of primitive 0-automata in the form of graphs, as is the usual practice.


The vertices of the graph shall be depicted by circles. Within the circle we shall write the state which denotes the given vertex. The initial state is depicted by a circle drawn with a double line.

9. The number of accessible states of a primitive n-automaton for n > 0 may be infinite. It is precisely for this reason that we have called n-automata infinite for n > 0.

The fact that a primitive 1-automaton having an infinite number of states is possible shall be shown in the next example. We shall soon be provided with the possibility of proving the same thing for automata of rank higher than 1 as well.

Let us consider the 1-automaton ~w in the one-letter alphabet A = { I} having the initial state A. Under these conditions 121< 0> is a 0-automaton in the alphabet {I , *}, which has the graph shown in Fig. 2. In other words, 2f<O> is a 0-automaton in the alphabet { I, *} , which has the initial state A and the following collection of Substitution formulas:

Let us prove that this 1-automaton has infinitely many accessible states. For this purpose we shall show that

(PQA). (1)

The latter statement derives (by induction from the length of the word P) from the relationships

(2)

and

(PQA). (3)

The first of these relationships derives directly from the definition of a transition from state to state due to the action of an empty word. In order to substantiate the second, it is necessary to consider the functioning of the automaton ~<o>. For the latter the following relationship holds:

(PQA), (4)

(PQA). (5)

From this (3) and (1) derive.

Thus, due to the action of various words Pi and P2 (Pi, P 2QA) the automaton 2rm enters various states Pi and P 2• Since there is an infinitely l~rge number of words in the alphabet A = { I } , there are also infinitely many accessible states of the automaton ~m.

§ 2. Examples of n-Automata

1. In this section we shall consider a number of examples of n-automata. Basically these will be n-automata which carry out the same transformations as do normal algorithms given in §4 of Chapter I in [5]. Almostall of these transformations may be realized by means of 0- and 1-automata.

In view of the unwieldiness of the formulations, we shall not present the constructions of the corresponding automata; it is assumed that where necessary the reader can carry them out himself. As an example, a description of one 1-automaton is given at the end of the section.


2. A 0-automaton 2I<X>. A may be constructed in the alphabet A, which is applicable to any word in A and which attaches the word AS1A to this word on the left; i.e., it functions thus:

(PQA).

3. An empty n-automaton (n = 0, 1, 2, ... ) may be constructed in the alphabet A (i.e., an automaton which is not applicable to any word in A).

4. One may constructt a 0-automaton ~~: A in the alphabet A, which attaches the word A at the right to any word in A; i.e., it functions thus:

(PQA).

5. One may construct a 0-automaton Q,~;"' in the alphabet A, which manipulates any word P in the alphabet A into the word obtained from P by discarding all a (a E A).

6. One may construct a cancelling 0-automaton ~~~> in the alphabet A, i.e., an automaton which operates thus:

Q,~> (P) o A (PQA).

7. A 0-automaton G:1 in the alphabet A may be constructed which operates thus:

G};1 (P) o A (l'QA, AQA).

8. A 0-automaton 'IlA in the alphabet AU {I} may be constructed which manipulates any word P in A into the length of this word:

(PQA).

9. Assurne that the Ietter a does not belong to the alphabet A. Truncating 0-automata 3A, a and @lA, a may be constructed in the alphabet AU {a} , i.e., automata which operate thus:

3A, a (PaQ) o P @lA, a (PaQ) o Q

(P, QQA),

(P, QQA).

10. Assurne that the letters a, ß do not belomg to the alphabet A. An excising 0-automaton S\:A, a, ß may be constructed in the alphabet AU { a, ß} , i.e ., an automaton that operatues thus:

S\:A, a, ß (PaQßR) o Q (P, Q, RQA).

11. A 0-automaton 2IA, A, B, c may be constructed over the alphabet A, for which

2IA, A, B, c (A) 0 B

2IA, A, B, c (P) o C

(AQA),

(PQA, P -:t A).

t The role of * , which terminates the manipulated word, is evident in this example; if we were to agree to apply the word P, rather than the word P* , to the automaton input in calculating m (P) it would be impossible to construct the indicated automaton.


12. A 1-autornaton ~~) rnay be constructed which rnanipulates any ward in A into the inversion t of this ward:

(PQA).

13. A doubling 1-autornaton ~A rnay be constructed in the alphabet A, which operates thus:

~A(P) o PP (PQA).

14. Let us juxtapose each letter ~ of the alphabet B with a specüic ward Af in the alphabet A. Then, replacing each letter ~ by the ward A6 in an arbitrary ward P in the alphabet B, we obtain a certain ward in the alphabet A - the result of replacing the letters ~ by the ward As (~ E B) in the ward P.

If B = { a 1, a 2, ... , an}, then the result of replacing the letters ~ by the ward A6 (~ E B) shall be denoted by the syrnbol

where B;-;- Aa. (O < i :s n). - l

One rnay construct a 0-autornaton st~Ji',".". ·:, '1':, over B which is such that

st {P)-;- sal, ... ' an p I**). _ B1, . •. ,Bn

The discarding of certain letters frorn a ward is a particular case of the Substitution of words for letters; this operation has already been partially considered in subsection 5.

The discarding of certain letters frorn a ward is a particular case of replacernent of letters by words for which A = B, A~ o A for certain ~ and A< o ~ for the rernaining ~ .

We shall apply the following terrninology. Assurne that the alphabet A is an expansion of the alphabet B , i.e., B s A. The result of discarding the letters of the alphabet A "-B frorn the ward P in A shall be called the projection of the word P onto the alphabet B and shall be denoted by [ pB.

Assurne A={a~o ... , an}, B={a~o ... , ak}, the condition k :5 n being valid. Then, obviously,

[pB-z- Sa!, ... ,ak, ak+!•· .. ,a"P J _ a 1, ... ,ak,A, ... ,A .

15. Assurne A and B are alphabets without cornrnon letters, the following equalities being valid:

t The inversion of the ward P is called the ward A if P ~ A; ü P "$:. A and P 2.. ~~~2 ... ~k (k > 0),

then the inversion of the ward is called the ward ~ k ... ~2 ~ 1 • Theinversion of the ward P is denoted by [Pu.

t Here and later on we shall drop the indices of Sf. for purposes of brevity.

FINITE AUTOMATA AND PRIMITIVELY RECURSIVE FUNCTIOT\!S 137

For each word P in the union of these alphabets - A U B - one may construct both the projection onto the alphabet A- [PA andthe projection onto the alphabets E-[PE.

One may construct a 1-automaton ~A. 0, which manipulates each word P in AUE into the word [PA [P0 .

16. Let us now consider a series of automata which transform natural numbers or systems of natural numbers into natural numbers. A natural nurober is a word in the alphabet q = {I }. The system of natural numbers is a word in the alphabet C oc~ { 1. a }.

17. One may construct a 0-automaton 2!1, which manipulates any natural nurober N into the remainder from the division of this nurober by 5.

18. One may now construct a 0-automaton 2!2, which manipulates any nurober N into the integer part obtained by the division of this nurober N by 5 (this integer part is denoted by [N/5]).

19. One may construct a 0-automaton 2Ia, which manipulates each natural nurober N into a pair of natural numbers KaL, where L = N- 5[N/5] (i.e., the remainder of the division of N by 5), while K = [N/5].

20. A 1-automaton 2! 4 , may be constructed which manipulates a pair of natural numbers MaN into the product M · N of these numbers.

21. One may construct a 1-automaton 2ls, which manipulates each natural nurober N into its square; i.e ., the automaton is such that

2ls(N) o N · N.

22. One may construct a 1-automaton 2Is, N which manipulates the natural nurober M into the nurober NM (N isanatural number, N > 0).

23. One may construct a 1-automaton 2!7, that manipulates any pair of natural numbers MaN into a nurober equal to the modulus of their difference; i.e., this is an automaton for which

21 7 (MaN) o IM -N [.

24. Let us now consider several automata which perform more special functions.

One may construct a 1-automaton 2ls, which manipulates each natural nurober N into a nurober equal to the distance to the largest complete square that does not exceed N. t

25. One may construct a 1-automaton 2!g, that manipulates a natural nurober to zero if it is the complete square of a certain natural number, while it manipulates it to 1 if this nurober is not a complete squareJ

26. One may construct a 1-automaton 2!10, which manipulates a pair of natural numbers MaN into their arithmetic (or allowed) difference.

t The corresponding primitively recursive function is denoted in [7] by quadres ~) and expressed as:

quadres (n) = n-'-- IVnP.

t The corresponding primitively recursive function is denoted in [7] by quad (n) and may be defined thus:

quad (n) =sg (quadres (n)).


27 o One may construct a 1-automaton ~11 , which manipulates a pair of natural numbers MaN into the remainder from the division of M by N (usually the nurober obtained is denoted thus: rm(M, N))o

28o One may construct a 1-automaton ~12 , which manipulates the pair of natural numbers MaN into the integer part of the quotient from the division of M by No The nurober obtained is usually denoted thus: [M/N]o

29. As an example, let us consider the construction of the 1-automaton ~8 from 24o

ills is constructed thus: ~s=~s,20~s.1·

The 1-automaton is ~s. 1· The initial state is A o

The 0-automaton is ills, 1· We assume that Ws, 1 =Ws, 1, 2 o Ws, 1, 1·

Now we stipulate the 0-automaton Ws, 1, 1· The initial state is ßßo

lßß~A *ßß~aa l~a

*~IJJO Jaa~aa

Oaa~aaa

*aa~A

ja~J

oa~lll laaa~y

Ws, 1. 1: * aaa ~A 11~1 01~01

*I~A llll~ß lv~v *'\'~A IOI~I *OJ~A lß~ß *ß~O

The graph of this automaton is depicted in Figo 3o As the 0-automaton Ws. 1, 2 we choose the 0-automaton

~{!, 0, a, ß, '\'}, 1. 0, a, ß, '\' {I,O,v},J,O,A,I,v •

Let us consider the operation of the 1-automaton Ws, 1 on the nurober No Assurne N = Oo Let us calculate ~8 , 1 (0):

and since

it follows also that

FINITE AUTOMATA AND PRIMITIVELY RECURSNE FUNCTIONS 139

Fig. 3

Thus,

2ls, 1 (0) o A.

Then let us consider the following relationships:

since

lli"s, 1,1: ßß l-1 A (.\) 1-* 1110 ( 111 0) and

Ws, L 2 ( Ii I 0) o III 0.

From here on we reason by induction. However, we shall first introduce a certain nota-tion.

Assurne N is a natural number. N shall denote the nearest natural number which is not larger than N and is a complete square. t

N shall denote the nearest natural number which is not smaller than N and is a complete square.t

The inductive assumption resides in the following: if N 2: 1, then

Let us determine the state in which the automaton 2ls, 1 arrives due to the action of the number N + 1.

tin the notation of [7] we have

lY_=rVNJ2.

t The corresponding primitively recursive function has the following definition: N = ([.JN'J + sg (N- [IN]2)) 2, or

N --- { N, - ([V~H-1)2 ,

if N = N,

if N =!= !!·


Forthis purpose it is necessary to calculate:

2fs, 1, 2 (2fs, 1,1 ((1 + (([VNJ + 1)2 -=-- N)) 0 (N-~!!_))).

Let us begin by considering the case in which N + 1 ~ N. In this case, as is easily seen,

oV~1 + 1)2 -=-- N > 1.

Then the word 1 + (([V.NJ + 1)2 -=-- N) begins with the word 111· Carrying out the calculations, we obtain

2fs, 1, 1: ßß l-1 A (.\) 1-1 a ( a) 1-11 ( j) I= <<rV~J+l)2~N)~ 2 1 (((([V"El + 1)2 -=-- N) -=-- 2)) 1- o 0 I (0 I) I=

l=:v~~ I (N-=-- !!_) 1- * A (A).

Thus,

2f8, 1, 1 (((1 + ([~] + 1)2)-=-- N) 0 (N-=-- N)) o a ((([V_N] + 1)2 -=-- N)-=-- 1) 0 ((N-=-- :Y> + 1). (1)

Since we are considering the case N + 1 ~ N, it follows that N + 1 = N.

Therefore from (1) we have

2fs, 1, 1 (((1 + OV ~I+ 1)2)-=-- N) 0 (N-=-- N)) o a (([V N + 1] + 1)2 -=-- (N + 1)) O((N + 1)-'-N + 1)).

Now applying the 0-automaton 2fs, 1, z, we obtain

ms, 1: (([V~J + 1)2 -=-- N) 0 (N-=-- :Y_) l-1 (([V N + 1] + 1)2 -=-- (N + 1)) 0 ((N + 1)-=-- N + 1).

Moreover, the calculations yield

Thus,

Now applying the automaton Ws,t,z, we obtain

This completes the induction step for the case in which N + 1 ~ N.

Let us now consider the case N + 1 = N. In this case, as can easily be seen,

OVNI + 1)2 ...!... N = 1. (2)

Then the word (1 + ([V"El + 1)2) . .!.. N begins with the word II· Carrying out the calculations, we obtain


Thus,

It can easily be seen that by virtue of (2) we obtain the following result by applying the automaton Ws, 1, 2 :

~8 1: (([l/JV] + 1)2 _:__ N) 0 (N _:__ N) f-1 (([l/ N + 1] + 1)2 _:__ (N + 1)) 0. ' - - --

Further,

where K = ([l/N}_!l + 1)2 _:__ (~ -i-1).

Thus, applying the automaton W8, 1, 2 , we obtain

The inductive proof has been completed. Let us now choose the automaton R't~}. 1.A;~,u1 , 1• 11

as the automaton ~8 • 2 ; we obtain the required equality:

~8 (N) o N _:__ N.

CHAPTER II

The Construction of n-Automata

In this chapter we consider a number of constructions that allow new n-automata to be constructed on the basis of gi ven m -automata (m ::::; n).

§ 1. The Extension of n-Automata

1. Assurne ~ is an m-automaton in the alphabet A, while B is an expansion of the alphabet A. Weshall say that an n-automaton ~in the alphabet B is an extension of the m-automaton ~ on the alphabet B, if

~ (P) ~ )S (P) (PQA).

Just as in the theory of normal algorithms, we shall consider certain special forms of extensions of m-automata.

2. Assurne ~ is an m-automaton in the alphabet A while B is an expansion of the alphabet A. Let us stipulate the m-automaton 18 in B, having taken the Stipulation of the mautomaton ~, as its stipulation; this, of course, is allowable, since each word in A is also a word in B at the same time.

Expansion of the alphabet A in no way affects the operation of the automaton on a word in A. Therefore, the constructed automaton ffi is the expansion of the automaton ~ to the alphabet B.

3. The extension of the m-automaton ~ to the alphabet B , which was described above, shall be called the direct extension of this m-automaton to the alphabet B.


It is obvious that if m is the direct extension of the automaton ~ to the alphabet B, then m may not be applied to a word P for which PQB, but PQA is invalid. Thus, the automaton Q:, in the alphabet B is such that it is the extension of the automaton ~ to the alphabet B , while at the same time it is applicable only to words in the alphabet A (and thereby only to those words to which ~ is applicable).

§ 2. The Theorem of Rank Elevation

1. In this section we shall show that, roughly speaking, all transformations of words in the alphabet A which can be performed by means of automata of rank n can also be performed by means of automata of rank m (n :s m), i.e., we shall show that in a certain sense m-automata are no weaker than n-automata.

1.1. The Theorem of Rank Elevation. For each automaton ~(nJ in the alphabe t A one m ay c ons tr uc t an au toma ton lS(nHJ in the s ame alphabet, which is such that

m(nJ (P) ~ jbln+!J (P) (PQA).

2. Pro o f. For each n 2:: 0 an identical n-automaton m~J will be constructed in the alphabet A, for which

(PQA).

Further, the automaton mln+IJ that we are required to construct in the alphabet A for the proof of Theorem 1.1 shall be defined thus:

Thus, the proof has been reduced to the construction of the automaton m~J (for n == 0, 1, 2, •.. ). Assurne that the initial state of the automaton m~J is A. We shall construct the automaton m~J in such a way that it operates on the word P o ~ 1~2 ••• ~k (k> 0) in the following manner:

Thus, the following relationships must hold:

ill"A (~1]) o ~

ill"A (s) 0 ~

ill"A(*~) o A

WA (*) 0 A.

(s, 11EA),

(SEA),

(SEA),

(1)

(2)

(3)

(4)

Let us denote the automaton satisfying the conditions (1)-(4) by m. It is obvious that if we construct the (n- 1)-automaton mtn-lJ (n 2:: 1), then one can determine the m~J by choosing WA tobe the automaton m(n-IJ; the initial state of the automaton ~~> is A.

We inductively construct the automata m~> and m(nJ.

For n == 0 the automaton m~lJ was constructed in [Chap. I, §2, subsection 2 ]; this is the automaton m~.) A· The automaton m<OJ is stipulated thus: the initial state is Cl/Cl/. (It is assumed that a:EA.)


~aa~~ (~ E A) t *aa~A

\:5: ~11~A (~,TJEA). 1.0,. *~~A (~EA)

~~A (~ E A) *~A

Fig. 4 The graph of this automaton for A = { 1, 0} is displayed in Fig. 4. As a we have taken 1.

We further assume that the automata 21~) and m<n) have already been constructed. Let us construct the automata 21~+ 1 ) and m<n+1J. As has already been indicated above, the automaton 21~nf- 1 > may be constructed thus: its initial state is A. As WA we choose the automaton m(n).

If we now construct 21~n-f 1 ), it follows that m<n+ 1J may be constructed thus:

Since 21~+ 1 > is an identical automaton, the same relationships (1)-(4) are fulfilled for the automaton mcn+ 1> as are fulfilled for the automaton Q)co>. Thus, the construction of an identical n-automaton 21~J has been completed for any n (and thereby the proof of Theorem 1.1 has been completed).

§ 3. The Branching of Automata

1. Sometimes it is necessary to construct instructions in the following form: "the automaton 21: or the automaton Q) is to be applied to the original word, depending on whether or not the original word begins with a given letter."

Thus, a certain new algorithm is stipulated which is a combination of the automata 21 and m. The question naturally arises as to whether the calculations instructed tobe performed by the algorithm described may be performed on some automaton. The answer to this question is given by the following theorem.

1.1. The Branching Theorem. Assurne 21 and lE are automata having the ranks na and nb, respectively, in the alphabet A. Assurne a is a certain letter of the alphabet A. Then one can construct an automaton ~ of rank n = max (na, nb) over the alphabet A such that

~ (A) ~ 21 (A),

{ 21(~P), ~ (~P) ~ \8 (~P), (PQA),

(PQA).

Pro o f. The proof is carried out by induction with respect to n. The basis is n = 0. Let us define the automaton ~ as follows:

(1)

(2)

(3)

t Following [5], we use abridged notation for the Substitution formulas. The letters ~' Ti are arbitrary letters of the alphabet A, as is stated in the condition written to the right of the formula. These letters take the values of any letters of the alphabet A. Thus, for example, the first row replaces only that number of substitution formulas which is equal to the number of letters in the alphabet A.

144

Fig. 5

V. A. KOZMIDIADI

Under these conditions the automaton Q: 1

operates thus in the alphabetAU {ß, y} (ß ~ A, 1' ~ A): its initial state is A.

*--?'\'

s _,.'l's (sEA "'-{a:}) 0: -?ßa:

sßa __,. s (s EA) ~~: s'l'TI __,. s (SEA,TIEA "'-{a:})

*ßa: _,.A * 'l's _,.A (sEA"'-{a:})

sTI __,. s (s,TIEA) *s_,.A (s EA)

The graph of this automaton (for the alphabet A = { 0, I, a}) is depicted in Fig. 5. For this automaton the following holds:

Q:t(A) o y,

Q: (t.. P) --;;- { ßsP, 1 "' - 'l'sP,

(PQA),

(PQA).

Let us construct the automaton Q:2 in the alphabet AU AU {ß, y, ö} (ö ~ A). The initial state is A.

ß -?ß 1' --71'

Q:2: s'l' __,. Q~ (SE AU{*})

sß _,.R; (s E A)

21' 18

Let us clarify how the words Q f and R f are chosen. Assurne that the initial state of the automaton ~I is S, while that of the automaton m is T; in >E (the collection of substitution formulas of the automaton m) we find the formula of the form ~ T - A, where s E A U {*} • If there is no such formula, the Substitution formula for the corresponding ~ is also absent in ~2 If there is such a formula, we assume that Q~ o A. The words R 6 are defined somewhat differently from the procedure described above. In ~ we seek the formula of the form ~ S- B (s E A). lf B o A , we assume that Re, o ö; if B ""*- A; then we assume that Rq, o ii, where B is the twin t of the word B.

Let us now clarify what ~, is. 21' is obtained fröm the table N as follows. Each formula of the form ~ P- Q from the table N is replaced by the formula ~ P' - Q', where the operation 11 1 11 is defined as follows:

tThe twin of a word is defined exactly as in [5]. Let us give the necessary definitions. Each let~r ~ of the alphabet A is juxtaposed with a new Ietter - the twin of the Ietter ~ (denoted by ~),different letters of A being juxtaposed with different new letters. The twins of letters from A comprise an alphabet of twins X containing the same number of letters as the alphabet A but having no letters in common with A. The twin of a word in the alphabet A is defined obviously.

FINITE AUTOMATA AND PRIMITIVELY RECURSIVE FUNCTIONS

{ ö, if A--;;- A.

A' o A ( the twi:f the word A),

Finally, ~ 2 operates as follows:

~2 (~I (A)) ~ y~I (A)

~ (~ (tP)) ~ { ß~. (~P), 2 1 " l'~ (~P),

if ~ o a if ~ ~ a

if A o A.

(PQA),

(PQA).

And, finally, there is the automaton ~3· As this automaton we shall take

S{A 1jÄl_){ß,y,ö},r.:q, ... ,am.ät, ... ,äm, ß, y,ö, A, cq, ... ,Ct.m,O:l, •.. ,am,A,A,A

(let us assume thatA = {a1, ••• , am}).

This ends consideration of the basis.

145

T h e In du ct i o n S t e p . Let us assume that the theorem is valid for automata having a rank not exceeding n. Let us show that it is also valid for automata of rank (n + 1).

We shall construct the automaton ~ in the form

As the automaton 'll1 let us take the 0-automaton ~~ from the basis. Then let us define the (n + 1)-automaton 'll2 over the alphabet AU {ß, y}. The initial state is A. Basedon the inductive assumption, let us construct the n-automaton ~2 :

{ ßS,

'®z (~P) ~ yT, a; (~P),

if ~ 0 ß, if ~ 0 l'•

if ~~ßand ~~y;

here S, T are the initial states of the automata ~ and ~ , respectively.

The n-automaton (!; is defined as:

Under these conditions the 0-automaton a:1 operates thus:

We omit the details of the construction of this automaton.

Once again, we define the n-automaton Q::2: on the basis of the inductive proposition:

J (.(o'.:>\•':'• ... ,~m (Ö· (stAUNJ{ß),al, ... ,am.äl .... ,am. ß(tP))) pJlA a a .J\. A, cq, ... , am, a1, ... , am, A ":) ,

~ (~P) "-' , 1, ... , m 2 · l iE(S\'AIJ{V},al •... ,am,V(tP)) 'V A, ar, ... ,am,A S '

if ~ 0 ß,

if ~ 0 l'.

Let us finally construct the last automaton '!l3, in the capacity of which we choose the 0-automaton


As is easily seen, the (n + 1)-automaton (§; constructed in this way satisfies the conditions of the theorem.

This completes the proof of the branching theorem.

§ 4. A Union of n-Automata

1. For many constructions it is necessary jointly to consider the results of the operation of two or several automata on the same initial data. In such cases the construction of a system of all of these results is useful, since this system may itself serve as the initial condition for the operation of some other automaton.

In general, if the automata ~ 1 , ~ 2 , ••• , ~I~< are given in the alphabet A which does not contain the Ietter a, one may consider the algorithm ~I, satisfying the condition

We shall call this algorithm the nunion of the given automata ~~> 2l2, ... , ~k>>.

It is natural to ask whether a union of automata may be calculated on a certain automaton. The answer to this question is given by the following theorem which constitutes the basic content of the given section.

1.1. The Union Theoram. Assurne ~j, ~2 , ••• , 21k (k::::: 1) are automata in the alphabet A which have the ranks n 1 , n 2 , ... , nk, respectively. Then an automaton 21 over AU{a} of rank n may be constructed, where n = max(n 1 , n 2 , ... , nk, 1), suchthat

(a~A, PQA).

2. We begin with the proof of the following fact.

2.1. Regardless of the automata 21 and m in the alphabet A which have the ranks n1 and n2, respectively, one can construct an automaton (§; over A of rank n, where n = max(n1, n2, 1), suchthat

(§; (P) ~ 21 (P) lS (P) (PQA).

In the beginning Iet us construct a 1-automaton which operates as:

where P is the twin of the word P, while a ~ A U Ä. It is not difficult to construct such an automaton. Then we define the automaton (§; thus:

(§;2 is stipulated as follows. The initial state is S1, where S1 is the initial state of the automaton \2{.

if ~EA,

if ~ o a,

if ~ {A. U {•}.


Here S1 is the twin of the initial state of the automaton ~l. In constructing ~2 , the brauehing theoremwas used. As (l:3 we choose the automaton

S\AUX,cq, ... ,a.m.Ut • ... ,Um • A, a1, ... ,arn,CI.!, ... ,am·

N ow let us prove the following statement,

2.2. Regardless of the automata ~I and \B in the alphabet A having the rank n1 and n2 ,

respectively, one can construct an automaton 0: of rank n over the alphabet AU {a}, suchthat

(l: (P) c::,: 2I (P) aQ\ (P).

Under these conditions n = max(n1, n2, 1).

This statement can easily be obtained from the previous one, 2.1, in the following way:

(l: (P) c::,: 2I (P) 2l~CJ{a}, a ('.S (P)),

where 2I~u{a}. a is a 0-automaton that attaches the letter 0! to any word in the alphabet AU {a} at the left.

From (2.2) the union theorem can easily be obtained by induction with respect to k.

§ 5. Repetition of an n-Automaton

In constructing n-automata it is sometimes necessary to use the following construction: it is required to manipulate the system of words in the alphabet A:

into the word 2I (P 1) a2I (P2) a ... a2I (P,.), 2I being a certain n-automaton in the alphabet A. The natural question arises as to whether the operation described above can be carried out on some automaton, The answer to this question is given by the following theorem.

1.1. The Repetition Theorem. Assurne 2I is an n-automaton (n 2:: 0) in the alphabet A, while a is a letter that does not belong to A. Then one can construct an n-automaton Q\ over the alphabet AU{a} such tha t

.m (P 1aP 2a . .. al'r) c::,: \li (P 1) a2I (P 2 ) a ... a~I (P,.)

Proof. Assurne

(k> 1),

where 2Ii (0 < i :s k) areprimitive automata in the alphabet A which all have (according to the theorem of rank elevation) the rank n. Assurne that the initial state of 2Ii is S0• Let us define the n-automaton 0:i in the alphabet AU AU {a, ß, 'V} (ßEtA U Ä, '\'EtA U A). Its initial state is S0• ri, is defined as:

if

if

if

~ o a or ~ o *,

~ c ß, ~ EA.

(1)


Let us now show how to construct the (n - 1)-automaton 1>; (or the collection of Substitution formulas, if n = 0):

'Il;, 1 is a 0-automaton suchthat

'Il;, 1 (~P) o P~.

The (n- 1)-automaton 'Il;, 2 is constructed as follows:

f ~ · ('Il· 3 ('YlR)) 'IJ· ( R rv J I I, ., '

z,2 lJ )-I ili.(ct\· ( R)) l <:< 1 IU1, t, lJ ,

under these conditions

if

if

lJ..::... y'

11 E A;

(see [1, § 2.14] and [1, § 2.4]), while the 0-automaton '1>;, • is such that

'Il;, .(P~) o ~p.

As is easily seen, the following relationship holds:

Let us now introduce the 0-automata m1 and m2 • m1 is the automaton

while m2 is the automaton

~AU{a, ;>}, a1, •.• , am, a, v AU{a, fl}, tx1o ••• , am, afl, aß,

~AU{V}, a1, ... , am, V AU{a}, !Xlo ••• , am, a•

Now we may define the n-automaton m thus:

It can easily be checked that (1) holds for it.

§ 6. The Continuation of n-Automata

1. Sometimes the necessity arises of "supplementing" a function stipulated by a certain n-automaton in the alphabet A in some fashion so that as a result a function is obtained that is defined on all words in the alphabat A. The question arises as to whether the function obtained in this manner can be calculated on an automaton. In order to answer this question, we shall formulate the following theorem.

1.1. The Continuation Theorem. Assurne W is an n-automaton (n 2: 0) in the alphabat A. An n-automaton l8 may be constructed such that in the same alphabet A it is applicable to all words in A, and if !W (P), then

W (P) o jß(P).

We omit the proof of this theorem.


CHAPTER III

Automata and Primitively Recursive Functions

In this chapter we consider the problern of the equivalence of automata and primitively recursive functions. In order to compare these notions it is necessary either to find a method which allows primitively recursive functions tobe used for manipulating the word in any alphabet into a word in the same alphabet, as automata do, or to consider only those automata which manipulate systems of natural numbers into numbers. In this chapter we consider both approaches.

Throughout this chapter the word "function" denotes a partial arithmetic function (i.e., a function whose argument values are natural numbers and which takes natural values).

§ 1. m-Automaton Functions

1. The function j(x1, x2, ••• , Xn) is called an m-automaton function (an mfunction, an automaton function) if there exists an m-automaton Ir over the alphabet C such that

(1)

The function j(x1, x2, ••• , xn) is called fully m-automaton (fully automaton) if there exists an m-automaton Ir over the alphabet C which is applicable to any ward of the form x1ax2a .•. axn, suchthat (1) holds. The class of completely m-automaton functions shall be denoted by A T.

2. We shall assume that the notion of a primitively recursive function is known.

The rank of primitively recursive function is called the length of its shortest primitively recursive description.

3. Letustakeuptheproofofthefactthat any primitively recursive function is an automaton function. Theproofisbasedonseverallemmas.

3.1. Assurne g(x1, ••• , xm), h1(x1, ••• , xk), •.• , hm (x1, ••• , xk) (m 2: 1, k 2: 1) are respectively n0-, nc, ... , nm-automaton functions. Then the function

f(x1, ... ,xk)~g(h1(x1 , ... ,xh), ... ,hm(x1, ... ,xk))

is a max(n0, n1, ••• , nm, 1)-automaton function. (The lemma on Substitution.)

Assurne @l, ~ 1 , ..• , ~m are automata suchthat

@l (x1ax2a ... axm) ~ g (x1, ... , Xm),

.'f;\1 (x1axza ... axk) ~ h1 (x1, x2, .•. , xk),

On the basis of the union theorem one can construct an automaton SJ having the rank max(n1,

n2, ••• , n m' 1) such that

tWe identify natural numbers and words in the alphabet q={[}.


Now it is obvious that the autornaton '% having the rank rnax(n0, n1, ... , nm, 1), which is defined as

satisfies the condition

whence the lernrna derives.

3.2. Assurne the functions g(x1, ... , xk) and h(x1, ... , xk, xk+l' xk+2) are respectively nc and n2-autornaton functions. Then the function f (x1, ... , xk, xk+l), which is defined by the systern

{ f(x1, ... ,xh, O)~g(x1 , ••. ,xh),

f(xh ... ,xk, y+i)rvh(f(xt. ... ,xh, y), x1, ... ,xk, y)

is a rnax(n1, n2 + 1, 2)-autornaton function.

Assurne @l and ~ are autornata such that

@l (x1a ... ax") ""'g (x~> ... , xk),

S:J (x1a ... axkaxk+1axk+z) ~ h(xt. ... , xk, xh+i, Xk+z).

Let us construct the autornaton 6 as follows:

Here 61 is an autornaton constructed by rneans of the union theorern and having the rank rnax(n1, 1). Under these conditions

~ being a 0-autornaton that operates thus:

one can also construct a 0-autornaton 6z, such that

Let us now define the (~ax(n2, 1) + 1)-autornaton ~3 thus. The initial state is A. The rnax(n2, 1)-autornaton %3 is stipulated as:

{ "P<, if ~ :t ß, - m (~tJP), if ~ :t *• t] 0 ß, %3 (6tJP) ~ ~ (6tJP)' if ~ 0 *•

6tJP, if 6 ° ß

(PQ {I, a, ß} ).


Let us describe the automata m and ~. The automaton m is described by:

The 0-automaton m1 operates thus:

931 (I ßP) -o P (PQ {I, a}) (see [I. § 2.9]).

The automaton lE2 is defined by means of the union theorem. Its rank is max(n2, 1)

m2 (Xoax1a ... axkay) ~ fJ (XoaX1a ... axkay) axoax1a ... axkay.

The 0-automaton 583 operates as follows:

Thus,

Now we stipulate the 0-automaton ~:

It can easily be shown that

It is obvious that the rank of an automaton tr constructed in this way is equal to

max (max (n1, 1), max (n2 , 1) + 1) = max (n1 , n2 + 1,2).

Since the original functions are automaton functions, we arrive at the following statement.

3.3. The Theorem of Primitively Recursive Functions. Any primitively recursive function is completely an automaton function.

§ 2. Numeration of the Words in the Alphabet A

151

1. The correspondence r between the words in the alphabet A (as usual, it is assumed that *~A) and natural numbers is called the m-numeration of the words in the alphabet A, provided that there exists n-, n0-, ncautomata @1, @1 0 , and @1 1 , respectively (m = max(n, n0, n1)), such that:

a) @1 0 is applicable to all natural numbers and manipulates into an empty ward those and only those numbers x for which !@1 1 (x) and @1 1 (x) QA.

b) @I is applicable to each ward P in the alphabet A and manipulates such a ward P into a natural number; the corresponding natural nurober @l (P) is suchthat @1 0 (@1(P)) o A, while @11(@1 (P)) o P.

If @1 0 (n) o A and @ldn) o P, then the natural nurober n is called the nurober P in the m-numeration r.

It is obvious that if PfJA, QfJA, where P -:1:- Q, it follows that if n1 and ~ are respectively the numbers of the words P and Q, n1 ~ n2 (i.e., the numbers of different words in the alphabet A


are different). However, from the definitions it does not follow that each ward in the alphabet has only one number. On the contrary, examples can be constructed of m-numeration suchthat at least certain words have more than one number.

For any ward P the natural nurober @l (P) is called the principal nurober of the wo rd P. Thus, each ward has a unique principal number. The m-numeration of the words in the alphabet A is called c o m p 1 e t e if any natural nurober is the nurober of a certain ward in the alphabet A.

Assurne there exists an alphabetAsuch that * ~ A, and also a certain complete m-numeration r of the words in the alphabet A.

Weshall say that the automaton m in the alphabet A is a primi ti vely recursi ve automaton if there exists a primitively recursive function f for which the following holds:

a) if m (P) o Q, (PQA), @l 1 (n1) o P (i.e., n1 is the nurober of the word P), then @ld/(n1)) o Q (i.e.,f(n1) is the nurober ofthe ward Q).

b) if f (n1) = n2, then m (@31 (nt)) 0 @31 (n2)·

2. The problern of this section is to construct an example of the 1-numeration of words in the alphabet A.

Assurne A = { a 1, ... , a } (k > 0). Let us assume, furthermore, that the letter 0 does not belang to the alphabet A.

We shall assume that the letters 0, a 1, .•• , ak are digits in a (k + 1)-ary numbering system; under these conditions 0 denotes 0, a 1 denotes the natural nurober 1, a 2 denotes 2, ... , a k denotes k.

Each ward in the alphabet A is then the notation for a certain natural nurober in the (k + 1) numbering system. Let us juxtapose a certain natural nurober with each ward P in A; if P o A, then we juxtapose the nurober 0 with P; if P .2- A and P o a;0a;1 ••• a;s (s > 0), we juxtapose the nurober

with it.

Each natural number, of course, has its notation in the (k + 1)-ary numbering system. This notation R is a word in the alphabet AU {0}. The ward [RA is already a word in the alphabet A. Thus, each natural nurober corresponds to a completely specified ward in A.

Let us now construct the n-automata @l and @3 1• Under these conditions @l must manipulate A into the nurober 0, and the nonempty ward PQA into a natural nurober whose notation (in the (k + 1)-ary system) is the word P. @31 must manipulate any natural nurober x into the word [RA, where R is the notation of the nurober x.

Let us begin with the construction of the automaton @l. We split up this construction into a series of stages. First, one may construct the 1-automaton m in the alphabet A such that

m (A) o A

~ (6162 ... st-1s1) o (t----1)st(l----2)s2 .. ·lst-tst (l> 0; s; EA, 0 < i-<.l)

FINITE AUTOMATA AND PRIMITIVELY RECURSNE FUNCTIONS

(we assume that I~ A). Then we take the 0-automaton

, ~AU{J},a1 , ... ,a",l . st =Si; } IR I" R I' {l,a,ß,ap, ... ,a p,

then the following holds for l > 0:

Si;(~ (a;1a;2 ••• a;)) o (l-'-1) ai 1ß (l-'-2) ai2ß ... I aiz- 1ßaizß;

for l = 0:

st (~ (A)) o A.

For any k 2: 0

h (x, y) = (k --!)'"· y,

153

(1)

one may construct the automaton m", which calculates the function (1); i.e ., one can construct .an automaton such that the following relationship holds:

'i!.\ (xay) o (k+ 1)"·y.

It is not difficult to see that the automaton Ql" has the rank 1.

Now Iet us use the repetition theorem [II. §5.1.1]. However, we are not able to apply it immediately.

The cause of this lies in the fact that the word st (~ (P)) ends with a separating Ietter ß, and for the repetition theorem this should not be clone. It is easy to construct the 0-automaton ~ in the alphabet A U {I, a, ß} , which is such that

(§; (Pß) o P, where PQA U {I, a, ß}.

Now we do have the right to use the repetition theorem; using the 1-automaton ~"' we construct the automaton m thus:

Now we apply the 0-automaton l§:\1>, ß}, ß• to the results and discard all entries of the Ietter ß from the word to which the automaton is applied (see [I. §2.5]). As a result we obtain (for l > 0):

Let us assume that

l

l§:\1>, ß}, ß (Ql ((§;(Si:(~ (ai1ai2 .•. a;1))))) o Lj ij (k+ 1)1-i. i=l

®=(§;{I). ß}. ß o m o (§; o st o ~.

It is not difficult to check the fact that for l = 0,

Having traced the construction of tlle automaton @3, one can verify the fact that the rank of @J

is equal to 1.

Let us now go over to the construction of the automaton @lt.


One may construct a 1-automaton 21 in the alphabat { J, a, ß}, which is suchthat

21 (A) o A,

21 (x) -~ xaxß (x ~ 1) axß .•. ß1axßaxß (x > 0).

Let us now determine the two-placed partially recursive function fk (x, y) (k > 0), which causes a natural number z (0 :::::; z :::::; k) equal to the digits having the weight (k + 1)X in the notation of the number y in the (k + 1)-ary numbering system to correspond to each pair of natural numbers x and y.

The function fk(;x., y) may be stipulated thus:

/J,(x, y)=[y/(k+1)"']_:_(k+1)[y/(k+1f'+1]. (2)

Assurne m is an automaton which calculates the function f k(x, y). Let us estimate the rank of the automaton m, considering the fact that the function f k (x, y) calculated by the automaton is stipulated by the relationship (2). The functions (k + 1)x, [x/y], xy, x.!.. y, x + 1, can be calculated on 1-automata (see, respectively, [I. §2.22], [1. §2.28], [I. §2.20], [I. §2.26]).

From this it is easy to obtain the fact (using the theorem from Chapter II) that the rank of the automaton ~ is equal to 1.

Making use of the repetition theorem, we construct an automaton ~. suchthat

holds. Then we construct the 0-automaton 'll in such a way that

Finally, we obtain

Thus, we have constructed the automaton @3 1 • Its rank, as is easily verified, is equal to 1.

So, the construction of an example of 1-numbering r of words in the alphabat A has been completed.

§ 3. Primitively Recursive Automata

1. Weshall say that an n-automaton 21 is a primitively recursive automaton if the function F defined by the relation (1) is primitively recursive:

F (x) ~ @l (21 (@! 1 (x))).

Our problern now is to prove the following:

1.1. Theorem. Any everywhere-determinate n-automaton 21 in the alphabat A is a primitively recursive automaton.

(1)

2. In the beginning we shall deal with the construction of certain primitively recursive functions.

Let us consider the alphabet A = { a 1, .... , ak} and the numeration r A of the words in the alphabat A, which was constructed in the preceding section. Each word P in A corresponds to at least one natural number - the number of the word - in this numeration. Let us consider

FINITE AUTOMATA AND PRIMITIVELY RECURSNE FUNCTIONS 155

the alphabet B = {a1, .... a1, • ••• , au 1} (Z :::: 0) which is an expansion of the alphabet A; for words in this alphabet we construct the numeration f 5 . in accordance with §2. Each word P in A receives at least one number in the numeration rE (as a word in B). We shall call this number the quasinumber of the word P.

Let us construct a prirnitively recursive function f such that for every nurnber of any word in the alphabet A the function gives sorne quasinumber of this word.

It is stipulated as follows:

X

f(x)= ~ (k-l--1)ifh(i, x). (1a) i=U

The function fk appearing here was constructed in [§2.2]. If x is the principal number of the word P, then weshall call the number j(x) the principal quasinumber of the wo r d P . It is likewise easy to construct the prirnitively recursive function g that uses each quasinumber of any word P in the alphabet A to produce the nurnber of this word. This function is defined thus:

X

g (x) = I (k --l)i IR.+! (i, x). i=O

(2)

It is evident that if x is the principal quasinumber of P, then g(x) is its principal number.

Assurne P and Q are nurnbers in the alphabet IJ. while x and y are their numbers in the nurneration rE. Let us construct a primitively recursive function h such that h(x, y) is the number of the wor~ PQ in the nurneration r E· Assurne r (y) is the nurnber of the highest nonzero digit of the notation of the nurnber y in the (k + l + 1)-ary nurnbering systern.

The function r(y) rnay be defined in a prirnitively recursive rnanner thus:

{ 0, if y=O,

r(y)= . -(flXx~ y ((k-;- l- 1 r > y)) __:_ 1.

(3)

Now, of course, one can define the function h(x, y) thus:

h (x, y) =X· (k --/-c-f)'(y)-,1 _._ y. (4)

Under these conditions it is clear frorn the construction of the functions h and r that if x and y are the principal nurnbers of the words P and Q, respectively, then h(x, y) is the principal nurnber of the word PQ.

Finally, let us construct the last prirnitively recursive function a that we need, which uses the nurnber (in the nurneration r A) of the word P (PS1A) to produce the principal nurnber of this word.

Forthis purpose we begin by constructing the auxiliary function b which for y > 0 produces the nurnber of the x-th nonzero digit in the (k + 1)-ary notation of the nurnber y frorn x.

The function b rnay be stipulated as follows:

If y = 0, then

b (x, y) = 0 for all x :;:> 0. (5)

If y > 0, then

b (0, y) = flio:Si:Sy (fk (i, Y) > 0), (6)

b (x _._ 1, Y) = [fliu (x, y)<i:Sy7z(fR (i, y) > 0)] sg [y __:_ (k ~ 1)"<-'· Yl+!J + (y __;__ 1) sg (y __:_ (k + 1)b (x, Yl+ 1].


It is obvious that

b (x, Y)<Y+ 1. (7)

Now one can easily go over to the function a:

X

a(x)= ~ (k+i)i·fk(b(i, x), x). i=O

(8)

In order to prove the Theorem 1.1 we shall consider the alphabet B tobe AU{*}. Thus, l == 1.

3. Let us now go over to the proof of Theorem 1.1. Weshall conduct this proof by induction with respect to n- the rank of the automaton ~-

Basis: n == 0. Assurne ~ is a primitive 0-automaton having the initial state S0• The substitution formulas from ~ have the form ~ S- T, where s E AU{*}, SQA, TnA. We shall juxtapose each such formula with a pair of natural numbers (s, t) such that s is the principal quasi number of the word ~ S, while t is the principal quasi number of the word T. As a result, we obtain a finite collection of pairs (s ~ to), ••• , (s p, tp). This collection is such that if s i == Sj,

then ti = ti• This provides the possibility of defining the primitively recursive function cp 0 in the following manner:

In other words,

p

!Jlo (x) = ~ t; ·Sg I x-s;j. i=O

AssurnePis a word in the alphabet A having the number x. Then its principal number is equal to a(x) (see [2.(5)-2.(6)]), and the principal quasi number is f(a(x)) (see [2.(1)]), and the principal quasi number of the ward P* is equal to h(j(a(x)), yo) (see [2.(4)]); here y 0 is the principal number (in the num.eration rAU{*}) of the ward * • Let us now construct the auxiliary function x0(x, y) which uses the pair of natural numbers (x, y) to produce the principal quasi number of the state into which the automaton ~ transfers due to the action of the ward having the quasi number [y!(k + 2)'<Yl~x]. If the ward having the principal quasi number y is P *, where P * o Sr<yJSr<Y>~1 ... s1s0 (so o *), then the word having the quasi number [y!(k + 2)r(y)~x] is the ward

(O<x<r (Y)).

Let us define the functions x 0(x, y) by means of the recursion

{ n0 , if fhH (r (y), y) = 0,

Xo (O, y) = !Jlo (h (fk+t(r (y), y), no)), if fk+dr (y), Y) > 0;

here n0 is the principal quasillumher of the initial state S 0 of the automatoll W,

{ Xo(x, y), if x+i>r(y) orif x+1<

Xo (x + 1, y) = < r(y) and- fk+dr (y)-'- (x + 1), y) = 0,

!Jlo(h(/~t+dr(y)-(x+i), y), Xo(x, y))),

if x+i<r(y) and fkH(r(y)-'-(x+i), y)>O.

(1)


And, finally, let us define the function w0(x, y) which uses the pair of natural numbers (x, y) to produce the principal quasi nurober of the words that the automaton ~ prints out by manipulating the word having the nurober [yi(k+2)r<v>~x]. This function is defined thus:

ru0 (0, y)== { 0, if fk+t(r(y), y)=O, Xo(O, y), if fk+t(r(y), y)>O,

{ ru0 (x, y), if x+i>r(y) orif

Wo (x+ 1, Y)= x+ 1 <r (y) and ik+dr (y)-'- (x+ 1), y) = 0,

h(w0 (x, y), Xo(x+1, y)), if

x+1<r(y) and h+t(r(y)~-(x+1), y)>O.

It is obvious that the functions Xo and w0 are primitively recursive.

Fora primitive 0-automaton W the function F 0 may be defined thus:

F0 (x) = g (w0 (h (! (a (x)), Yo), h (! (a (x)), Yo)))

(2)

(3)

(here the function g was constructed in [2.(2)], h was constructed in [2.(4)], f was constructed in [2.(1)], a was constructed in (2.(8)]; from this it follows that the function F 0 is primitively recursive.

Assurne \21 is a 0-automaton in the alphabet A such that

(l> 1),

the automata \21~> •.• , \21 1 being primitive 0-automata in the same alphabet. Assume, moreover, that the functions F 0, 1, ••• , Fo,z havebeenconstructedforthemaccordingtothediscussion above. Then the function

Fe (x) = Fo, z (Fo, 1-1 (. . . (Fo, 1 (x)) •.. ))

obviously proves Theorem 1.1 for all 0-automata.

Th e Ind uc ti on S tep. According to the induction proposition each everywheredeterminate automaton having a rank not exceeding n in the alphabet Ais primitively recursive.

In exactly the same way as in the basis we shall begin by considering only primitive (n + 1)-automata in the alphabet A. For them we define the functions Xn + 1 and w11+1 in a manner completely analogous to the c ons tructions in the basis. The sole difference will be the fact that in the basis it was necessary to construct the function cp 0 on the foundation of the collection of substitution formulas. Now, however, the function CfJn+i may be defined thus:

(jln+! (x) = f (Fn (g (x)));

under these conditions the function F 11 is constructed for ~. having the ranknon the basis of the induction proposition.

In exactly the same way as in the basis one may carry out the transition from primitive (n + 1)-automata to all (n + 1)-automata.

This completes the proof of Theorem 1.1.

§ 4. Completely-Automaton Functions

1. In this section we consider the second approach which was mentioned in the introduction to the present chapter: namely, we shall consider only those automata which manipulate


systems of natural numbers into natural numbers. The functions stipulated by such automata shall be called n-automaton functions in accordance with §1, subsection 1 of this chapter.

Our problern is to prove the following:

1.1. Theorem. Any completely n-automaton function is primitively recursive.

Assurne ~ is an automaton in the alphabet A, where {1, a} s::; A. Basedon the continuation theorem it may be assumed that ~ is applicable to any ward in the alphabet A. Let us introduce the nurneration of words in the alphabet A (see [§ 2]). Assurne x is the number of the ward x1ax2a ... O!Xm· As was proved in [§3.1.1.], the function F, which is defined as

F (x) = @l (~ (@3 1 (x))), (1)

is a primitively recursive function.

We shall assume that A = { a 1, a 2, ... , ak}, (k:::: 2); assume that the principal number of the ward I is a, while the principal number of the ward a is b. We shall construct the primitively recursive functions em (m:::: 1) which use the m-tuple of natural numbers x1, ... , Xm to produce the principal number (in the nurneration r of the words in the alphabet A) of the ward

The one-place function e1 is defined as

{ 81(0)=0,

81 (x+ 1) = (k+ 1) 81 (x) +ao (2)

Assurne that the function em (x1, ... , Xm) is defined. Let us define em+i (x1, ... , Xm+i):

{ 8mH(xb o o o' Xm, 0)=(k+1)8m(Xb oo o' Xm)+b, 8mH(x1, ooo' Xm, Xm+1+1)=(k+1)8m+dxb .. o, Xm, Xm+l)+ao

(3)

After this we construct a primitively recursive function -x. which uses the number (in the numeration r of the words in the alphabet A) of a natural number to produce the nurober itself. This function is constructed thus:

X

x (x) = ~ sg I !k (i, x)- a I, i=O

(4)

where fk is the function constructed in [§2.2]. And, finally, we define the new primitively recursive function:

for this function the obvious equation

holds, from which Theorem 1.1 derives.


CHAPTER IV

n-Automata, Türing Machines, and the

A. Grzegorczyk Classes

159

A. Grzegorczyk [9] considered the classification of the class .32 of primitive recursive

functions, having defined the dass ~n of primitively recursive functions which are such that CD

\R= U ~n, l}nc~"+l, but ~"=i=-c/'+1 (n2:: 0), n=O

In this chapter we introduce for consideration the classes Y1 of Türing machines whose length of calculations in operating on words of length x (x 2:: 0) is majorized by a certain function f (x) from 'fjn, and "111 of TÜring machines whose operating zone for operation on words of length x is majorized by a certain function g(x) from ~".

On the other hand, we have classes of n-automata available. The purpose of this chapter is to establish the following facts:

Al= c'1::._ 12 = UZL1+ 2 (n > 1),

A~ = g-~+2 = uu~+z = e;n+2 (n > 1).

The subscript "f" shows that a dass of arithmetic functions is being considered.

§ 1. Determination of the Grzegorczyk Classes and

Türing Machines with a Constrained Operating Time

and a Constrained Operating Zone

1. Let us present the definition of the classes ~n (n 2:: 0) given by Grzegorczyk [9].

Let us construct the sequence of functions fn (x, y) for n 2:: 0:

/ 0 (x, y)=y+1, / 1 (x, y)=x-Ty, /2 (x,'y)~=(x+1)(y+1);

for n 2:: 2:

{ /n+I(O, y)=fn(Y+1, y+1),

fn+dx+-1, Y)=fn+l(y, fn+i(x, Y)).

The class 'G" (n 2:: O) is the least class of functions which contains the functions S(x) = x + 1, U1(x, y) "'x, U2(x, y) = y, fn (x, y) and is closed relative to the operation with Superposition and bounded recursion.

Following Grzegorczyk, we shall say that the class % is closed relative to the Superposition operation if it is closed relative to the following three operations:

a) The operation of Substitution of functions. If

f(x 1, ... , Xk-l, Xk, Xk+l, .•• , X1)E% (1-<k<l), g(y,, •.. , Ym)E% (m>1),

then

f(x 1, ••• , Xk-1> g(y,, ... , Ym), Xk+f, ••• , x1)E%.


b) The operation of identification of variables. If

then

f(x1, •.• , Xj-b y, Xj+b .•. , Xh-l• y, Xhth ••• , Xz)E:Jt:.

c) The operation of substitution of a constant. If

(L(.k<. l),

then

The class ~ct· is called closed relative to the operation of bounded recurs i o n if the function f is defined as

{ f (x11 ••• , Xz, 0) = g (x" ... , Xz),

f(x1, ... , x1, x+1)=h(x1, ... , Xz, x, f(xl> ... , Xz, x)),

where

belongs to the class ft'.

2. Now let us go over to the definition of certain classes of Türing machines.

We shall consider Türing machines which operate on a two-way type and have an external output at A, including the empty letter A. (machines in the alphabet A).

As usual, a machine over the alphabet A shall be called a machinein any extension of the alphabet A.

The machine may be in any of a finite list Q = { q0, ••• , q k} (k 2: 1) of states. In this list the initial state q0 and the concluding state q k are isolated.

A s i tu a t i o n may be described thus. Assurne the machine is in state q and accepts the letter ~ with its head. Assurne PL o-sr1Sq_1 ••• s1 is a word printed on the tape and situated to the left of the head, r 1 being the minimal nurober such that all cells are empty to the left of the cell containing the letter Sr 1• Analogously, we define PR o lJ 1lJ 2 ••• lJr2 as a word printed on the tape and situated to the right of the heads. Then the situation of the Türing machine may be stipulated by the word S o PLqsPR; here P1 , PR may be empty words. Under these conditions itis assumed that Q n A = 0. The operating zone of the given situation is called the minimal zone containing the scanned cell plus all cells in which nonempty letters are printed. The length of the operating zone of the given situation is called the nurober [PLsP~, if sPR ':j:. A and the nurober [Pt+ 1, if sPR o A. Weshall denote the length of the operating zone of the situation S thus: [S3.

In order to describe an elementary action we shall make use of expressions of one of the following three types:

sq; -J>l]Lqj,

sq; _,.lJCqi (SE A, 11 E A, q; E Q, qi E Q), sq; _,.l]Rqj;

FINITE AUTOMATA AND PRIMITNELY RECURSNE FUNCTIONS 161

L, C, R indicate the fact that the readout head is moved to the left, remains in place, or is moved to the right.

Assurne the machine :t operates on the ward P. It begins to operate from the initial situation So o qoP and successively passes through the situations S1, s2, ....

This sequence of situations is called the process of manipulating the ward P; the manipulation process either terminates (at the very first appearance of the situation containing qk) or continues without limit. If the process of the manipulation of the ward P ends, then we shall characterize it by the following quantities:

[t (P)" is the duration of the process (i.e., the nurober of its situations);

[t (P)3 is the length of its operating zone. The operating zone of the process is the minimal zone encompassing the operating zones of all of its situations.

We shall consider only those Türing machines over alphabet A for which the process of manipulating any ward P!JA ends, a ward in A being obtained as the result. Let us define the classes 31, n:::: 0 of machines over the alphabetAsuch that for any machine t from the class 31 one can find a one-place function f E -on, which is such that

(PQA).

Analogously, we define the classes '1l1 (n:::: 0) of Türing machines over the alphabet A such that for any machine t from the class U1 one can find a one-place function f E '0", which is such that

(PQA).

By analogy with the definition of the classes A~ of fully n-automaton functions one can introduce the classes of 3~ and "11~ of functions that are calculable by means of Türing machines from 3~ (from 91~), correspondingly). The class of n-automata over the alphabet A shall be denoted by A1.

§ 2. Basic Theorems

1. Let us begin by dealing with the relationships between 31 and U1. Let us prove the following:

1.1. Theorem . .31~'111 (n > 3).

We shall show first that

51 s;; 911. (1)

Assurne that the Türing machine t over the alphabet A is such that t E 31. This means that one can find a function f (x) E '0", such that

(PQA).

But since in the process of operating on the ward P the machine t cannot scan more than f ([P8) cells in the course of making no more than f ([P8) steps, it follows that

(PQA).

Further, the function x + y belongs to w,n (n :::: 1). From this it follows that if f (x) E o", then g (x) = f (x) + x E 0" (n :::: 1) also. Thus, the function g (x) E ö" such that

(2)

(3)


(PQA) (4)

has been constructed. The relation (1) derives from this.

Let us go over to the proof of the opposite

(n>3). (5)

Assurne that (4) holds. This means that the Türing machine completes its Operation on the word P without departing from a zone of length g([P 8). Let us calculate the maximum nurober of situations through which the machine might pass during the process of manipulating the word P. These situations can be described by words of the form

The inequality

holds. It can easily be seen that the nurober of different situations for a length g ([P0) of the operating band does not exceed

where (k + 1) is the nurober of plates of the Türing machine; r is the nurober of different letters of the alphabet B (A ~ B). From this it follows that

(PQA),

where

f (x) = (k+ i)·g(x) ·rg(x).

But from [9] it is well known that h1 (x, y) == xy and h2(x, y} == xY belong to ~n (n 2: 3). Therefore, if g (x) E ~n, then f (x) E ~n (n 2: 3} also. And this means that (5} holds. Thus, the proof of Theorem 1.1. has been completed.

2. Now we shall consider one auxiliary (but important} lemma:

2.1. Lemma. Assurne the function j(x} is a completely m-automaton function (m 2: 1}. Assume, moreover, that !t is a Türing machine over the alphabet A and is such that !t is applicable to any word PQA and

Then one can construct an m-automaton ~ over the alphabet A, which is such that

~ (P) o !t (P) (PQA). (1)

We shall assume that !t is a Türing machine in the alphabet B (Ac B) which has the separate states Q; here BnQ= 0, and the letters a and I aresuchthat .a, I~BUQ. Further, we shall assume that B = {ß1, •.• , M (r 2: 1), while B is the alphabet of twins.

We shall construct the automaton ~ in the form of the composition


The automaton ~~ performs the following transformation of the word P:

here rr is an m-automaton (m 2: 1) corresponding to the function f. The automaton ~I may be constructed thus:

The description of the 1-automaton ~;{> may be found in [I. §2.12], that of the 0-automaton ~A may be found in [L § 2.8], and that of the 0-automaton mAU{q0}. q0 may be found in [!.§2.4].

Further, the automaton 2!1 is constructed from these automata on the basis of the union theorem [II. §4.1.1]. From this theorem it follows that the rank of the automaton ~~ is equal to max(m, 1); since m 2: 1, the rank of ~I1 is equal to m.

Let us now consi.der the 0-automaton ~r, which performs the elementary action of a Türing machine !:. Assurne !: is a Türing machine in the alphabet B, which has the set of states Q. Assurne that the machine !:, being in the active SituationS, performs one elementary action and transits to the situation R. The 0-automaton ~T in the alphabet B U Q operates as follows:

If S is a passive situation (i.e,, a situation containing a concluding state), then

The construction of the automaton ~T presents no difficulty, and we shall not dwell on it.

Now let us construct the 1-automaton Wz. Its initial state is A. Making use of the brauehing theorem [II. §2.1.1], we construct lllz thus:

Here R is a word in the alphabet B U Q. The automaton 21.2 is applied to words of the form [Puqoarr ([P0). Assurne

(l )> 0).

Then the automaton illz manipulates the word

thus:

illz: A 1- ~ 1 sz f-- ~l-ISl-ISl 1- · · · 1- ~zSz · · · SI-! SI 1-51

1- ~tS!S2 ... SI-I SI 1- qoqoP 1- a qoP I-I SI l-j Sz 1- ... 1-, s~(l) 1- J[l!; (P) Rz (Rb RzQ {~}).


Here qoP, 81, 82, ••• , 8q is the process of manipulating the word P by the Türing machine ~ (q<~ (l)). The inequality

q<~ (l)

derives from the fact that from the condition of the theorem,

[~(P)"<~(l).

8q isapassive situation; Sq+i-2-Sq (O<i<~(l)-q). Now the choice of the automata l213-1ll5 is obvious. As the automaton 1ll3 let us choose the 0-automaton from [I. §2.14]

We obtain

Now it remains only for the words R1, R2 tobe "removed." Forthis purpose let us use the 0-automaton m in the alphabet B, which can be described as follows: its initial state is A,

(

1-->A ~ __,. ~

~: ~'Y] -~s *__,.A

* ~ __,. ;\

(sEB"{A}) (s, 'YJ E B)

(s E B).

It is clear that if ~r, is chosen tobe the automaton }]), then

Then we also choose ~~5 and IJl1 tobe the 1-automaton .\2t1l (from [I. §2.12]). As the automaton \!I s we again use ~ · Then we obtain

Thus, an m -automaton W, has been constructed for which (1) is fulfilled.

3. Our next problern is to show that the following theorem holds:

(n>1).

We need to prove that for each n-automaton W over the alphabet A one can construct a Türing machine !: over the alphabet A, which is such that

~ (P) o !; (P) (PQA).

Under these conditions there exists a function f (x) E ~"+2 , which is such that

[~ (P)3 <f ([Pfl).

We shall assume that A ~ A, and also that if ~ is an automaton in the alphabet B. it follows that A. El: E •

We shall show that furthermore the function f may be made nondecreasing.

We shall prove this theorem by induction with respect to n.


The Basis, n = 0. Tobegin with, let us consider the case in which 21 is a primitive 0-automaton. Let us describe how the corresponding Türing machirre operates. At first the machirre determines whether the word that it is manipulating is empty. If it scans an empty cell in the initial situation, then the original word is empty. It writes * into the scanned cell. If it scans a nonempty cell in the initial Situation, then it moves its head (without altering the word written on the tape) until it finds the first empty cell, and then writes * into this cell. Then the machirre returns to the cell into which it wrote the first letter * of the word P. After this, cyclic operation of the machirre begins. The machirre scans the next letter of the original word (at the beginning this letterwas the first one). Moreover, it remembers the state in which the automaton is. The machirre erases the letter in the scanned cell, remembers the state into which the automaton W transits, and, if the word which the automaton W is about to print is not empty, it moves the head to the right until it encounters the first empty cell. Beginning with this cell, the machirre writes into the subsequent cells that word which the automaton W prints. After this the machirre moves the head in such a way that it scans the first nonempty cell to the left. If the word that the automaton W must print is empty, then the head simply moves one step to the right. The operation is repeated from the beginning of the cycle. The machirre goes out of this cycle for the case in which it reveals * in the scanned cell. It remembers this fact, and, in performing its operation according to the cycle for the last time, it

· returns the head to the first nonempty cell to the left and stops.

Let us note that the machine constructed is such that its head moves only one cell to the left of the first letter of the original word during the process of its operation. We shall call this property of the machine property A.

It is obvious that for the case in which W is a nonprimitive automaton and the relation

holds, where wi (1 :Si :S k) arealready primitive 0-automata, the machine does not stop after it has operated for 2!1 but continues to operate for the automaton Wz and so forth. The machirre constructed in this way also has the property A.

It is not difficult to see that for the case of a primitive 0-automaton

Simple operations show that this same inequality also holds for the case in which W is a nonprimitive automaton. The function in the right side of the inequality is nondecreasing.

On the other hand, the functions xy and x + y belang to the class '#, 2• Therefore the theorem for the case n = 0 has been proved.

The Ind u ction Step. Let us assume that the theorem holds for n-automata. Let us show that in this case it also holds for (n + 1)-automata. Tobegin with, we consider the case in which W isaprimitive (n + 1)-automaton (n:::: 0). Once again let us describe the operation of the machine ~ on the word PS1A.

The initial stage of operation of the machine ~ is exactly the same as it is in the basis: namely, it attaches * to the word P at the right. In the cells which follow the one occupied by *, the machine writes in the word A.aS0 (S 0 is the initial state of the automaton 21; we assume that a Ef IJ). After this that portion of the operation of this machine begins which will be repeated many times. The head returns to the beginning of the word, the machine remembers the first letter on the left, erases it, and writes this letter into the cell in which the letter a was written previously. After this the control is transferred to that portion of the machine which simulates the operation of the automaton W". According to the inductive assumption,


this portion ofthe machine satisfies the property A. When this operation ends, the following word will be written on tape:

Now the machine proceeds thus: first it obtains the word

on the tape. Mter this it shüts the word S1 until the empty cells within the word vanish.

As a result it obtains

Then it attaches the word A.O!S1 to the word obtained on the tape; as a result the word

is formed on the tape. Finally, the machine deletes the first entry of the letter 0! and again shifts the word:

(1)

(2)

(3)

(4)

(5)

For all of these operations the machine, as can easily be seen, is not required to depart beyond the zone required to write the Iongest of the words given in (1)-(5).

Mter this a word of the same type as that with which the description of the cyclic part of the operation began turns outtobe on the tape. The cycle is repeated until the machine erases the letter * which follows the original word P. After this the machine performs the next cycle; however, at the end of it, it merely shifts the newly obtained word while "removing11 the empty cells; now it stops.

Let us estimate the operating zone required by the machine ~ in order to perform the operation described.

Since the automaton ~ has the rank n, it follows that according to the induction proposition there exist a Turing machine ~ and a nondecreasing function 1 (x) E ~n such that

f (P) a ~ (P) (PQB U {*}),

Let us define the primitively recursive function g thus:

g(Ü}=s0 +2, g (x+ 1) = f(g (x) + 1).

This function is nondecreasing, since the function j (x) does not decrease. This function obviously belongs to the class ~n+1, since f(x) E ~n and since one recursion applied to the functions ~n may yield only a function from ~n+1.

Before the machine ~ began to operate on the word P, an operating zone of length [Pa was used to write the original word. Before the beginning of cyclic operation a zone of length

was used, where s 0 is the length of the initial state.


Assurne that the function h(x) yie1ds the 1ength of the operating zone used after the x-th cyc1e has been carried out (O<x< [Pa+ 1). In all, the machine carries out [Pa+ 1 cyc1es. Before the beginning of cyclic operation(i.e., for x = 0),

h (0) = [P 8 +s0 +3= [Jß + 1 +g(O).

Further,

h (x+ 1)-<:h (xH- 2g (x+ 1) +2.

Since the function g(x) is nondecreasing, it follows that

X X

h(x)<[Pa+1+2 2J (g(i)+2)= [Pa+ 1+4(x+1)+2 2J g(i). i=O i=O

Since the Summationoperation does not extend beyond the c1ass ~n (n :::-:: 2), while g(x)E~n+l,

it follows that the function appearing in the right side of the inequality be1ongs to ~"H. Finally, in order to calcu1ate I2I (P) the Türing machine :t uses an operating zone which is bounded as follows:

[PiJ+i

[:t(P)3 <:C1 2J g(i)+C2[Pa+Ca, i=O

where c1, C2, C3 are certain constants determined by the automaton 2I. Once again we have constructed a nondecreasing function. Using a method analogous to the one used in the basis, one can go over from primitive (n + 1)-automata to nonprimitive ones. This ends the proof of Theorem 3.1.

4. Let us now prove a series of auxiliary statements.

4.1. Lemma on Superposition. The c1ass of arithmetic functions '11~ i s c 1 o s e d r e 1 a t i v e t o t h e s u p e r p o s i t i o n o p e r a t i o n .

In accordance with the definitions of the Superposition operation (§ 1, subsection 1) the proof breaks down into three points.

a. Assurne that the functions j(x1, ••• , xk-i• xk, xk+1• ... , xz) (1:::; k:::; l) and g(y1, ••• , Ym) be1ong to the c1ass UU~ and are ca1cu1ated on the machines % and @l , respective1y.

Let us show how to construct a Turing machine ~ from UZL~, which calculates the function

h (xl, .. ·, Xk-b Yb · · ·, Ym, Xk+b • • ·, Xz) = f (xb · · ·, Xk-1, g (yb ... , Ym), Xk+b ••• , Xz).

Before the beginning of operation the word

is written on the tape of the machine. In the beginning the machine ~ operates in almost the same way as the machine @l in manipu1ating the word y1a ... O!Ym. The differences reside in the following: for the operation of @l empty cells are situated to the 1eft and right of y 1a •.. ay m;

for the machine ~ this is not so. Therefore, the machine ~ starts by noting the boundaries of the word y1a ... O!Ym (i.e., it writes the word x1a ... axk-1ßy1a ... aymyxk+ia ... ax1 on the tape (we shal1 assume that ß and y do not be1ong to the alphabets of the machines % and @l).

In the process of operation the machine S7J, in simulating the activity of @l, constantly monitors it, and in the case in which @l attempts to write some 1etter in a cell containing ß, it moves the word ending in the 1etter ß one cell to the left, thereby freeing one cell for the


operation of @l. Analogously, the machine moves the word beginning with the letter y one cell to the right when this is necessary. It is obvious that the operating zone required for these calculations does not exceed the following magnitude:

z + g ([y1a ... ay?,.),

where g is a function belanging to ~n suchthat

m

[@l (y!a. ·. CGYm)a <g ( ~ Yi +m-"'--1), i=l

while z = [XjCG ••• r:XXn-ICGYjCG ••• CGYmCGXß+jCG ••• axr. Since [YICG ... ay?,. < z' the zone may be bounded by the function of the following type:

It is clear that the function h1 belongs to the class ~n.

Then the machine s:J, after the calculation of g(y 1, ••• , y m) has been completed, shifts the word that has been obtained so as to achieve a situation in which the word

is formed on the tape. Now the machine Sj operates on this word in exactly the same way as the machine ~; as a result, h(x1, ... , xk-t• y1, ••• , Ym• xk+t• ... , xz) is calculated. The zone which would be required for this calculation does not exceed

h (z) = T(ii1 (z));

here f is a function from ~n such that

l

[%(x1a ... axz) 3 <f(~ x;+l-'-1). i=l

It can be understood that h belongs to the class ~n, which proves point a) of the lemma.

We shall not consider points b) and c) by virtue of their obvious nature.

4.2. Lemma on Bounded Recursion. The class of arithmetic functions U~ is closed relative to the operation of bounded recursion.

Assurne the function f (x1, ... , xz , x) is obtained by the operation of bounded recursion from the functions g(x1, ••• , xz), h(x1, ... , xz, x, y), j(x1, ... , xz, x) in accordance with [§1.1]. Assurne that the functions g, h, j belang to the class u;p, the functions g, h, T from ~n bounding the zones required for calculating the functions g, h, j on the machines @l, Sj, 3, respectively.

As the function j (x1, ... , xz , x) one may always choose a nondecreasing function. Actually, from the function j (x1, ... , xz, x) one can construct a function of the following type:

Since the summation operation does not extend beyond the limits of the class ~n (n 2:: 2) (see [9]), the function j' E ~n. It is obvious that the function j' is nondecreasing and

j(x11 ••• ,xz,x)<j'(x11 ••• ,xz,x).

Therefore,furtheron weshall assume in proving this lemma that the function j is nondecreasing.


Let us construct the machine ~ from 91'({,, which calculates the function f. Before the beginning of operation, the word

is written on the tape of the machine. Assurne z is the length of this word:

l

Z= 2j Xi+x+ l. i=l

First we double the word written on the tape; i.e., we obtain the word

Under these conditions ß is a letter that does not belong to the alphabets of the machines @!, ~. 3. For this purpose we require an operating zone having a length 2z + 2. After this we calculate j (x1, ••• , xz , x) in such a way as to obtain the word

x1rx ••• rxx1rxxßj (x~> .•. , x1, x)

on the tape. For this we require a zone no larger than

2z+ 1 + ](z).

Then we construct the operation of the machine ~ in such a way as to obtain the word

x1rx ... rxx1rxxßOrxOrxP0, 0ßOrx I rxP0o1ß ••• ßOrxj (xh ••• , Xz, x) rxPo, i(x1, ... , x1, x)ß I rx0rxP1, 0ß ...

• • • ßxrxOrxP x, oß ... ßxaj (xit ... , Xz, x) rxPx, J<x1o ... , x1, x),

P 1, k a x1rx ... rxx1rxirxk (O<;::i<;::x, O<;::k<;::j (xt> ... , Xz, x))

on the tape. Obviously, in order to construct this word we require a zone which does not exceed

[(x+ 1) j (xh ... , x1, x) + 1]-(z+2j (xit ... , x1, x) +x+4) <!(z+ 1) · i (z) + 1] [2z+2i (z) +4],

where i (z) = j (z, ... , z, z).

Since the function j is nondecreasing, while z :::: xi, (1 :::s i :::s l ), z :::: x, it follows that

j(xit •.. , Xz, x)<i(z).

After this the machine ~ applies the machine ~ to each word Pi,k. As a result the word

x1rx ••• rxxzrxxßOrxOrxh (x1, ••• , x1, 0, 0) ß Orx I rxh (xh ... , Xz, 0, 1) ß ... ßOrxj (xh ... , Xz, x)

rxh (x1, ••• , x1, 0, j (xh ... , Xz, x)) ß ... ßxrxOa.h (xit ... , x1, x, 0) ß ... • • • ßxrxj (x17 ••• , Xz, x) ah (x~> ... , Xz, x, j (x1, ••• , x1, x))

is obtained. For this calculation we require a zone no larger than

This zone may be bounded as follows:

(z + 1) · i (z) · (h (2z + i (z)) + 2z + 3] + z + 1.


Now the roachine ~ transforros the word by applying the roachine @I to the piece x1a ••. axz. As a result the word

g (xh ... , Xz) axßOaOah (x11 ••• , Xz, 0, 0) ß Oajah(x11 ••• ,xz,O, 1)ß ..• ßxaj(x11 ••• ,xz,X)ct

h(x17 ••• ,xz,x,j(x1, ••• ,xz,x))

turns outtobe written on the tape. The zone required for the calculations indicated is bounded as follows:

g(z) +(z+i)·i (z) [h (2z+i (z)) +2z+3J +z+ 1.

For the subsequent calculations we already require no expansion of the zone. The roachine ~ operates further in the following way. It clarifies whether or not it is true that x = 0. If x = 0, then the calculated value is equal to g(x1, ... , x1 ). The roachine leaves this value on the tape, while it erases everything else.

Let us now consider the case in which x > 0. Let us show how the roachine, knowing f (x1, ... , xz, k) and k, calculates f (x1, ••• , x z, k + 1). For k = 0, f (x1, ... , x 1, 0); as we have already said - this is g(x1, ... , x1). This nurober written on the tape is tagged in soroe way by the roachine so that the roachine can find it. Moreover, it also tags the nurober x, the tag denoting that the case k = 0 is being considered. Then the roachine seeks an entry of the word ßPaQaRß (P, Q, RQq), suchthat P = 0, Q = g(x1, ... , xz). Such an entry can certainly be found, since

f (xt. ... , x 1, x)-<. j (xt. ... , x1, x),

while the word written on the tape is such that for any P ::::; x, Q ::::; j (x1, ... , xz , x) an entry of the word ßPaQaRß can be found in it.

Thus, the necessary entry has been found. Then

R = h (xt. •.. , x 1, 0, g (xt. ••. , x1)) = f (x1, ••• , x 1, 1).

The roachine tags this ward (and erases the tag an the preceding value of the function j). Furtherroore, it shifts the tag an the word x, noting that the calculated value corresponds to k = 1. Then it checks whether or not k and x coincided. If k < x, then the operation described above is repeated. If k = x, the roachine erases everything on the tape with the exception of the tagged word.

Considering the constraints on the length of the operating zone which we gave during the description of the operation of the roachine ~. it can easily be seen that

m (XttX •,. CGXzCGX) 3 ~}(z),

where f{z) E ~n. Thus, the proof of the leroroa has been coropleted.

5. Let us consider the sequence of functions fn (n :::::: 0) froro [§1.1]. The description of the function / 11 +i (n:::::: 2) [§1.1(4)] is not priroitively recursive (fn+ 1 is defined by roeans of so-called recursion with insertions; see [7]). Let us give a priroitively recursive description of f n +1• Note initially that the functions j 0, j 1, f2 are obviously priroitively recursive. Let us now construct the auxiliary function gn+i (n:::::: 2):

{ gnH (0, Y) =Y, gn+1 (x+ 1, y) = fn (gn+l (x, Y)+ 1, gn+1 (x, y) + 1).

(1)


It is not difficult to see that the function fn +i (n ::::: 2) may be expressed in terms of the function gn+1 thus:

fn+! (x, Y) = gn+t (2"', y).

Then we prove primitive recursiveness of the functions gn+1 and f n+ 1 by induction with respect to n on the basis of the fact that since h is primitively recursive, g3 is also primitively recursive.

5.1. One may construct a Türing machine from §~,such that it calculates the function fa(x, y) (i.e., h(x,y) E§~).

First let us show that the function g3(x, y) belongs to §~. Actually, this function is defined as follows in accordance with (1):

ga(O, Y)=y, ga (x+ 1, y) = (g3 (x, y) +2)2 •

From Theorem 4.4. of [9] it follows that

Thus, obviously,

y+2< 2Y+2.

Therefore,

22X ga (x, Y) < 2(y+ 2)o2 •

The funr:tion appearing in the right side of the latter inequality belongs to A~ , since A~ belong to the functions N x, x • y, x + y and this class is closed relative to Substitution in accordance with [III. § 1.3.1].

22X

Since, as proved in Theorem 3.1, A$ s 3$+2 , it follows that the function 2<v+ 2J2 be-longs to §~. Thus, applying the lemma on bounded recursion 4.2, we find that g3 (x, y) E §~. Further, the function 2x belongs to A~, and therefore it belongs to §~. Applying the lemma on Superposition 4.1, we find that /3 (x, y) E §~.

6. Weshall say that the function f (x 1 , ... , xz) majorizes the function g (x1, ••• , xz) (or the function g is majorized by the function f) if the function f is nondecreasing and for any collection of values of the variables x1, ... , xz the inequality

g(x1 , ••• ,xt)<.f(xl, ... ,;"t)

holds.

Firstlet us prove severallemmas.

6.1. Lemma. Assurne that the functions h 1 (x 1, ... , xk_ 1, xk, xk+i' ... , Xz) and h 2 (y 1 , ... , Ym) are such that for them h1 and il2 from A$, exist which majorize them. Then for the function h obtained by substituting the function h 2 into the function h 1 in place of the variable xk, there exists in A$ a function h that majorizes it.


Thus,

h (x!t ... , Xn- 1 , Y~t ... , Ym, Xn+l, ... , xl) =ht (xt, ... , Xn-b h2 (Yt. · .. , Ym), Xk+!t ... , x1);

h(x!t ... ,Xk-~tY~o ... ,ym,Xk+t> ... ,xl)<.ht(Xt> ... ,Xk-~th2 (Yt, ... ,ym),xk+l• ... ,x1)<.

<.ht (xh ... , Xk-!t h2 (Yt> ... , Ym), Xk+l, ••. , Xz).

Since the class A~ is closed relative to Superposition of functions, it follows that a function h has been constructed that belongs to A$ and majorizes the function h.

6.2. Lemma. Assurne that the function h 1 is suchthat a function h 1 f r o m A~ e x i s t s f o r i t w h i c h m a j o r i z e s i t. T h e n f o r t h e f u n c t i o n h , obtained from h 1 by the operation of identification of variables, there e x i s t s in A~ a f u n c t i o n h t hat m a j o r i z es i t.

The function h is derived from the function i11 by identification of those same variables that were identified in the transition from the function h1 to the function h.

6.3. Lemma. Assurne the function h 1 is suchthat for it there e x i s t s a f u n c t i o n h 1 f r o m A~, w h i c h m a j o r i z e s i t. T h e n f o r t h e f u n c -tion h obtained from h 1 _Ey the operation of substitution of 0 there e x i s t s i n A~ a f u n c t i o n h t hat m a j o r i z e s i t •

The function h is obtained from the function n1 by substituting 0 in place of the very same variable as in the transition from the function h1 to the function h.

6.4. Lemma. Assurne that the functions h 1 (x 1 , ... , xz), h 2 (x 1 , ... ,

xz, x, y). h 3 (x 1 , ... , xz, x) aresuchthat for them the functions i11,h2, h3 ,

r e s p e c t i v e 1 y , f r o m A~, e x i s t w h i c h m a j o r i z e t h e m . T h e n f o r t h e f u n c -tion h obtained from the function h 1, h 2 , h 3 by the operation of bounded r e c u r s i o n ( h ( x 1 , ••• , x z , x ) ::::; h 3 ( x 1 , ••• , x z , x) ) t h e r e e x i s t s i n A~ a function h that majorizes it.

It is obvious that h is majorized by the function hso Let us now prove the following theorem.

6 . 5 . Theorem • a) F o r e a c h f u n c t i o n g (x 1 , o •• , x z ) f r o m ~n+z o n e c a n f i n d a f u n c t i o n f ( x 1t ... • x z ) f r o m A~ ( n 2:: 1 } t h a t m a j o r i z e s i t ;

b) Y1+2 =A1;

c) fn+2 EA~.

We shall carry out the proof of the theorem by induction with respect to n.

The b a s i s , n == 1. By analogy with the definition from [III. § 1.2] one can introduce the concept of the rank r of a function from ~3 • Let us prove statement a) of the theorem (for the case n == 1) by induction with respect to r o

The b a s i s, r == 1.

It is required to show that the original function can be majorized by certain functions from A~. For the functions

f(x)=x+1, U1 (x, y)=x, U2 (x, y)=y

this is obvious:


The function X+ y belongs to A~l - this is the automaton QW, a), a from [II. §3.5].

Let us deal with the function f 3(x, y). This function is defined as follows:

{ la(O, y)c=(y-j--2)2 ,

Ia (x-1-- 1, y) =Ia (x, Ia (x, Y)).

If we now define the function g3 according to 5.1 as follows:

( ga(O,y)=y, l ga(x-j--1, y)=(ga(x, y)-j--2)2 ,

it follows that the equation

Ia (x, Y) = ga (2x, y)

holds. 22X

In §5.1 it was proved that the function 2<y+Zl·Z majorizes g3(x, y) and belongs to the class A~. It is not difficult to see that this function is nondecreasing. Further,

where the function appearing in the right side of the inequality belongs to A~.

The induction step (in the basis). This step is carried out on the basis of Lemmas 6.1-6.4. Thus, for the basis (n = 1) the statement a) of the theorem has been proved. But then, taking account of Lemma 2.1, it can easily be seen that

However, from Theorem 3.1 it follows that

§i = A1_ a11d §~ = A~.

Since Ia (x, y) E §~ (see 5.1) 1t follows that Ia (x, y) E A~.

The Ind uction Step. Let us begin by proving statement c) of the theorem. In accordance with the inductive proposition fn+2 (x, y) E A~ (n 2: 1). The function g 11-1-ß(X, y), which is defined according to [5.(1)], belongs to the class A~+ 1 (see [III. § 1.3.2]). The function y = 2x belongs to any class A~ (i 2: 1). Since A~+ 1 is closed relative to the Substitution [III. § 1.3.1], it follows that fn+a (x, y) E A~+ 1 •

Now in order to prove statement a) of the theorem we again carry out reasoning by induction (induction with respect to the rank r).

The b a s i s, r = 1. Only the case of the function f 11 + 3(x, y) is of interest. Thus, as proved above, the function f 11 + 3(x, y) belongs to the class A$+1 • Therefore, the majorizing function fn+3(x, y) may be constructed as follows:

T h e In du c t i o n S t e p . The inductive step is conducted in exactly the same way as in the basis. Reasoning as in the basis, we obtain

§';t+3 =AAn+1.

With this the proof of the theorem ends. Let us go over to the next statement.


6.6. ~n s§~.

Actually, according to 6.5, fn (x, Y) E §~, other original functions of the class 'On obviously likewise belong to §~. Moreover, from Lemmas 6.1-6.4 it follows that §~ is closed relative to the operations of superposition and bounded recursion. From this we obtain 6.6.

7. The next problern of this section is to show that A$+1 c ~n+3 (n :::: O).

For this purpose we carry out the arithmetization of the automata from A'l,+1 , using ~n+3 as the sole resource.

Let us modify the definition of a primitively recursive automaton given in [III. §3.1] as follows.

An n-automaton '2l is an m- r e c ur s i v e a u t o m a t o n if the function f (x) defined by the relationship

f (x) ~ @3 (W (@3 1 (x))),

is a function from ~m.

Now we have the possibility of formulating such a theorem.

7.1. Theorem. Any everywhere-determinate (n + 1)-automaton in the alphabet A is an (n + 3)-recursive automation, n:::: 0.

For purposes of the proof we will have to show that certain functions constructed in [III. §2] and [III. §3] belong to ~3 , i.e., they are elementary in the Kalmar sense.

First, the function fk (x, y) defined in [III. §2.2(2)] is obviously elementary. But then the function f (x) from [III. §3.2(1)] is likewise elementary, since the class ~3 is closed relative to the operation of bounded summation. Further, the function g(x) from [III. §3.2(2)] likewise belongs to '03 • It can easily be shown that the function r(y) defined in [III. §3.2(3)] is likewise elementary. And, finally, the elementarity of the functions h(x, y), b(x, y), and a(x) from [III. §3.2(4)-(8)] is obvious. Let us now show that the functions Xo (defined according to [III. §3.3(1)] and x1 are elementary, just as are the functions w0 (from [III. §3.3(2)]) and w1, while the functions x i and w i belong to the class 'Gi+2 (i :::: 2). Actually, the function x 0 uses the pair of natural numbers (x, y) to produce the principal quasi nurober of the state into which the primitive 0-automaton W transfers due to the action of the word having the nurober (y/(k + 2) r (yf~x]. Therefore,

Xo (x,y) < C 1,

where c1 is a constant which does not depend on x, y.

Further, w0(x, y) yields the principal quasi nurober of the word that the automaton W

prints due to the action of the word having the nurober [y/(k+2)'<Y>~x]. If the presented word P has the length x, then it is obvious that the automaton prints a word whose length does not exceed

But then

It turns out to be possible to use bounded recursion to define the functions xo and w0 defined in [§3.3(1)] and [§3.3(2)] by means of recursions. Therefore, Xo E '03 and w0 E ~3•


Let us go on. The function F0, as follows from its definition [III. §3.3(3)] and what has been said above, is likewise elementary. We now go over to the functions x1 and w1•

Due to the action of the word g1 ••• gx (x > 0) the primitive automaton l.li transits from the initial state S0 to the state Sx. Let us estimate [S~.

For a certain constant C3 the following inequality is fulfilled for the automaton ~ :

Let us define the function q (x) thus:

{ q (0) = [S~, q(x+ 1) = (C3 +2) (q (x) + 1).

Then

(S~<;:q (x). (1)

On the other hand, by induction it can be proved that

(2).

Further, the length of the word that the automaton W prints when the words g1g2 ••• gx (x > 0) are presented to it does not exceed the quantity

X X

~ q (i)<;: ~ ([S~ (Ca+2)i+ (Ca+2)i+l)<;:x·[S~ ((C3 +2t+ (C3 +2)x+I). i= 1 i= 1

(3)

Let us assume that

(4)

If the length of the word does not exceed z, then the principal quasi number of this word does not exceed (k + 2!2 +1•

Let us now return to the definition of the function x1• This function is defined by means of recursion via the elementary functions fk+ 1(x, y), r(y), h(x, y), etc., and likewise via the functions cp 1• However, the function cp 1 is likewise elementary, since

1P1 (x) = f (F0 (g (x))),

while the elementarity of the functions f, g, and F0 has been proved.

Further, the function x1 may be bounded by an elementary function as follows by virtue of (1)-(2):

Thus, x1 is elementary.

Now concerning w1• According to the definition [III. §3.3(2)], w1 can be defined in terms of elementary functions by means of recursion. On the other hand, w1 may be bounded by an elementary function in the following way by virtue of (3)-(4):

wl(x, y)<;:(k+2)p(x)tt.

Therefore, w1 is elementary; then F 1 is also elementary.


Reasoning inductively, we then go over to automata having higher ranks. Let us consider the functions Xn+1 and w11+1• The function Xn+i can be defined by recursion in terms of elementary functions and the function <Pn+1 that belongs, according to the induction assumption, to the class ön+z; then the function Xn+i will belang to the class ö"+3 • Estimating the function W11 +1 as in the basis, one can bound it by means of a function from 0n+3 • Therefore, w11 +1 and thus F 11 +1 belongs to the class en+a too.

With this the proof of Theorem 7.1 ends.

Now Iet us prove the following:

7.2. Theorem. Any completely m-automaton function belongs to the Grzegorczyk class lt"+ 2 (m 2: 1).

The proof of this theorem is analogaus to the proof of the theorem [III. §4.1.1]. It is merely necessary to show that the functions () i (i 2: 1) and x constructed there are elementary (belang to Ö3).

Elementarity of the function x derives directly from its definition [III. §4.1(4)], it being necessary to consider the fact that, as shown above, the functions fk (i, x) are elementary.

In order to prove elementarity of the function ()i defined in [III. §4.1(3)], it is necessary to construct the elementary function that majorizes it.

It is not difficult to check the fact that

the elementarity of e1 derives from this.

In exactly the same way one can immediately write the formula for calculating () m. Actually

Thus, Theorem 7.2 has been proved.

8. Let us make several remarks. Basedon the Grzegorczyk result from [9] to the effect that an (s + 1) -place function (s 2: 0) exists from the class ön+l (n 2: 2), which is universal in the class of s-place functions from 'Ibn, one can prove the following statement.

8.1. AssurneAis analphabetsuchthat aEA, while öEfA. One can construct an (n + 1)automaton m<n+il in the alphabet AU {ö} (n 2: 0) which is such that regardless of the nature of the n-automaton ~<n>, one can find a ward ~ ~<n> l ( ~ ~<n> j Q {a}), in the alphabet A such that the following equation is valid for it:

m<n+o ( ~ ~<n> j öP) ~ ~<n> (P) (PQA).

The automaton m is called an ( n + 1) - a u t o m a t o n , w h i c h i s u n i v e r s a I i n t h e class of n-automata.

Using the arithmetization presented above, it is easy to obtain the following statement:

8.2. Regardless of the nature of the n-automaton W (n 2: 1) in the alphabet A, which is applicable to any ward P in A, the function f (x) defined as


f (x) = max [~ (P)0 , [P0=x

belongs to the class ~'!+2 •

Then it may be shown that:

8.3. For each n 2: 0 one can construct an (n+l)-automaton ~<n+o in the alphabet A, which is such that no n-automaton )S<nl over the alphabet A exists for it such that the relationship

~<n+ll (P) ~ m<nl (P) (PQA)

holds.

Literature Cited

177

1. V. M. Glushkov, "Abstract theory of automata," Uspekhi Matern. Nauk, Vol. 16, No. 5 (101), pp. 3-62 (1961).

2. S. C. Kleeny, Introduction to Mathematics [Russian translation], IL, Moscow (195 7). 3. V. A. Kozmidiadi, "On sets which are decidable and enumerable by automata," Dokl.

Akad. Nauk SSSR, 142(5):1005-1006 (1962). 4. V. A. Kozmidiadi, "On sets which are enumerable and decidable by automata, 11 in: Prob

lems in Logic, Philosophy Institute, Academy of Seiences of the USSR (1963), pp. 102-115. 5. A. A. Markov, Theory of Algorithms, Transactions of the V. A. Steklov Mathematics

Institute, Academy of Seiences ofthe USSR, Vol. 42 (1964). 6. V. A. Trakhtenbrot, Türing Calculations with Logarithmic Delay, Algebra and Logic,

3(4):33-48 (1964). 7. R. Peter, Recursive Functions [Russian translation], IL, Moscow (1954). 8. V. S. Chernyavskii,On a Certain Class of Normal Markov Algorithms, in: Logic Inves

tigations, Philosophy Institute, Academy of Seiences of the USSR (1959), pp. 263-299. 9. A. Grzegorczyk, "Some classes of recursive functions," Rozprawy Matematyczne

(Warsaw), Vol. 4 (1953).

GENERAL LINEAR AUTOMAT At

A. A. Muchnik Moscow

Introduction

Many problems in the theory of control systems reduce to the objects considered in the present paper - general linear automata. The necessity of studying them is manifested most clearly in the problern of reducing finite probabilistic automata [12, 13, 15, 19] which are an important particular case of general linear automata. On the other hand, many properties which apply to finite deterministic automata can be generalized successfully for the case of general linear automata, their proof turning outtobe sometimes simpler than well-known proofs of analogaus theorems for finite automata.

§ 1. The Definition of General Linear Automata

A general linear automaton \!I over the field P is called the system

\!I= (V, M, 2":, Z, IJl, 'ljJ),

where V and Z arelinear spaces over the field P; M is a certain set of points in V~ M <;:;:;V;

L = {a1, a2 , ••• , a"} is an input alphabet; IJlcri (;) = IJl (;, ai) is a linear transformation of the ... -+ ...,.

space V which maps the set M in itself and depends on the input symbol ai, z='ljlcri (v) = 'ljJ(v, ai)

is a linear operator which maps V in Z. The points (vectors) v of the set M are called s ta te s of the general linear automaton W while the vectors z = lf!(v, ai), where ; E M are called out p u t s of W. The entire system operates in time that is considered to be discrete:

tc=Ü, 1, 2, ...

If w is in the state v(t) at time t and a signal a = a(t) is applied at that time, then the output of the general linear automaton W at time t is

-;(t)='ljJcr(t)[;(t)]='ljJ[;(t), a(t)], (1)

while the state of \!I at the next instant in time is

(2)

t Original article submitted October 26, 1967.

179

180 A. A. MUCHNIK

As we see, this definition is analogous to the definition of a finite deterministic Mealy automaton [4], while generallinear automata are deterministic Mealy automata which arefinite or infinite (this depends on the power of ~, M, and V).

In i t i a 1 g e n er a 1 li n e a r a u t o m a t a are called general linear automata having an isolated initial state 7;0 E M.

The dimensionality r (~) of a general linear automaton ~ is called the dimensionality of the minimal hyperplane rM in V, which contains the set M. The di-m e n s i o n a 1 i t y o f t h e out p u t o f a g e n e r a 1 1i n e a r a u t o m a t o n ~ is called the dimensionality of the space Z. Further on we shall be interested chiefly in general linear automata having finite-dimensioned r M and Z. We shall call such generallinear automata finite-dimensioned general linear automata.

Two cases are possible: 1) rM = L(M), where L(M) denotes the linear hull of the set M, and 2) r M c: L (M) (the symbol c: denotes rigorous insertion throughout this paper).

For example, if V is a three-dimensional space, while the set M consists of two points a = (1, 0, 0) and b = (0, 1, 0), then rM is the straight line x + y = 1 in the plane Oxy, while L(M) coincides with Oxy. Now assume M contains, besides a and b, an additional point c = (xc, Yct 0), which does not lie on the straight line x + y = 1. Then rM = LM = Oxy.

In the first case r (~) is equal to the dimensionality of L (M).

In the second case (for finite-dimensioned general linear automata) r (~) is equal to r[L(M)] -1, where r[L(M)] is equal to the dimensionality of L(M).

Actually, having chosen from the vectors M the basis v1, v2, ••• for L(M) and having expressed all the coordinates of points M in this basis, we obtain: either (a) a point x = (x1, x2, ••• )

can be found in M such that ~x i ;-.! 1, or (b) ~x i = 1 for all points (x17 x2 , ••• ) E M.

It is not difficult to see that cases a) and b) coincide with cases 1 and 2 considered above, respectively.

General linear automata for which case 1 [a) holds are called g e n e r a 1 - 1 in e a r a u t omata of the I type, while generallinear automata for which case 2 [b] holds are called general linear automata of the II type.

Since the set M goes over into itself for any mapping of cp";' it follows that the functioning of generallinear automata depends solely on the set M, and on ~, Z, and cp, while it is independent of the choice of the linear space V, provided only that the mappings of cp and l/J on the set Mx~ remain unchanged. The least of possible V containing M is obviously L (M), and therefore hereafter weshall not distinguish between ~(V, M, ... ) and ~ (L (M), M, ... ), and V will sometimes be dropped in the notation of a general linear automaton for the sake of brevity.

If M = L (M), then the general linear automaton ~ is called c o m p 1 e t e . If M = r M c: L (M). then the general linear automaton ~ is called w e a kl y c o m p 1 e t e .

The general linear automaton ~' = (M', ~, Z, cp', 'ljl') is called a sub a u t o m a t o n o f the general linear automaton ~. if M'sM and the mappings cp~i and 'ljl~i coin-

cide with C{Joi and '\jloi of the space L (M') respectively (it is obvious that L (M') s L (M )) . In this case the generallinear automaton ~ is called a hyperautomaton or supplement of ~'. The generallinear automation W =(M, ~. Z, cp, 'ljl) with M = L(M) is called the supplement of the general linear automaton ~(M, ... ). while the generallinear automaton ~ (fM, ~. Z, cp, 'ljl) is called the weak supplement of the general linear a u t o m a t o n ~ (M, ... ) , if r M c: L (M).

GENERAL LINEAR AUTOMATA 181

Since generallinear automata are automata (perhaps, infinite ones), one may speak of their automaton isomorphism, homomorphism, etc. Various generallinear automata may be isomorphic to one and the same deterministic automaton )B (S, ~. U, cp, 1/J); this depends on the choice of the field P and the representations of S and U as sets in linear spaces over P.

The mappings rta1 can be generalized naturally for words x:

cp A = e (e is an identical transformation, } A is an empty ward),

Cfxcr = (jlcr • (jlx·

(3)

If systems of coordinate vectors are chosen in the spaces L(M) and Z, then the linear transformations <fat and the linear operators 'IJ.'at correspond to the matrices A(CTi) and B (CTi). Equations (1) and (2) then yield

-; (t) =; (t) n [a (t)],

; (t+ 1) =; (t) .-t [cr (t)]. (2')

For general linear automata of the II type the sum of the elements of each row of each matrix A(CT) is equal to 1.

The system of coordinate vectors (the basis) in L (M) is called proper if it is contained in M. Sometimes it is necessary for us to choose linear dependent systems of coordinate vectors M and Z. In these cases the coordinates of the points M and the matrices A(CTi) and B(CTi) are ambiguously defined if additional constraints are not imposed on them.

Let us note that the term "linear automata over the field P" in the literature is currently assumed to define automata whose states, inputs, and outputs at time t are vectors over the fields P: s(t), x(t), and y(t), respectively, while the transitions and outputs are described by the equations

where A, B, C, and D are matrices which do not depend on time. t

Let us show that linear automata are a particular case of general linear automata. As the states of the generallinear automaton tll we shall take the vector ; = (;, 1, 1, ... , 1),

'-.".-' l ones

where l is the dimensionality of the input vector x(t). The transition matrix A1 (i') is assumed to equal

A (I

1 0 ... 0 -· B* (x) 0 1 ... 0

'0 0 ... 1

*- *- -where the elements B (x) are b ii (x) = xibii, while the output matrix B1 (x) is assumed to equal

tRecently, a paper by D. R. Deuel has appeared [J. Comp. Syst. Sei., 3(1):93-118 (1969)] in which linear automata with time-varying matrices A, B, C, and D are considered.

182 A. A. MUCHNIK

where d!i == xid ii. It is obvious that

-+ -+ -+ y (t) = v (t) B 1 (x).

Thus, the generallinear automaton 21 describes the operation of a linear automaton.

Although the dimensionality of the vector v is equal to the sum of the dimensionalities of the vectors s and x, the set M of all vectors v == (s, 1, 1, ... , 1) nevertheless lies in the hyperplane defined by the condition that the last l coordinates v are equal to 1. Therefore, the dimensionality of this hyperplane is equal to the dimensionality of s, and the dimensionality of the generallinear automaton 21 is the same.

One may define the notion of a Moore general linear automaton by choosing the linear operator ~ tobe independent of a.

The initialgenerallinear automaton 21 maps any string of inputs a;1 , a;2, ••• , a;k, ... into a string of outputs

-+ ..,. -+ z(a;1), z(a;1, a;2), ••• , z(a;1 , G;2 , ••• , aik), ... ,

this mapping being deterministic [5, 22].

Let us also introduce the linear space of inputs U :r: having the dimensionality m == I I: I , having juxtaposed each input signal ai with a basis vector "ö;. The elements of U :r: will be the vectors k1a1 + k2a; + ... + k mUm, where k1, k2, ... , km are arbitrary elements of the field P.

-+ _,. -t- -+ The input k == k1a1 + k2a2 + ... + kmam (hereafter weshall write: (k1, k2, ... , km)) corre-

sponds to the linear transformations

~ m ~~

qJ-> (v) = ~ ki<pa. (v) k i= 1 '

and the linear operators

-+ m _,.

'ljl-. (v) = ~ ki'ljla. (v). k i= 1 '

"h Instead of the matrices A(ai) and B (a i) it is necessary to take the vectors al = aih (ai) and bih=bth(ai) •

Due to the action of the vector r == rj» the state V== vk goes over into the state

(1")

and yields the output- the vector

(21t)

(the multiplication in (11t) and (2") is tensor multiplication; i.e., rß{h = ~ ria~ (ai); compare with (1') and (2')). i

In matrix form the transition and output equations will be the following:

; (t+ 1) =; (t) A (-;), ..... -+ -+

z ( t) = v ( t) B (r),

GENERAL LINEAR AUTOMAT A

~ ~

where A (r) and B(r) are defined as:

B{;) = 2J B (aj) ri. j

~ ~ ~

Now any input string of vectors r, r 2 , ... , rk, ... is already juxtaposed with the output string :Z1 , z2, ••• , zk by an initial general linear automaton, this mapping e being linear anct deterministic (i.e., without anticipation [4]).

183

We use 8__,. __,. __,. to denote the mapping of the set of strings {r;:} into the set of strings ~ sr,s2, ... ,s1

{ z k}, which is such that -+----+ -+-+-+ -+-+ -+-+-+

0 (s1, s2, .•. , Sj, r1, r2, ... , rk) = w1, ••• , Wj, z~> z2, ••.• zk,

The mapping 8__, __, __, is achieved by a general linear automaton 2f having the initial state s1s2···sj

v(-;1, 82, ••• , S:>, which is that state into which v0 transits due to the action of the input 81, 82, ••• ,

si. The set of mappings 8__,__, -• is thus a linear space having a dimensionality no higher than 81 82 . • 0 s j

the dimensionality of \21-. These considerations are analogaus to the investigation of finiteautomaton mappings by Raney [22]. Hereafter we, however, will be more interested in mappings connected with input symbols { (}' i } , rather than with their vector generalizations.

It is not difficult to prove the equivalence of Mealy and Moore generallinear automata from the point of view of the mappings which they produce. Only the dimensionality L (M) of a general linear Moore automaton may be, in general, higher than the dimensionality L (M1) of the corresponding general linear Mealy automaton by a factor m, where m is the number of input letters.

§ 2. Examples of General Linear Automata

A physical system considered in discrete time, whose space can be described by n real (or complex) numerical parameters which vary due to the input of external signals, these variations being linear transformations of the parameters, may serve as the interpretation of an ndimensional general linear automaton. A controllable Markov chain having n states may serve as an example of such a system.

Below these objects -finite probabilistic automata and sources -will be considered in detail, and it will be shown that they are a particular case of a general linear automaton.

Let us give a number of other examples of general linear automata.

1. Let us show that finite deterministic automata having states are (n- 1)-dimensional general linear automata over any field P.

Assurne m (S, ~. 8, «p*, 'ljl*) is a finite deterministic Mealy automaton, where S = { s 1, s 2, ••• ,

s 11 } is its set of states, L: is the input alphabet, 8 = (~1 , ~2 , ••• , ~m) is the output alphabet, cp* (s, (}') and 1/J* (s, (}') are the functions of transitions and outputs, respectively (4].

Let us consider the generallinear automaton 2f (V, M, ~. Z, «p, 'ljl), where M is the set of basic vectors P 11 = fe1, e2, ••• , e11}, Z is an rn-dimensional space over the field P having the

-+ -+ ~

basis (f1, j 2, ••• , fm ), and the linear transformations 'Po-; and the operators 'ljl11 ; are defined as follows:

184 A. A. MUCHNIK

when cp* (sha) = si (i = 1, 2, ••• , k; h = 1, ••• , n) and

when l/J (sh, ai) = ~z (h = 1, 2, ••• , n; i = 1, 2, ••• , k). It is not difficult to verify the fact that m is a general linear autoroaton of the li type.

Juxtaposing the state sh with the vectors eh, and the outputs ~l with the vectors lz' we thereby establish isoroorphisro of the autoroaton m and the generallinear autoroaton m • t

The diroensionality of m is equal to n- 1, since all of the eleroents belong to the hyperplane x1 + x2 + ••• + x11 = 1. t

2. In exactly the saroe way one roay construct a generallinear autoroaton having the diroensionality n- 1 which is isoroorphic to any probabilistic autoroaton having n states [13, 20]. One need only take the vectors of theprobabilitydistribution of the states of the probabilistic autoroaton as the "states" ofthis autoroaton, and the probability distribution vectors of the output syrobols of the probabilistic autoroaton as its "outputs."

After this the probabilistic autoroaton becoroes an autoroaton (having an infinite nurober of states and outputs), and one can speak of its isoroorphisro relative to a general linear autoroaton (of the li type).

3. Probabilistic sources roay be interpreted by roeans of a generallinear autoroaton. However, instead of considering probabilistic sources, it is better to consider probabilistic autoroata having a Carlyle output which generalize these sources; such autoroata are essentially Shannon cororounication channels having a finite nurober of states [7, 13, 14]. They are defined by stipulating sets of inputs X, sets of states S = (s 1, s2, ... , s 11), sets of outputs Y, and the conditional probabilities p(y, s'/s, x) that the systero will go over froro state s to state s 1

due to the input x (a channel transroitting the signal x) and produce y at the output.

(Finite probabilistic sources correspond to the case in which X contains one syrobol.)

Each string of inputs-outputs (x1y1), (X2J2) ••• (xkyk) and each state s corresponds to a nurober p5 (y1y2 ••• yk/x1x2 ••• xk) - the probability that the autoroaton in the state s, having accepted the input word x1x 2 ••. xk, will deliver the word y 1y2 ••• Yk.

Let us now construct a generallinear autoroaton m (M, X x Y, <p, 'ljl), where the set of states M is the set of points p= (p1, p2, •.• , pk), 1 =:: pi =:: 0; X x Y is the set of inputs of m; the set of real nurobers is the space of outputs of W ; the roapping cp(x, y) of the set M is deterroined by the matrix 11 aii (x, y) 11, aii (x, y) = ~ p (y, si/s;, x), and the linear functional 1/J(x, y) = B ('P) = b · p will be defined by the vector b(x, y) = (b1, b2, .•. , b11), where bi (x, y) = ~ p (y, s' Ist. x).

s'ES

The number p i (i = 1, .•• , n) is the probability that the probabilistic automaton (starting from a

certain state) delivers the word y due to the input of the input word x, and arri ves at the state si. n

Therefore, the suro ~Pi roay be less than unity. A Carlyle probabilistic autoroaton is a gen'=1

erallinear autoroaton of the I type.

The input-output pair (x, y) of a probabilistic Carlyle automaton corresponds to the input of a generallinear autoroaton m. Froro the generallinear autoroaton m one may restore the

t Above we noted that general linear automata are automata, and therefore one can speak of isoroorphisro of 58 and m.

t Such a representation of finite deterministic automata was first given in a paper by M. L. Tsetlin [11].


Carlyle automaton (communication channel). In this sense generallinear automata allow Carlyle automata tobe interpreted, since the latter are not in general automata even when they have an infinite nurober of states and outputs, if Xis taken as the input alphabet; this is true because the probabilities of the transitions of states depend not only on the input symbols but also on the output symbols, and a transition to probabilitic Moore automata requires a time shift of the outputs. We shall discuss the difference between Carlyle automata and other probabilistic automata in greater detail in the chapter on probabilistic automata.

4. Assurne that there is a finite set of identical finite automata m (S, 2:, U, cp, 'ljl).

At each time identical input signals CJ(t) are applied to the inputs of all the automata (which may beindifferent states). The output of such a system is the vector (t1, t2, ••• , tm), where ti is the nurober of automata producing the symbol ui at the output. Obviously, ti is an integer

n

nonnegative nurober 2J t; = d. i=1

The operation of such a system may be described by means of a general linear automaton 2f. Let us assume that the vectors (a1, a 2, ... , an) are its states, where a i is the number of automata Q~, that are in the state si at the given time.

The function <Pa shall be defined by means of the transition matrix II aii (CJ) 11, where

{ 1 for <p (s;, a) = sh au(a) = .

(i,J=t, ... , n) 0 otherw1se;

1/Jb shall be defined in terms of the output matrix B(CJ) = II b i/CJ) ~,

{ 1, if 'ljl(s;, a)=th bu (a) c .

(i=1.~ ..... ,n; 0 otherw1se. j~1,2, ... , m).

A more general case holds in the consideration of a finite set D of finite automata among which there may be automata of different kinds ( D = D1 lJ D2 LJ .•. lJ Dh where Di is the set of automata of the i-th kind) but with a common input and output alphabet.

Then the coordinates of the vectors of state will be partitioned into eight groups in the corresponding generallinear automata; the coordinates of the i-th group give the distribution ofthe set of automata Di over the states s; = {qlil, ... , q~;}, while the matrices A(CJ) will have a diagonally cellular structure in which the i-th cell of A(CJ) will contain the matrices Ai(CJ) that are transition matrices of the automata of the i-th group.

Thematrices B(CJ) of the outputwill be constructed from the matrices Bi(CJ) formed in the column

An interesting scheme is obtained if we allow for the possibility of pro du c t i o n and 11 an n i h il a t i o n 11 of automata. Assurne that the finite automaton ~I in the state s i is caused by the action CJ to generate a ik (CJ) automata in the state sk. The states of the corresponding generallinear automata will be vectors whose coordinates ai constitute the nurober of automata 2f in the state si. The elements of the matrix A(CJ) areinteger nonnegative numbers.

186 A. A. MUCHNIK

Let us e:xpand this model by allowing the generation of both automata and "antiautomata." The corresponding matrices A(a) may have any integer numbers as their elements. Let us define the notation of an antiautomaton 21- (S, ~. u-, ({', 'ljl) for a finite automaton 21. This finite automaton differs from 21 only in that the symbols u1, u2, ... , u7n serve as its outputs, and zero is obtained for "addition" of the same outputs ui + u"f of the automaton and antiautomaton. Then one may assume that the automaton 21 and the antiautomaton W- "annihilate 11 if they are in the same state.t

From theorems on e:xperiments with general linear automata there will derive basically the same estimates for 11 automaton media11 formed from finite automata of a singletype having n states, both in examples analyzed in this subsection and for finite automata with n states.

5. Let us consider the four-pole in Fig. 1. Alternating currents of specified amplitude and phase are applied to poles 1, 2 at each instant of discrete time. The instants of discrete time may be represented as intervals of a certain length ~' the gaps between them being of sufficient duration so that the transients in the network can establish a specified state at the next instant. At the initial instant t0 the pulses i1 (0) and i2 (0) are applied to pol es 1, 2, respectively.

The state of the network is characterized by the amplitude and phase of the currents i1 and i2 which are applied to the poles 1, 2 (Fig. 2).

Each of the currents ik (k = 1, 2) may be characterized by a complex nurober rkeiqJ", where rk is the amplitude and CfJk is the phase of the current.

The block of the network Ukl (k, l = 1, 2) transmits the current from the pole k to the pole l, amplifies or attenuates it, and changes its phase. Consequently, the action of each block may similarly be characterized by a complex nurober akl. Thus, the current applied to the pole j' is equal to

Using relays, one can make the blocks ukz controllable. Then the numbers akz will depend on the input signals a.

From the pole 11 (21) the pulse of alternating current is again transmitted to the pole 1 (2) after a delay d1 (d2). It is necessary to ensure synchronization of the pulse.

The state of the network at the instant t is the complex vector (i1, i2). Its transformation occurs via a complex matrix II akl (a) II (k, l = 1, 2). Thus, the network operates as a general linear automaton having the dimensionality 2 over the field K of complex numbers. A vector of state may serve as the network output (or it is necessary to use additional transforming blocks). Of course, the implementation of this network encounters certain engineering difficulties - it is necessary to ensure the accuracy of the amplitude and phase modulation. However, with a certain degree of approximation this network can be implemented even if it has a larger nurober of poles.

t Here analogies with quantum mechanics and the physics of elementary particles suggest themselves. E. Moore [18] indicated a certain analogy between e:xperiments on automata and on particles in quantum mechanics.

On the other hand, the state of any physical system can be described by a l/J-function (i.e., a vector of Hilbert space).

Transformations of l/J-functions due to the effect of an external influence are usually linear. Thus, quantum-mechanical systems are countable-dimensional generallinear automata (see [8, 7]).

GENERAL LINEAR AUTOMATA

Fig. 1 Fig. 2

Fig. 3

It is not difficult to construct the analogous network for a generallinear automaton having an arbitrary finite dimensionality over the field K.

187

The idea of using phase modulation for the transmission and manipulation of information in digital machines was stated by J. Neumann,

6. From the engineering standpoint, networks for any finite-dimensional generallinear automaton over a finite field P may be fully implemented.

As an example, Iet us consider a general linear automaton of dimensionality 3 having an output of dimensionality 2.

For a generallinear automaton of dimensionality 3 the network is shown in Fig. 3. The state of the generallinear automaton 21 consists of the vectors (v1, v2, v3) (signals traveling along conductors may take a number of values corresponding to the number of elements in the field P). The elements aii (i, j == 1, 2, 3) carry out multiplication of the incoming signal by aii (for a ii = 0 this element and the line passing through it may be dropped, while for aii == 1 only the element can be dropped; therefore, in fields having the characteristic 2, networks of autonomous generallinear automata have an especially simple appearance). The elements aii may be made controllable (i.e., dependent on external inputs).

For fields having the characteristic 2 and k binary inputs x1, x2, ••• , xk, where the input signal a == (x1, x2, ••• , xk), the function a ik (a) == a ik (x1, x2, ••• , xk) will, in general, be an ar-

188 A. A. MUCHNIK

bitrary logic-algebra function of x1, x2, ... , xk, while for any finite field P of m elements the function aii (a) will be an m-valued logic function. In order to obtain the output vectors of the network, we require the additional blocks bik (a).

In this lies the difference between a generallinear automaton and linear switching networks which use only EB and delay elements in the binary case, and multiplying devices aik in the multivalued case; however, in this case the blocks aik do not depend on the input.

Finite-dimensioned generallinear automata having a finite input alphabet over a finite field arefinite automata. The advantage of representing them in the form of generallinear automata lies in the fact that if one is able to code the states of a finite automaton (and there may be very many of them) by means of vectors over the field P (their dimensionality is of the order of log m n, where m is the nurober of elements of the field P, and n is the nurober of states of the finite automaton) and these vectors can be transformed linearly for transitions and outputs, it follows that in the network implementation it is possible to achieve a large economy of elements in comparison with the general case of a finite automaton.

Let us estimate the nurober of elements V, &, I in the implementation of a general linear automaton over the field GF(2). Only the dependences of aii on the input symbols a~ coded by the collection x1, x2, ... , xk of elements from P, will be nonlinear (in the binary case aii (x1, x2, ••• , xk) is an arbitrary logic-algebra function). However, for a small nurober of inputs and a large nurober of states the implementation of a ii <x1, x2, ... , xk) requires comparatively few elements, while the remaining part of the network is implemented simply. At the sametime for arbitrary coding of states in m-valued logic the nurober of elements will be considerably higher. A generalized linear automaton over the field P having the character-istic 2, which has the dimensionality n, k binary inputs, xb Xz, ... , xii and m binary out-puts requires n2 blocks au (x1, ... , xi<) and nm blocks bu (x1, ... , xk), each of which is im-

plemented with a complexity not exceeding (i~e)~ in the classical basis V, & and I; k

n + m adders modulo 2, each with n inputs, and n delay elements. Each adder is implemented with complexity Cn, where C is a constant. Thus, we find that it is sufficient to have

n (n+m) [ (1 +e) 2: +C]

elements V, & and I and n delay elements [5]. However, when an arbitrary finite automaton having 211 states, k inputs, and m outputs is implemented, it is necessary to use at least

elements /\, &. 1 and n delay elements [5, Ch. 8], where e- 0 for n + k- oo • Moreover, it is considerably easier to analyze t n-dimensional general linear automata and arbitrary finite automata having 211 states.

§ 3. Certain Possible Generalizations

1. N ondeterministic automata, which were considered by Rabin and Scott [21], are of interest. However, they do not fit directly into the scheme of a generallinear automaton. Nevertheless, if instead of linear operations on vectors, which use the operations of the fields

tThat is, to clarify the character of the mappings which they perform.

GENERAL LINEAR AUTOMATA

+ and x, we consider the operations + and x suchthat the operation + is commutative and associative, while x is associative, and distributiveness holds:

. . . . . (a + b) X c = (a X c) + (b X c)

and c X (a-i- b) = (c X a) --~ (c x b),

189

(4)

(5)

then it follows that analogs of generallinear automata may be considered over such an algebra; the states -; and outputs z of these automata are declared tobe vectors having coordinates that are elements of this algebra; the transition and output functions are determined by the matrices A(u) and B(u), where u is the input symbol, in the conventional manner:

while the multiplication of vectors by matrices and the multiplication of matrices in such an algebra is defined conventionally and has the properties of associativeness and distributiveness:

. . . . . . . . . . 1 X B XC= A X (/l XC), (.1 -r- /J) XC' (.-t XC) -f- (B >< C),

. . . . . C X (A -f- B) c (C X A) -I (C X B)

(see [1]).

Let us consider a Boolean algebra with V instead of +, and & instead of x, for which all of the required properties are fulfilled. Assurne 21 (S, F, ~. U, ~o, 1p) is a nondeterministic automaton [4, 21].

In the determination process the subsets S = {s1, s2, ••• , sn} are taken as its state. Each subset R c;; S is juxtaposed with a Boolean vector r = (r1, r 2, ••• , rn), ri = 1 being valid for s; ER and ri = 0 being valid for s; ER. We define the matrix of transitions as a Boolean matrix A(u), having placed aii (u) = 1 when and only when tr~sition to the state si is possible from the state si due to the action of u. The output vector b = (b1, b2, ••• , b11), where bi = 1 for s; E F and bi = 0 for s; E F, while F is the set of isolated states; b is independent of u (a Moo:t'e automaton). The output is binary (i.e., the input word is tagged or not), since we are speaking of the representability of events:

~,(t :--1) =-;(t) A (cr),

lJ (t) =,--; (t)l

y (t) outpur ( = 0, 1).

(6)

(6')

If the initial state was si, then the word x = u1u2 ••• u k will be tagged by the automaton 21, if and only if

where --;<il is a Boolean vector in which a single "one" appears in the i-th position.

An automaton whose operation can be described by Eqs. (6) and (6 1) (i.e., a Boolean automaton) adequately reflects the operation of the nondeterministic automaton 2L Therefore, nondeterministic automata shall be called Boolean. Here, however, an arbitrary choice of the 11baSiS II in the Set of State Vectors { r} iS impOSSible, and thiS greatly COmplicateS the finding of the structure of the Boolean 'lutomaton from an experiment on it. Therefore, gen-

190

Figo 4

Ao Ao MUCHNIK

erallinear automata are still the best generalizations of finite automata from the point of view of experiments o

2o Semilinear automata ~(NI, ~. Z, !Jl, R1) are another generalization of general linear automata; these automata are distinguished by the fact that the output vector

; (t) = (z 1 (t), z2 (t), ... , Zm (t))

is defined by the polynomials { Ri} (i = 1, 2, .. o' m)

Z; (t) = R;, a [s; (t), S2 (t), ... , Sn (t)],

where Ri,o is a polynomial of degree g(i, a) having coefficients that depend on i and the input ao The transformations of the transitions

are linear as previously o

Here one may apply the linearization method [3] by taking vectors having coordinates corresponding to monomials of degree not exceeding max g (i, a) = h from s 1, s 2, ••• , s 11 (i.e., with Coordinates of the form i, a

where the nurober of each coordinate is determined by the collection h1, h2, ... , h11 , h1 + h2 + ... +h 11 :S h) instead ofthe vectors (s1, s2, .. o, s 11 )o

-+ Since the vectors s = (s1, s 2, ... , s 11) can be transformed linearly, it follows that the vec-

-+ -+ tors q (q0,o, ., 0 , 0, ., 0 , qh, " 0 , hn, .. ) can similarly be transformed linearly, and the output z (t) depends linear ly on q (tJ.

Automata for which the transformations cp0 are any affine transformations of the space L(M):

; (t + 1) =; (t) A (a) +;:: (a),

-; (t + 1) = ;(t) B (a) + ;2 (a)

also reduce to the case of a generalized linear automaton (it is sufficient to add one coordinate equal to 1 to the vector s1, and one row equal to r1 (a) to the bottom of the matrix A(a), and a column containing all 0 elements except the last element which is equal to 1 to the right

-+ of the matrix obtained, while to the matrix B(a) it is required just to add the row r 2(a) on th(' bottom.

§ 4. Experiments with General Linear Automata

The notion of experiments with automata was introduced by E. Moore [18] and was studied in papers by B. Ao Trakhtenbrot, S. Ginsburg, T o Hibbard, A. A. Karatsuba, V o A. Dushskii, Yuo Mo Borodyanskii, A. A. Muchnik, et al. [5, 17, 2, 9].

lo Since general linear automata are automata, the notions of equivalence and distinguishability of states, the notions of prime-uniform, branched, and multiple experiments [18], as well as the notion of distinguishability of automata, are automatically extended to them.

GENERAL LINEAR AUTOMAT A 191

In the book [5] infinite trees (deterministic operators) are used to describe automata having the input alphabet I: and the output alphabet U. Each string of inputs a(1), a(2), ... , a(t), ... corresponds to one and only one string of edges of the tree D with its beginning in the root of the tree. Each edge of the tree is assigned a corresponding output symbol z. Assurne that the input symbols will be vectors z over the field P. T h e 1 in e a r c o m bin a t i o n aD 1 +ßD 2 of trees D 1 and D2(a, ßEP) iscalledthetreeDwhoseedgesareassigned the linear combinations az1 + ßz2, where z1 and z2 are the vectors assigned to the corresponding edges D1 and D2• Thus, the set of all trees over the input alphabet I: which have the outputz will be a linear space over P. We shall call the dimensionality of the minimal hyperplane in this space, which contains all branches of the tree D, the d im e n s i o n a 1 i t y o f t h e t r e e D • The d i m e n s i o n a 1 i t y o f a c e r t a i n s e t o f t r e e s { D i } shall be defined as the dimensionality of the minimal hyperplane in the space over all trees which contains all branches of the trees from { nJ.

It is obvious that each state of the general linear automaton 21 (M, I:, Z, <p, '\jl) corresponds to a certain tree D over I: having outputs from Z (and the initialgenerallinear automaton 111 corresponds to one tree Dw). The dimensionality of the set of trees corresponding to the state of the general linear automaton does not exceed (as we shall see below) the dimensionality of the general linear automaton 2l. The converse is also true. Each tree D having the dimensionality n defines a certain initialgenerallinear automaton having the dimension- · ality n. Its initial state corresponds to the root of D. Let us choose the basis in the set of branches of the tree D. Then each branch of D can be expressed as a linear combination of basis branches, and thereby the transformations of the transition and outputwill be defined. In exactly the same way the set of trees { nJ having the dimensionality n defines a general linear automaton having the dimensionality n.

Two states of the general linear automaton 21 are called e q u i v a 1 e n t if identical trees correspond to them. Two generallinear automata 21 1 and 21 2 are called equivalent if for each state of one general linear automaton one can find its equivalent state in the other, and vice versa. For a transition from the set of trees corresponding to 21 to a general linear automaton, we may obtain a generallinear automaton 21 1 having a dimensionality small· er than the dimensionality of 21 as a consequence of the splicing of equivalent states. Such a transition to a general linear automaton without equivalent states is called the operation of the red u c t i o n of the general linear automaton 21. The reduced form of the complete general linear automaton 21 is unique with accuracy up to isomorphism of the linear s paces L(M) which preserves the operators cp 0 and 1/!a• However, the matrixform of a reduced general linear automaton 21 always depends essentially on the choice of the basis L (M). Besides, the matrices A(a) of the reduced generallinear automaton go over into matrices of the form c-1A(a)C when the basis changes, where C is a certain nonsingular matrix that depends on the new basis and is independent of a, while the matrices B(a) go over into C - 1B(a)C, respectively.

One may consider the subtrees n( il, nC2 l, ... of the tree D having the height h = 1, 2, ... (i.e., with h stories of edges).

The next theorem is an analogy of the Moore theorem on the length of a prime experimentt which distinguishes the state of the automaton.

Theorem 1. For a general linear automaton 21 of dimensionality n the states v 1 and ~2 are equivalent if and only if identical trees D~1 l and D~1 ) of height n correspond to them (i.e., they are indistinguiE able by any prime experiment of length n).

tThe corresponding theorem for probabilistic automata was proved independently by Carlyle [13, 14] and by the author (the results have not been published previously).

192 A. A. MUCHNIK

Pro o f o The necessity of this condition is obvious; let us prove its sufficiency. Let us consider the set M of states of the generallinear automaton ~. Let us assume that one can find two states v1 and v2 from M such that the trees of height n for them coincide, while in the (n + 1) -st storydifferent outputs will be assigned to two identical edges. Let us prove that in this case the dimensionality rM of the minimal hyperplane containing M will be no less than n + 1 [i.e., r(~) = n+ 1 ]o Forthis purpose let us consider the hyperplane rM parallel to rM passing through the origin (the point 0; if OEfM, then fM=rM)·

f M is the minimal hyperplane which contains all differences of the vectors from M • ... ... Let us consider the difference v1 - v2• It corresponds to the tree D1 - D2 for which the

outputs are zeros on the first and second stories, while on the (n + 1)-st story there is at least one nonzero output z(n + 1) = z1(n + 1)- z2(n + 1). Assurne cri1, cri2, ... ,ain is a certain string of inputs that leads to a nonzero edge of the (n + 1)-st story of the tree D1 - D2•

Let us consider all branches (subtrees) of the tree D1 - D2

having roots at the vertices of this path. The k-th branch corresponds to the vertex x = cri1cri2 ... cr;"_1 (k = 1, 2, ... , n + 1) for k = 1, x = A. Zeros appear in the k-th branch on the n- k + 1lowest stories, while in the (n - k + 2)-nd story one can find a nonzero output. Therefore, all branches E1, E2, o •• , E n+t are linearly independent; ioe., one can find n + 1linearly independent vectors in the hyperplane r M 3 0 0 Therefore the dimensionality of r M is not less than n + 1, and consequently, r(~)>n+1, which contradicts the assumption of the theorem. Thus, if two trees D1 and D2 coincide at n stories, then they also coincide at the (n + 1)-st story, etco The theorem has been proved.

Co r o 11 ary 1 (E. Moore) o For a finite automaton having n pairwise-distinguishable states any two states are distinguishable by a prime experiment of length n - 1.

Co r o 11 a r y 2 ( E • Ca r 1 y 1 e) . For a probabilistic automaton with n states the necessary and sufficient condition for equivalence of two states (or the distributions of the initial states) is their indistinguishability by any experiment of length n- 1.

Co r o 11 ary 3 o If in an n-dimensional generallinear automaton W of the type I a tree of height n having a root in a certain state q is a linear combination of trees having a height n with roots in the states q1, o .. , qk, then all of the infinite tree Dq is the same kind of linear combination of the trees Dq1, ••• , Dq"·

For a generallinear automaton of the type II, Corollary 3 is true if one considers linear combinations with the sum of the coefficients of 1, which we shall hereafter call n o r m a 1 i z e d linear combinations.

The proof can be reduced to Theorem 1 if the generallinear automaton ~I is expanded to a complete generallinear automaton ~, if ~ is a generallinear automaton of the I type, or to a weakly complete generallinear automaton if ~ is of the II type.

Then all linear combinations ~ ai qi for a general linear automaton ~ of the type I and all linear combinations ~ a iq i having the condition ~ ai = 1 for a general linear automaton ~ of the type IT will be states of ~, which has a dimensionality equal to m.

Let us give another proof of Theorem 1, which has a clear geometric meaning and is closer to the ideas of E. Moore [18]. Instead of the generallinear automaton ~ we consider the expansion W: a supplement if r M = L(M), or a weak supplement if rM c L (M) ; then we prove the theorem for 21, and consequently for ~ also. Obviously, the dimensionality of W is equal to the dimensionality of ~.


-+ Assurne that the automaton 'QI is initially in the state s. Since we know nothing about

-+ -+ the initial state, we may choose s to be any point from r M. Each word transfers the state s to the state cpK(S). If the symbol (J is then applied to the input, the vector 'l!Jcr [q>x (s)J will appear at the output of the general linear automaton.

-+ • The value u of this vector 'll'a [q>x (s)] will become known to us in the experiment X(J.

Therefore, we obtain the linear equation

'Pa [q>x (~] = ;;, (7)

which isolates in r (M) the hyperplane r xcr (which may coincide with r (M)) containing the state s. If during the experiments (prime or multiple) the outputs of the generallinear automaton ~ , corresponding to several words x1(J1, x2(J2, ••• , Xk(Jk, becomes known to us, then this yields a system T of linear equations

(i=1, 2, ... , k). (8)

Such a system T likewise defines a certain hyperplane r r s r M· If the system T 2 derives from the system T 1, then 1'r1 s;:: r r 2 , i.e., a Iongerexperiment defines a smaller hyperplane or the same one. For various values in the outputs Üi various hyperplanes parallel to each other will be obtained. Thus, just the inputs of the experiment define the direction of the hyperplanes r (M) for us, while the experiment partitions r (M) into hyperplanes parallel to r T.

(The difference between a uniform experiment which does not use information on the output of the automaton and is connected with a consideration of the partitions of the set of states, and a branched experiment in which the value of the output immediately isolates the subset of states to which the consideration is restricted during the subsequent course of the experiment resides precisely in this .)

Assurne now that the generallinear automaton has the dimensionality n. i.e., r(I' (M)] = n. Let us assume further that r (M) = r for the sake of brevity. The multiple experiment E1 of length 1 isolates the hyperplane f1 s r. The multiple experiment Eb Ek_1, ..• , E1 (the length Ei is equal to i) respectively isolates the hyperplanes r" s l'h-1 = ... s f 1 (s f).

Assurne that this chain rigorously decreases to k- 1, while r k = rk-i (i.e., the experiment of length k + 1 does not add new information on s in comparison with Ek). Let us prove that then r" = rk+1 = r"+2 = ... ; i.e., states which are not distinguishable by the multiple experiment Ek arenot distinguishable at all. Let us take any input symbol (J and consider the hyperplane rk ((J) into which rk makes the transition due to the action of cp 0 • r"_t(a) II r"_b since if rn~ 1 (o) -tt- fn_ 1 were to hold then the multiple experiment of length k- 1 would partition r k _1 ((J) into hyperplanes parallel to rk _1• But then r k _1 - the prototype of r k _1 ((J) for the mapping C.Oa -would be partitioned by an experiment of length k, but rk = rk-1; i.e., rk-1 is not partitioned. Since r k = rk-1• it follows that rk ((J) = rk_1((J) and rk ((J)II rk-1• Therefore, r k ((J) is not partitioned by a multiple experiment of length k, and therefore r k is not partitioned by an experiment of length k + 1; i.e., rk+i = rk, etc., rk = rk+i = rk+ 2 = .... The states that are not distinguishable by a multiple experiment of length k turn outtobe indistinguishable altogether. The entire set r (M) can be partitioned in this manner into sets of equivalent (indistinguishable) states that form hyperplanes obtained by parallel shifts of rk _1 (i.e., into contiguous classes r M in rk _1; these hyperplanes may also be points).

N ow it remains for us to note that

From this it follows that k:::; n, i.e., states that are not distinguishable by experiments of length n are equivalent to r, etc.

194 A. A. MUCHNIK

From the proof given one may extract the following corollaries.

c 011 0 ar y 4. If the distinguishable states v1 and ;2 of the general linear automaton 2I belong to the hyperplane r having the dimensionality n, which due to the action of each input a goes over into the hyperplane r (a)ll r, then these states are distinguishable by a prime experiment of length n, regardless of the dimensionality of the general linear automaton 2I.

Co r o 11 a r y 5 • The state s of the general linear automaton 2I is equivalent to the state q of the general linear automaton \b if and only if these states are indistinguishable by any experiment of length r (21) + r (Q).

For a proof it is sufficient to consider the direct sum 2I (jj m of the general linear automata 2I and ~- The dimensionality of r (21 (jj Q:) is equal to r (21) + r ()Z). The states s and q are states of 2I EB ~-

2. If states from r k _1 are identified in the latter proof of this theorem, then we obtain the reduced "form of the general linear automaton 2I; i.e., we obtain a general linear automaton W that is equivalent to the given one and has states which are all distinguishable from one another. Under these conditions the dimensionality of W1 is equal to n- r(rk_1) and to the dimensionality of the set of trees { D s} of the general linear automaton W. The set of states of 2r, will be the factor-hyperplane fM/fk-t·

The reduced form of W 1 is unique, since the partition of r M into sets of equivalent states of the hyperplane rk-i is unique. However, its matrix e:xpression depends on the choice of the basis in L (M) and Z. Another expression for the reduced form of the automaton in the form of a graph of transitions-outputs is also possible, which is equivalent to the matrix e:xpression. Assurne that a ba.sis in the space of trees D8 of the general linear automaton W has been chosen - D1, D2, ••• , Dn. Let us depict the trees { D J by circle-vertices of a directed graph. n arrows issue from each vertex. These arrows connect it with the remaining vertices of the graph.

The arrow connecting the vertex Di with Dj is assigned the e:xpression

whose meaning resides in the fact that the state si is caused by the action of the input ak to go over into the state a ii (ak) • si, where a ii (a k) is an element of the field P (Fig. 5). E ach (i-th) circle-vertex of the graph is assigned an e:xpression

whose meaning resides in the fact that the generallinear automaton, having received the signal ak at its input in the state Di, produces the vector signal

~ b;h (ak) -;h (h= 1, 2, ... , p) h

at its output (where p is the dimensionality and Z is the space of outputs), it being assumed that the basis z1, z2, ... , Zp has been chosen in z. The matrices II rr.ij II and II b;h (ak) II are the matrices of transitions and outputs of ~ (with allowance for all of the inputs a).

3. However, let us return to experiments with generallinear automata. An e:xperiment is called uniform if it does not depend on the initial state (i.e., the next input signal is chosen independently of the output of the preceding portion of the experiment.

Fig. 5


Theorem 2. For any reduced general linear automaton W having the dimensionality n there exists a uniform experiment E of length no greater than n(n + 1)/2, which allows the state of W at the end of the experimenttobe established.

Pro o f. Since the general linear automaton W is in reduced form and has the dimensionality n, it follows that the chain of hyperplanes (see the preceding theorem) r,,[ =:J r 1 =:J r 2 =:J ••• =:J r k is not broken off be

fore rk, while rk is a point (the initial state). Each hyperplane ri is isolated from the results of a multiple experiment of length i. If multiple experiments are carried out in another state, the output will be different, and we shall obtain a different chain:

rM =:J r~ =:J r~ =:J ••. =:J I'k.

Here rk consists solely of the new initial state of the experiments. Each hyperplane r; + 1

is obtained by a parallel shift of the corresponding hyperplane r I, the inequality r (r I +1) < r (r f) being valid, and the dimensionality r (r i) :s n - i, while r (rk) = 0.

Thus, the multiple experiment E1 of length 1 partitions r = rM into hyperplanes { ri}, which are shifts of r 1; the multiple experiment of length 2 partitions each hyperplane r f and the entire r M into hyperplanes r; which are shifts of r 2, and each hyperplane r; and the entire hyperplane r M is partitioned by a multiple experiment of length 3 into hyperplanes { r t} each of which isaparallel shift of r 3, etc.

From this it follows that there exists a simple uniform experiment of length 1 (i.e., an input symbol), Ö1 = a, which partitions r into a set of parallel hyperplanes (shifts) { Bf } that due to the action of ö, go over into the parallel hyperplanes {B~ (ö,)}, while rM goes over into r (01). The dimensionality of each B7 (ö,) is rigorously less than r(rM) = n, i.e., it does not exceed n- 1. Further, the multiple experiment E2 of length 2 partitions rM into parallel hyperplanes having a dimensionality not exceeding n- 2. Therefore, if B: (o 1) has the dimensionality n- 1, the multiple experiment E2 partitions B7 (Ö1) into hyperplanes of dimensionality not exceeding n - 2, and therefore one can also find a simple experiment Öz

of length less than or equal to 2 that partitions each hyperplane B7 (~ 1) and r (ö 1) into parallel hyperplanes s; (ö'1) having a dimensionality not exceeding n- 2. In this case, when the dimensionality of B~ (~ 1) does not exceed n- 2 one may assume that the input word ö'2 is empty. while s; (01) = H7 (0 1). Due to the action of 02 the set of hyperplanes {B; (ö't)} goes over into the set of parallel hyperplanes {H; (~ 102)}, while r (ö't) goes over into a hyperplane r (6'16'2)

having a dimensionality less than or equal to n- 2. If the dimensionality of B; (G',G'z) is equal to n- 2, then a certain simple experiment Ö3 of length no greater than 3 partitions each hyperplane B; (0102) into hyperplanes { s: (016'2}} having a dimensionality not exceeding n- 3, which due to the action of 03 go over into {B; (o1Ö2Ö3)} , etc.

The sequence

(9)

must be broken off no later than at the n-th place, since the dimensionality of the hyerplanes B{ decreases rigorously with increasing i, while the dimensionality of Bf is no greater than n- 1. Assurne that the last system of hyperplanes in the sequence (9) will be {Bk (ÖtÖz ... Ök)}.

Each of the Bk (Ö1Öz ... Ön) is a point, and their union forms r (ö,ö'2 ... Ök), i.e., it forms a hyperplane into which the hyperplane r M goes over due to the action of the experiment

196 A. A. MU CHNIK

~~~2 ... Gk· Thus, knowing the outcome of the experiment ~ = 'e1'e2 ... Gk, we define the point-state into which the generallinear automaton ~ falls due to the action of the experiment ~.

The length of ~ is estimated thus:

k k

z (~) ~ ~ l (~;) ~ ~ i = k (k2+ 1) < n (nt 1) '

i=! i=1

i.e., l (~) ::::; n(n + 1)/2, which is what it was required to prove.

This theorem generalizes the theorems ofT. Hibbard [17] and Karatsuba for finite automata, and its proof is even simplified so that it takes on a clear geometric meaning.

4. The generallinear automaton ~ (M, ~. Z, cp, 'ljl) is called strongly connected if the dimensionality of the set of branches of the tree D5 is the same for all s E M and is equal to the dimensionality of 21 itself. t

At first glance it seems that this definition derives directly from the definition of strong connectedness of automata according to Moore [18], extended to infinite automata. However, this is not so. Strang connectedness of the generallinear automaton 21 (V, M, ~. Z, cp, 'ljl) düfers from strong connectedness of 21 treated as an infinite automaton. Let us present two simple examples:

1. Assurne V is the real plane Oxy, while the set of states M is the ensemble of points of the straight line y = 1. ~ = { a }; Z is the set of real numbers; the transformation <p 0 is the shift of the straight line y = 1: x- x + 1 by the matrix (! ~), while the functionallf!: 1/J(x,y) = x and m is a Moore general linear automaton.

Each state s = (x, 1) is transferred by the action of the ward u = ak into the state q = (x + k, 1), which differs from (x, 1), since 1/J(s) = x, while 1/J(q) = x + k. However, no word u1 transfers q back to s. On the other hand, for each state s = (x, 1) the dimensionality of the set of branches of the tree D5 = Dx (chains here) {x + k, x + k + 1, ... } is unity, since the set of points {x + k, 1} lies on the straight line y = 1,

Dx+i =Dx+i (Dx+t-Dx) = iDx+t-(i-1) Dx,

while Dx+1 and Dx are evidently linearly independent for all s. Therefore, m is a strongly connected general linear automaton.

2. Another example of a general linear automaton is m (V, M, L:, z, !Jl, IJl). V is again the real plane Oxy. M is the circle 0(1) having its center at the origin 0 and a radius I; ~ = { a}.

The linear operator cp 0 performs rotation of the plane Oxy about 0 through the angle 1ra, where a is an irrational nurober 1/J(x, y) = x. It is not difficult to see that the set of states which are accessible from any state s = (x, y) has the dimensionality 2, whereas from each state s - the points on 0 (1) - only those points q on 0 (1) are accessible which are such that the arc (s, q) is equal to 1rra - 2k7r, where r isanatural nurober and k is an integer.

tHere it may not be required that a path exist from each state si to any other state s·, since in the overwhelming "inajority of cases" (with the exception of a set of measure zeio near P = D, K) the transition matrices A(a) are suchthat each state s goes over into a linear combination with nonzero coefficients of other states. True 7 one could apply this conventional definition if generalized inputs were considered which are linear combinations of the input words (and not the input symbols) of the automaton. This latter definition is equivalent to the definition of strong connectedness of generallinear automata that we adopted.


From strong connectedness of 21 as an automaton there derives its strong connectedness as a generallinear automaton.

Let us now consider the properties of strongly connected general linear automata.

Assurne ~~ and Q; aregenerallinear automata having a general input alphabet and the output space Z. The generallinear automaton ~ is indistinguishable by the experiment ~

from the generallinear automaton Q1., if for any state s of the general linear automaton ~ there exists a state q of the automaton \.t', suchthat the results of the experiment ~ with the automaton ~ in the initial state s coincides with the result of the experiment ~ with the automaton \!.) in the initial state q. In other words, the indistinguishability of 2I from Ql means that the set of outputs mi'fo (~) of the experiment '@, conducted from various initial states ~, is contained in the analogaus set of outputs m~lß (~1) of the automaton Ql.

The generallinear automaton ~ is indistinguishable from the generallinear automaton 23 if ~l is indistinguishable from 2I by any prime experiment.

The state s of the generallinear automaton 2I is equivalent to the state q of the general linear automaton 23 if s and q are indistinguishable by any experiment. Let us recall that the generallinear automaton 21 is equivalent to the generallinear automaton \b, if for any state 121 there exists an equivalent state of the general linear automaton \b , and vice versa.

Theorem 3. If a strongly connected general linear automaton 2I is indistinguishable from the complete generallinear automaton ib == (M, L:, Z, cp, <f;), then for each state s of the automaton W one can find an equivalent state q of the automaton Q1.

Pr o o f. F or each state s 0 of the general linear automaton 2I and a rrime experiment (input word) '13 let us consider the set M (W, 23, so,~) = {q} (more briefly, M (o)) of states of the generallinear automaton lU, which are such that the experiment begun in any state M (o) has an outcome coinciding with the outcome of the experiment ~ on the automaton \!I

for the initial state s 0• It is not difficult to see that the set M (ü) is a hyperplane in the space V}!). For continuation of the experiment ~ the dimensionality of M (~) may only decrease. Since V}B has a finite dimensionality, there exists an experiment ~*, suchthat for continuation of '13* the dimensionality of M(tJ,*) nolongerdecreases. Thestate so@*, intowhichtheautomaton W goes over from s 0 at the end of the experiment ~*, is equivalent to each state M (@*). Actually, if there were to exist a certain experiment 'G,, distinguishing s0@* from M (@*), then the set M (0*0,) could be rigorously contained in M (@*) and the dimensionality of M (@*'131)

would be lower than the dimensionality of M ('@*). Let us take a certain state qo E M (15'*). Now for any state s of the generallinear automaton ~ one can indicate an equivalent state q of the general linear automaton ).ß. For this purpose let us consider the set of states 2I accessible from s0~*. Its dimensionality r is equal to the dimensionality of 121. Therefore, any state s of the automaton 121 is a linear combination r (121) of states accessible from s0@*,

r(IJO

i.e., s = LJ a;s;, where si = s0@*~; (i = 1, 2, ... , r (21)). i=l

The state

r(Q[)

q= LJ aiqi i=l

of the generallinear automaton ffi where q; = qo~*~; (i = 1, ... , r (2!)), exists by virtue of the CümpleteneSS of .Q3 and iS equivalent to S, Since S; =So~*@; iS equivalent to q; = qo'f3*'[;;.

198 A. A. MUCHNIK

For the case in which the set of states of 21: belongs to a certain hyperplane r that does n<t contain 0 (the origin of V) - i.e., when any state s of the automaton 21: is a normalized linear combination ~ aisi (lj a; = 1) of the basis states si- it is sufficient to require w e a k

c o m p 1 e t e n e s s of the general linear automaton 58 in the condition of Theorem 3.

Pr ob 1 e m 1. The author does not know the degree to which one may be released from the requirement of completeness (or weak completeness) in the condition of Theorem 3.

If one considers simply infinite automata 21: and 58, where the automaton 21: is strongly connected (in the general sense), then indistinguishability of 21: from 58 is insufficient for the validity of Theorem 3 of E. Moore [18], whose analog for a generallinear automaton is the theorem that has been proved.t

Co r o 11 a r y 1 . If two general linear automata 21: and 58 are strongly connected and 21: is indistinguishable from m, then they are equivalent.

Co r o 11 a r y 2 • If the general linear automata 21: and m are strongly connected and the state q of the automaton 21: is equivalent to the state s of the automaton 21:, then the generallinear automaton 21: is equivalent to m (21:""' jt).

For each generallinear automaton 21: there exists a unique generallinear automaton Wh which is equivalent to it and has pairwise-distinguishable states; this automaton is called the reduced form of the automaton 21:.

5. Let us now stata the problern of experiments which allow the structure of a general linear automaton 21 of the type I having the dimensionality n tobe established. It is natural to restriet the analysis to the case of strongly connected general linear automata, since in other generallinear automata there are states about which one may not obtain any information during an experiment (in finite automata and Markovian chains these states are called nonrecurrent). The class of all such generallinear automata of the type I in reduced form over the field P having the input alphabet ~ and the output space Z is denoted by TP. z, n, z·

Theorem 4. Assurne 21: is a strongly connected general linear automaton of the type I over the field P having a dimensionality not exceeding n. A multiple experiment of length 2n begun at any state s of the general linear automaton 21:, allows the structure of 21, to be established; i.e., it allows the transition and output operators of 2r to be found.t

Pro of. Let us find the basis of the generallinear automaton 21:. Let us lexicographically number the vertices of the tree Ds of the automata 21: which has the height 2n with its root in s.

Let us assume that q1 = s. Let us prove by induction with respect to h that on the first h stories of the tree Ds for h :$ n1, where n1 is the dimensionality of the generallinear automaton 21, one may choose no less than h linearly independent states. For h = 1 this is obvious. Assurne this is true for a tree D5 (h) having a height h < ni' Let us assume that in the tree D5(h + 1) one cannot find (h + 1) linearly independent states (i.e., the number of linearly independent states of D5 (h + 1) is equal to h, and they are all situated in D5 (h). Then each state

tThe corresponding example was constructed in the paper by Ch. Facey [10]. The author introduced the notion of strong indistinguishability of automata. Ch. Facey proved the analog of the Mooretheorem for strongly indistinguishable strongly connected infinite automata.

t It is natural that in general no experiment allows determination of the set M of states of the general linear automaton m ; therefore the Supplements of general linear automata are actually determined in Theorem 4 and the succeeding theorems in the experiments.


of the (h +2)-nd storywill be a linear combination of states from the first h + 1 stories, and therefore from just the h first stories (considering the linear dependence of the states of the (h + 1)-st story on the states of the first h stories), etc. Thus, we find that in Ds there are only h (and not n1) linearly independent states, which is untrue. Therefore, in Ds (h + 1) there are at least h + 1linearly independent states o

From what has been proved it follows that all n1 linearly independent states of 121 may be chosentobe on the first n1 stories of Ds. Let us number all vertices of the tree D8 in the sequence of increasing stories, and lexicographically on one story. Let us choose the first n1 linearly independent states in Ds tobe on the first n1 stories of D8 • Let us denote them by q1=s1, q2 • ... ,qn1· Any state q in the n1 + 1 first stories isalinear combination q~>q2 , ... ,qn1· But for q17 q2 , ••• , qn1 and q we know the branches Dq17 Dq2 , ••• , Dqn and Dq ofthetree D5 having a height n, since n1 < n, while the length of a multiple experiment is equal to 2n. Recalling Theorem 1, we note that the expression for q may be found as a linear combination { q i} if Dq; and Dq are known. In finding the expression for all states of the first n1 + 1 stories in

terms of ql, q2, ••• , qn17 we thereby "close" the tree D5 in the graph of transitions-outputs and define the operators cp and ~ in the basis q1, q2, ... , q 11 , which is what was required tobe proved.

If we were to begin a multiple experiment in another state s* of the generallinear automaton 121, then, in general, we would obtain another basis q~, q;, .. . ,qi,1 having the same dimensionality in accordance with the definition of a strongly connected generallinear automaton, and the expression for cp and ~ in this new basis. Thus, we have defined the structure of a generallinear automaton 121 with an accuracy of up to the choice of the basis. Finding expressions for the state q of a generallinear automaton 121 as linear combinations of basis states is essentially the reduction of the generallinear automaton 121 • Thus, if we were given a nonreduced generallinear automaton 121 having a dimensionality not exceeding n in Theorem 4, a multiple experiment of length 2n on it would allow us to find its reduced form m1.

6. Let us now state the problern of constructing a simple experiment allowing the determination of the structure of any generallinear automaton 121 from TP, };, n, z. Let us note that the analogaus problern for finite automata (the class R n,m,p ) is sol ved by E. Moore [18] with sorting of all automata from R n,m,p. But in TP, };, n, z with an infinite field P there are infinitely many elements. Therefore, the Moore method is not tenable here.

Hereafter the field P is fixed, and in place of TP, 1:, n, z we simply write T };, n, z or even T 11 o

Theorem 5. There exists a prime brauehing experiment E which allows the structure and terminal state of any generallinear automaton ~ETn tobe determinedo

Pr o o f. Let us determine the structure of the general linear automaton 121 - the Supplement of ~.

Let us consider the experiment U 1u whose input consists of all strings of input letters a1, a2, ... , a m of length n(n + 1)/2 taken in some sequence. The length U11 does not exceed

( +1) n(n+1> i m ~ n (n2+1) (it can be proved that the length U may be chosen equal to m-2- + n (n2+ ) 1

[9]).

Such an input word U11 for each generallinear automaton ~ E Tn allows its terminal state tobe determined according to Theorem 2.

We take F 1 = U 11 to be the first place of the input of experiment E. Let us renumher all prime experiments of length 2n: a 1, a 2, ... , Cl!~. Let us carry out the experiment E1 = a 1 after

200 A. A. MUCHNIK

U11 • Assurne that the experiment-input F1E1F2E 2 ... FkEk has been constructed, where E1 = a;1 , E2 = CXi2, E3 = a;3 , ••• ,En = a\ , while Fi = U11 (i = 1, 2, ... , k).

Let us place Fk +t = U 11 • If the outcome of the experiment F k+1 is linear ly independent of the outcome F1, F2, ... , Fkt then we take Fk+1 tobe a 1•

In the converse case we consider all different groups of experiments (Fiu Fi2 , ••• , Fih) having linearly independent outcomes, on which the outcome Fk+1 depends linearly. (Note that since the dimensionality of ~ is equal to n, it follows that h :::; n, since the initial states of Fit, Fi2 , ••• ,Fih must be linearly independent.) Let us place Ek+1 equal to the first ai whose

outcome from the initial state (which is a final state for Fk+1) may not be determined from the previous part of the experiment (i.e., it may not be determined as a linear combination of outcomesforthewords Eit=a;,Ej2 =a;, ... ,Eih=a; foracertaingroup (Fit, Fi2 , ••• ,l\). In other words, let us assume that Ek+1 is equal to a ai suchthat additional information is obtained. The process is broken offwhenan Eg = ai is determined suchthat for the outcome of the experiment F g = U 11 the outcomes of all of the experiments a i begun in the terminal state sg of the experiment Fg may be determined from the experiment E = F1E1F 2E 2 ... FgEg.

In order to conclude the proof it remains to note that the outcomes of all experiments a i having the length 2n are determined in the state sg, and the structure of the generallinear automaton I2I may be determined in accordance with Theorem 4.

Let us note that each a i is encountered in E no more than n times, and therefore it is

1 n(n+!l

necessary to take g = n • m 11 • Since l (F;) = n (ni ) m-2- , while l (Ei) = 2n, it follows that

n(n-H)

z (E) = g ( n (ni 1) m-2- + 2n). Hereafter we shall be able to construct a substantially shorter

experiment ~, which recognizes the structure of a general linear automaton ~ for certain additional information on ~.

For the time being we note that the uniform experiment U211 allows any automaton 2I E Tn tobe distinguished from any other g) ETn. In fact, let us consider the direct sum 12I 8:) )!.),

which is a general linear automaton of reduced form having the dimensionality 2n. The experiment U211 , which is begun in a certain state ~ 8J ~, allows the terminal state 2I ct) >.0 tobe determined (and therefore the initial state also) along with that automaton from ~ and ~. to which it belongs (i. e., it allows ~ to be distinguished from ~). However, from this reasoning it does not follow that the experiment U211 yields the algorithm for determining the structure ~ E Tn , since here it is necessary to compare 2I with an infinite set of general linear automata from T11 •

Pr ob 1 e m 2 • Does such an algorithm exist?

Pr ob 1 e m 3 • It is required to construct the shortest possible (uniform or non uniform) experiment U 11 , which allows the terminal state of any general linear automaton 2I E Tn tobe determined. t

From E. Moore's example ofafinite automaton having n states (the "secret lock" [18]), which evidently is a generallinear automaton from T 11 _ 1, there derives the lower bound for the length U11 - 1 equal to M11- 1 (i.e., l(U11 ):::: m 11 ).

For finite automata from Rn,m,p we were able to obtain an upper bound of order n2 In (ne) m11 for En [9]. However, our method of constructing Un for Rn,m,p rests on the sorting

tBeginning here, U11 denotes an arbitrary experiment with such a property, while U~in denotes the shortest experiment among Un.


of automata from R n,m,p , while the length Un depends on the power of R n,m, p; therefore, for the case of a generallinear automaton with, in general, an infinite TP, n, s, z, this method is unsuitable.

An experiment U11 for R n,m, p, which is Ionger than the experiment in [18] but does not have sorting of all automata from R n,m,p, was constructed in the paper by Yu. M. Borodyanskii [2].

Let us now state the problern of constructing a simple uniform (nonbranching) experiment which allows the structure of any IJ ETn tobe determined.

Theorem 6. There exists a uniform experiment F which allows the structure of any general linear automaton 21ETn to be determined.

Proof. Assurnexis an arbitrary word in the alphabet 2::, x"=x·x· ... ·x • Then, "--.-'

k times

knowing the output corresponding to the word x211 for the initial state s 0 of the n-dimensional generallinear automaton m, we may determine the output of any word xk for any k for the same initial state. Let us place sk = soXk (k = 1, 2, ••• ). Let us denote the outputs corresponding to the initial state s and the input string xxx ••• , by y(s). Let us take the first state St in the string { s k } (k = 0, 1, 2, •.• ) , which is such that y (sk) depends linearly on the preceding y(si). It is obvious that t :s; n, since the maximum number of linearly independent states in {sk} does not exceed n. Two outputs y(s) and y(g) coincide if they coincide on the input word x 11 • This is proved in the same way as Theorem 1 (for autonomaus automata having an input consisting of the word x).

Assurne y11 (s) is the initial piece of y (s), which corresponds to the input word xn. Let us represent Yn (s1 ) as the linear combination {y11 (si)} (i = 1, 2, ••. , t- 1). y(st) is expressed by the samelinear combination {y(si)}. Since we know the outputs y211 (s11 ) (1. :s; i < t), one can determine the output y211 (St); then all outputs y311 (sd (i < t) become known, and the output y311 (st) can be determined, etc.

Let us place

where { a i } (i = 1, 2, .•• , u = m211 ) is the set of all words of length 2n in the alphabet 2::. Assume s is the terminal state of the experiment F. Finite pieces of F will be words of the form

2n ( 2n)2n ( ( 2n)2n)2n ( ( ( 2n)2n)"n)"n CXu , CXu-1CXu , CXu-z CXIL-!CXu , ••• , CX1 CXz • • • CXu-iCXu " " •

Therefore, we may determine the outcomes of the experiments

( ) 2n ( 2n)zn ( ( 2n)2n )2n CXu, CXu-1 CXu , CXu-2 CXu-1! CXu ' • • • CX1 CX2 • • • CXu-!CXu • • • •

from the state s taken as the original state. But thereby all prime experiments a i of length 2n will be defined ins (i.e., the generallinear automaton W will be defined).

The length of the experiment Fis fantastically great, but on the other hand the algorithm for determining the structure of W from F, given in the notation presented above with parentheses, is fairly simple.

Theorem 7 (see below) allows a considerably shorter experiment than F tobe constructed, but the algorithm reconstructing the structure of the generallinear automaton from such an experiment turns out to be considerably more complex. Here we come up against a frequently

202 A. A. MUCHNIK

encountered phenomenon: a reduction of the length of the calculation accompanies the complication of its program (see the collection of translations: "Problems of Mathematical Logic," Moscow, Mir (1970), Introduction).

Pr ob 1 e m 4 • It is required to construct the shortest possible uniform experiment which allows the structure of any generallinear automaton W E T n tobe determined.

7. Usually, before an experiment is conducted with an automaton W, we have certain information available on it. Otherwise, as E. Moore [18] showed, it is impossible to determine either the subsequent behavior of the automaton or its structure. This information may be incorporated in the upper bound of the number of states of a finite automaton or the dimensionality of a generallinear automaton, in the indication of strong connectedness of an automaton, the type of generallinear automaton, the length of an experiment that establishes the final state of a generallinear automaton, or directly in the indication in this experiment

-+ -+ -+ of the complete set n of words x1, x2, ••• , xt (i.e., a set suchthat the states vx1, vx2, ••• , vxt

-+ form a complete system of vector-states in W for any state v).

We shall consider a Moore generallinear automaton for convenience.

The set of words Y = {y 1, ••• , y r } is called the r e f er e n c e s e t f o r th e g e n e r a 1 linear automaton W if distinguishability (indistinguishability) of any two states of W -+ -+ v1 and v - may be established by a multiple experiment having the inputs y 1, ••• , Yr.

Rem a r k • It is obvious that for a complete general linear automaton the following -+ -+

statement holds: if the vector u whose coordinates ui are the outputs ofthe states vyi (i = 1, ••• , r) [i.e., ui = :;A(yi )b] is a linear combination of vectors t:"< 1 ), ••• , t;"<k) corresponding to the states v<1), ••• , -;<k), then the vector -; is the same linear combination of vectors v(1 ), ••. , v<k) • For a weakly complete generallinear automaton the analogaus statement is valid for normalized linear combinations.

The reference set Y is called the mini m a 1 r e f er e n c e set if none of the words y 1, ••• , y r may be replaced by its own final piece in such a way that the resulting set of words remains a reference set.

Knowing the complete set n and the reference set Y of a reduced strongly connected generallinear automaton ~. we may construct a uniform (i.e., independent of the initial state) multiple experiment which allows the structure of the general linear automaton W to be established. Let us call the sum of the lengths of the words that determine the multiple experiment 9, its resultant length, and let us denote it by L (9), while the number of words of the experimentshall be called its multiplicity K (9). Let us use L(A) to denote the sum of the lengths of the set A = { a1, ••• , aq}, while Ji.A denotes the power of A.

Lemma 1. Ass ume m is a c omplete c onne cted gene r al 1 ine ar automaton W in a reduced form having the input alphabet ~ =

{ a 1, ••• , am}, l1 = {x 1, ••• , xt} is a complete set, and Y = {y1, ... , Yr} is the reference set of the words of W. Then there exists a uniform multiple experiment having the resultant length

L~. Y, TI= (m + 1) (tL (Y) + rL (TI)) +mrt

and the multiplicity K~.Y.II=(m+i)!LY·!LII, which establishes the structure of W.

Pr o o f, Let us consider a multiple experiment having the set of input words

(i=1, ... ,t; j=i, ... ,r; k=i, ... ,m).


~ ~ ~ ~

Assurne v0 is a certain state of Wo The states VQX1, VQX2, ... , VoXt form a complete sys-tem of vectors in the space of states of W o Knowing the outputs of the states v oXiYi for all i, j, we may, according to the remark preceding the lemma, isolate the complete linearly independent subsystem of the system of vectors { VoX J (ioeo, the basis of the general linear automaton}, while from TI we may isolate the basis set of words B o Knowing the output of the states ~ ~

VoXiakyi for all j, we may determine how the vector-state VoXiak is expressed linearly in terms of the basis vector VQXi, x; E II, and thus determine which linear transformation carries out the entry of ak in the basis {%xi}, xi E B. Having performed this operation for all ak E ~ , we determine the matrices of the transitions and outputs in the basis fvoXJ (ioeo, the structure of the generallinear automaton m} 0

The sum of the lengths of all words :lS_Yj is equal to tL(Y} + rL(ll}, while the sum of the lengths of all words xi ak Yi is equal to

mtL (Y) +mrL (II) +mrt.

Thus, the resultant length of a multiple experiment would be

(m + 1) [tL (Y) + rL (II)] + mrt,

while

K'i,,Y,rr=(m+1)rt.

The lemma has been provedo ~

If we were to desire likewise to determine the initial state v0, then it would be required to perform experiments on the inputs y i (j = 1, .. o, r} having a totallength L (Y} in the state v0o

Let us note that for any general linear automaton W E T n the complete and reference set of words is the s et of all words having a length no Ionger than n - 1. Therefore the resultant length of a uniform multiple experiment for a generallinear automaton of class T n is (2n - l}m2n -i; ioeo, it is a multiple experiment of length 2n (if, according to the tradition going back to E o Moore, the length of an experiment is assumed to be the length of the chain of states corresponding to the input word, which is one greater than the input word} o Since, however, f.lne may select the reference and complete sets of words for a generallinear automaton 2! E T n having a considerably shorter resultant length (as we shall see further on, the first has a resultant length <n2/2, while the second has a resultant length < n3/2 -the nurober t ~ n2,

r ~ n), it follows that the estimates given by the lemma for this case will be:

while

As far as the lower bound for a universal uniform multiple experiment in the class T n is concerned, constructions analogous to "the secret lock" may be used to prove that it coincides with the upper bound; ioeo, the bound (2n- l}m211 -l for the class T11 cannot be improvedo

Let us consider the case in which we know only the reference set of words Y = {y1, .. o, Yr} for a complete generallinear automaton ~ E Tn, and this time let us construct a branching multiple experiment which determines the structure of ~.

204 A. A. MUCHNIK

Lemma 2. There exists a branchingt multiple experiment which allows determination of the structure of the complete general linear a u t o m a t o n ~ E T n, 1: h a v i n g t h e r e f er e n c e s e t Y = { y 1 , ••• , y r } o f r e -sultant length no greater than

and apower Kl:,Y<(m+1)nr.

Pr o o f • Initially let us carry out all prime experiments with inputs y 1, ••• , y r and inputs akYi (k = 1, ••• , m; j = 1, ••• , r). Let us assign the state v0 to the basis and use B 1 to denote the set {\i0 }. Among the states v0ak (k = 1, ••. , m) one can find at least one that is linearly independent of v0• Let us choose from among the states v0, v0 a1, ••• , v0 a m the maximal linear ly independent system containing v0• This may be done, since we lmow the outputs of the states ..... v0akyi by virtue of the remark preceding Theorem 5 (on p. 199). Let us denote these vectors by ;b ;2, ... , ~1 , their corresponding inputs by akp ak2 , ••• , ak;1, and the sets {;0, ... , ;:1}

by B2• For each of these vector-states from (B2 "'- B,) we carry out all experiments with inputs akyi and find from among

-+ -+ _,.. -+ v 0, vb ..• , v;l' v;ak (i=O, 1, ... ,i1; k=1, ... ,m)

..... the maximal linear independent system containing ;o, 7;, ... , 7;;1 • Among the vectors viak

one can find at least one vector that is linearly independent of the vectors ; 0 , ;,, ••• , ;;1 , provided only that these latter vectors do not form the basis for ~ (i.e., if their num:ber is less

than n). Let us denote the vectors added to B2 by ;;1+t, ... , ;:2, while the systeni of vectors obtained is denoted by B3, etc., until at a certain step we obtain

For states from (Bq "'- Bq-1) let us perform all experiments Uk Yi. Thus, for each basis state vi we will know how the vectors ~ ak are expressed in terms of v i (i.e., we will define the transformations (jlcrk of the vectors V i). The outputs in the states vi are determinate» and therefore the structure of the initial state of ~ can be determined.

Let us estimate the resultant length of the experiment. For each basis state we perform all experiments with inputs akyi' while for v0 we likewise perform all experiments Yj. The sum of the lengths of the corresponding words is

(mn+ 1) L (Y) +mnr.

Here we see that the nurober of all words in our experiment is equal to

mnr+r= (mn+ 1)r.

Moreovar, the resultant length of the experiment includes the sum of the lengths of the words required to attain the state vi' multiplied by mr, since each word is encountered with mul-

t A branching multiple experiment, unlike a unüorm multiple experiment, is constructed so that the next input word of this experiment is chosen as a function of the output of a general linear automaton in words defined earlier, while the input words are defined and applied to the input one after the other in one and the same initial•state.


tiplicity mr. Since the length of a word leading from v0 to ;i E (Bz "'- Bz- 1) is equal to l- 1, this sum is equal to

q-1

mr ~ l~-t(BZ+ 1 "-Bz), 1~1

where J.tC denotes the power of the set C. Since all sets (Bz+t "'- Bz) and B1 arenot empty, it q-1

follows that q :::: n, and the expression ~ l~-t (Bz+t "'- B1) attains a maximum for q == n when 1~1

ll (Bz+t "'- Bz) = 1 • This maximum is equal to

n-1

~ l= n(n;,-1).

1~1

Thus, we find that the resultant length of the experiment does not exceed

(mn-+- 1) L (Y) -t-mnr-t-m n (n2-1) r=mn ( L (Y) + r+ n;-1 ) -t-L (Y),

which is what was required to be proved.

It is not difficult to see that the multiple experiment constructed in Lemma 2 is brauehing. For the chosen initial state -;0 we constructed a certain set of words X leading from v0 to the states -;0, ... , vi, ... , v 11 _ 1 forming the basis in 1}1. A knowledge of the number n is not required to construct the experiment here either, since the experiment stops as soon as it is discovered that all vectors viuk for all vi from a certain set Bq can be expressed linearly in terms of ;i E Bq. However, the set of words X may not be complete, since the system of vec-

-+ -+-+ -+ ~ tors {vx}xex , being the basis for v == v 0, may not be such for another initial state v == v ~· For example, assume that the general linear automaton W having the input alphabet ~ == { 0, 1} is stipulated by the transition matrices

(cyclic permutation),

a;. o. ~= { 1, if j == i -1- 1 (mod n), 1 ( ) 0 otherwise

{ 1 for i = j = 1 ,

a·· 1 -' 1 ( ) -- 0 otherwise.

_. _. The output vector is defined thus: b == (1, 0, ... , 0), i.e., in the state v1 the output is equal to 1, while in v i (i > 1) the output is 0. Obviously, all states of 1}1 are distinguishable. Then the set of words X == { A, 0, 00, ... , 0 n-1} for each basis state -;i will generate a basis { ;i }, while it leaves the vector-state of the form v == (a, a, .... a) as is (i.e., the setX is complete, so to speak, r e 1 a t i V e t 0 any Vi, but it is not complete for \}1). lt can be proved that for COmpleteneSS of the set X of the input words of a complete generallinear automaton 1}1 in reduced form it is sufficient if the set of transition operators for the words <Px is complete in the linear space generated by the transition operators <Pu for all words u E F C~). Therefore, one can always choose a complete set of words containing no more than n2 words for a complete general linear automaton 21 having the dimensionality n, since the dimensionality of the set of all n2 matrices is n x n. In the example given above, the dimensionality of the set of matrices II aii (u) ~ for all words uE{O, 1}* is equal to n2•

Definition. Let us call the set ofwords X perfectly complete for the general linear automaton W, ifthesetofalltransitionoperatorsqJ for xEX iscomplete

206 A. A. MUCHNIK

in the linear state generated by the operators cp for all words E F (~). The perfectly complete set of words for a given generallinear automaton m can be found in the same way that we found the basis set of vectors in the proof of Lemma 2 (i.e., we being with A, then sort ak, which are all words of length 1, choosing from among them a maximal set such that the corresponding operators CflA, Cflcr~< are linearly independent, etc.).

It can be shown that a perfectly complete set is likewise a reference set. The following problern develops.

Prob 1 e m 5. Do n-dimensional complete generallinear automata W exist for which any complete set of words is perfectly complete? For which fields P may this hold?

Note that if the set of words X is such that for any ward x1 EX one may indicate a state v of the generallinear automaton m suchthat vx1 cannot be expressed linearly in terms of the remaining vectors 'Vx, where x EX, then the set of operators Cflx (x EX) is linearly independent.

Finally, we take account of the fact that each of the sets of words: the complete set relative to the given initial state v, the complete set for the generallinear automaton m, and the perfectly complete set for the general linear automaton m may be chosen so that

1) The minimality processes will be fulfilled; i.e., from the set one may not discard a singleward in such a way that the set remains complete relative to v, complete, and per-... fectly complete, respectively. Weshall call such sets the basis set relative to v, the basis set for the general linear automaton m, and the perfectly b a s i s s e t, respectively.

2) Along with each ward, the set X contains all of its initial pieces.

3) (The completeness criterion.) If a certain set of words R contains A and has the

property that for each ward ~ER all states ; ~cr~< (k = 1, ... , m) can be expressed linearly in ~- - ~

terms of v x (xE R) for the state v of the generallinear automaton m, it follows that the set R is complete relative to v.

... If the set R has the indicated property relative to each state v, then the set is obviously

complete.

4) (The criterion of perfect completeness.) If the set of words R contains A and for any ward ~ER the operators Cfla~< or Cfla x (k = 1, ... , m) of the transitions can be expressed

h

linearly in terms of the operators Cflx (xER), then R is a perfectly complete set.

The following metbad of constructing the complete set of words n for a general linear automaton 121: may be based on these criteria.

All words in Fa;) are ordered lexicographically.

Assurne x1 = A. Assurne x1, ... , x 5 are defined. x5 + 1 is equal to the first ward, which is suchthat for a certain state Wthe state ;Xs+t cannot be expressed linearly in terms of the state vxi' i = 1, ••• , s.

This process will be broken off, since the number of words x cannot be greater than n2

because the system of operators {Cflxi} is obviously linearly independent. When the process breaks off, the set n will have been constructed. The properties 1), 2), and 3) are fulfilled for it, and therefore it will be complete. Since for each ward x and each state v the state Vx: can be expressed linearly in terms of the state Vx:i, where xi lexicographically precede x (or coincide with x) while the set of all words of length not exceeding n - 1 form the complete set, it follows that the length of each word from n does not exceed n- 1. Thus, in n there are


no more than n2 words of length not exceeding n- 1. Therefore, by virtue of the property 2)

L(H)-:Snn(n;-1)< ~3.

Pr ob 1 e m 6 . May a perfectly complete set R of ..vords for ~ be constructed in this way? It is required to give the upper bound of L(R).

Let us now consider the case in which we know the complete set TI = { x1, ... , x t} for the generallinear automaton ~ E T n. The reference set y ={y 1, ... , Yr } is called the minimal reference set.if none of the words y1, ... , Yr can be replaced by its own final piece in such a way that the resulting set of words remains a reference set. Here it is necessary for us to solve the dual problern to the one that we solved in the proof of Lemma 2, namely: to construct the minimal reference set of words.

Lemma 3. Assurne that the complete general linear automaton ~ E T n, }; an d TI = { x 1' ... , x t } i s t h e c o m p 1 e t e s e t o f w o r d s f o r ~ .

Then there exists a brauehing multiple experiment that determines the structure of W and has a resultant length no greater than

L};,rr-:Sm(n-1) (L(I1)+ ~)

and a multiplicity KJ;,rr-:S(m+1)nt.

Let us denote the initial-state vector by Vo, while VoXi is denoted by vi (i = 1, ... , n).

We shall consti:'uct the set of reference words Y by steps: we carry out all experiments xi (i = 1, ... , t). We place Y 1 = {A}; A is an empty word; s1 = {ir"}, where the vector Ü = (u1, ... , ut); ui = ~\vi) is the output of the states v, and ~ is the output operator of the general linear automaton ~ (a Moore automaton!).

From among the vectors 7t (a1), ... , 7t (am), where ~ (ak) = (ul (ak), ... , Ut (ak)), while ui(ak) = 'ljl (;;a") (i = 1, ... , t; k = 1, ... , m) we choose the maximal set R1 of vectors which form (along with S1) the system R 1 = {~, ~ (Yt), ... , -;_; (Yh)}, where Yz denotes the corresponding letters ak.

Let us place Y2=Y1 U{Yt} (l=i, ... , jt), while S2=StUR1.

Further we consider the vectors ;; (akYt) (l = 1, ... , it; k = 1, ... , m) , where u; (akYt) =

'ljl ("J;akYt) , and choose from among them the maximal set R2 of vectors ;;(akYt), suchthat the union S3 = R2 U S2 is a linearly independent system. The corresponding words {akYt} are denoted by Yh+t, ... , Yi2:

Then we consider the words ahYt (j 1 < l <. j2; k = i, ... , m) and the vectors ;; (akYt), etc., until in a certain set Sq there turn outtoben linearly independent vectors. It can be proved that Y i (i = 1, ... , q) is supplemented on each step, whence it follows that q :s n. The corresponding set af words Yq will be a minimal reference set. The point is that from th~ con,"'t.-ruction of Yq it follows that for each Y;EYq (i=i, ... , jq) and each a"E~ the vector u(aky;) can be expressed linearly in terms of the vectors Ü (yz) (l = 1, ... , jq), whence it follows that in

-+ each state v of the generallinear automaton ~, we may define the output on the word akYi in this state knowing the outputs on all Yz E Yq • Further we define it the same way on all words of the form ak1ak2y1, etc. In this way the outputs in the state v on all w0rds y E ~* ara defined.

208 A. A. MUCHNIK

.... In view of the arbitrariness of the c hoice of the state v, it follows from this that Y0 is a refer-ence set. The minimality of Yq derives from the fact that for each Yi E Yq (i = 1, ... , jq) the output in the state ryi for an arbitrary state v may not be defined as a linear combination (independent of ~) of outputs of a generallinear automaton in states vy1 (l =I= i, Yz E Yq), while the set Yq ""- {Yi} contains all intrinsic final pieces of the ward Yi. Thus, y i may not be replaced by any of its intrinsic final pieces. Thus, in the process of constructing Yq all experiments xiyz, xiakYz (i=1, ... , t; 1=1, ... , jq; k=1, ... , m) are carried out; i.e., the structure of a general linear automaton ~ is determined according to Lemma 1.

The reader will establish without particular difficulty the fact that the resultant length of the constructed experiment does not exceed

m(n-1)(L(II)+ t;),

while its multiplicity is

K};, n<(m+ 1) tn.

Now we shall formulate and give the outlines of the proofs of three the.orems on prime experiments which allow the structure of a generallinear automaton ~ E Tn to be established, provided that the inputward E allowing the determination of the concluding state of the general linear automaton ~ is known, along with either the complete or reference set of words ~,

or both. Let us begin with the latter case.

Theorem 7. If the complete generallinear automaton ~ETn, then, knowing E, the complete set of words II, and the reference set Y, one may construct a prime brauehing experiment ~. that determines the structure of ~ and has a length no greater than n(K};.r.nl(E} +L};,Y,n), where LI.,Y,n and K};,Y,n are the resultant length and power of the multiple experiment 9, constructed in Lemma 1 for recognition of the s t r u c tu r e o f a g e n er a 1 li n e a r a u t o m a t o n, w h il e l ( E ) i s t h e 1 eng t h o f the ward E.

It is necessary to repeat the proof of Theorem 5 and Lemma 1 in their general features:

where q::::; n(m + 1)rt. F1 is the firstward of the multiple experiment 9. The next application of a ward E allows determination of whether or not the terminal state of the experiment EF1E coincides with the terminal state of E; if it coincides, then we assume that F 2 is equal to the second ward of 9. Otherwise, F2 is equal to the firstward of 9, etc.

R e m a r k 1 • The construction of the experiment ~ does not require knowledge of the nurober n (the dimensionality of ~) but is required for the upper bound in which n is included linearly.

Rem a r k 2 • As we shall see in the optimal selection of E, Y, II,

n2 n2 n3 l(E)< 2 , L(Y)< 2 , L(Il)< 2 , t<,n2 , r=n,

KI., Y, n<(m+ 1) n3, LI., Y, n< (m+ 1) {n4 +n3)

(see Lemma 1). whence

l(~)<n[(m+1)n3 ~2 +(m+ 1)(n4 +n3)l<(m+1)(n6t2n°+2n4).


Theorem 8. If a complete WETn, Y ={y1, ... , Yr} is the refer-ence set for W, while E is a uniform terminal experimentt for W, then there exists a prime brauehing experiment lb, which establishes the structure of 21, and has a length no greater than n 3(m + 1)(l (E) + L(Y) + n) + n3r.

Pr o o f. We shall construct the experiment lb again according to steps:

where w1 = y 1, x1 = A, II1 = { x1 }, W 1 = {A}. Then assume the experiment lb P = Ew1Ew2 ••• Ewp has been oonstructed and the sets IIP = { x1, ••• , x s} and W 1, ••• , Wp have been defined.

-+ Assurne v0 is the state of the generallinear automaton W at the beginning of the ex-

periment lb. Here Üp+1 denotes the state into which the generallinear automaton W has arrived at the end of the experiment GpE. From the output of the generallinear automaton on the last piece E of the experiment E one can determine via which linear combinations of

-+ -+ -+ vectors u1, ••• , up the vector up+i is represented (and whether it is represented at all). Let us define the set of words W P+i: :1 E W p+1 if and only if there exists a linear combination

I -+ -+ I

;~ a;uz; = up+f , where 1 :o; l i :o; p, such that x E ;~1 W l;. Let us consider the set of words xiyj

and xiakyj (i = 1, ••• , s; j = 1, ••• , r), which is orderedas follows:

while if i1 = i2, then

x;yj-< x;akyj for j 1 < j2 for all i, j, k;

X;U1<1Y j <( X;Uk2Y j for k1 < k2;

whence it follows from transitivenessthat

If one can find just one input word XiYj or xiakyi which does not belong to Wp+i• then we assume that Wp+l is equal to the first of such words in the sense of the order which has been introduced. Let us place

It is not difficult to establish the fact that Wp+i consists of all words for which the outputs in the state ;p+i have been determined from the experiment 'GpEWpH·

_... If after this the outputs on all input words Xj_Yj and xiakyi have been defined in the state up+1 (i.e., they belong to Wp+1), then we consider the vectors

-+ -+ up+!Xf and Up+!Xiak (i = 1, ... , s; k = 1, ... , m).

-+ -+ If the vectors up+lxiak can be expressed linearly in terms of the vectors upxi (and this

can be found out from the outputs of the state up+1 on the words xi akyi and xiyi, since Y = {y1, ••• , Yr} is a reference set), while Ilp contains A and all the initial pieces of its own words, then from the criterion of completeness relative to a state the set Ilp is complete relative to Üp. t A terminal uniform experiment E is understood to be an input word which defines a terminal

state.

210 A. A. MUCHNIK

-+ From IIp one may isolate the basis set of words B relative to u P, while in the basis

{;px} (xEB) one can define the matrices of transitions A(ak) and outputs in the states

{;;px} (xEB, k=1, ... , m),

i.e., one can determine the structure of the generallinear automaton ~. Then ~ = ~pEwP+I· -+

Re mar k. If for up+t the experiment ~PE has been used to define the output on the words xiyi, xiakyi (i = 1, ••• , r; k = 1, ••• ,m) .... for those i suchthat all vectorsüp+txiak can be expressed linearly in terms of the vectors up xi, then one can again find the basis among the vectors Üpxi and go on to find the transition matrices A(ak) and outputs in this basis. In this case ~=~11E. One may avoid the sorting of all possible subsets IIP with the object of clarifying their completeness relative to \rp+t by requiring that for Üp+t the output must be defined on all words x1yi and xiakyi, where x; EIIP.

-+ -+ However, if a certain vector up+txzak cannot be expressed linearly in terms of u +txi (i = 1, ••• , s), then assume xi1 is equal to the least of such l, while k1 is the least of such k for a given i 1, and p!ace

Xs+i = Xi10'h1, ll P+l = {xh •.. , Xs+!},

Wp+i=Xs+iO'tYi! Wp+! = Wp+i U {wp+i},

while ~P+i = wpHE , etc.

Let us prove that this process will break off.

Actually, the set IIP may not contain more than n2 words: from the construction it follows that each word x; E IIp for i > 1 is such that for a certain Pi

llpi "'- IIP;-1 = {xi} and X; =Xi10'h1

-+ -+ for certain i1 < i and k1, and upxi cannot be expressed linearly in terms of upxz (l < 1); from this it follows that the transition operator CJlx; cannot be expressed linearly in terms of the operators CJlx1 (l < i) , and therefore the system of operators CJlx1 is linearly independent for xz E llp , while the dimensionality of the system of operators CJlx, xE ~*, of an n-dimensional generallinear automaton does not exceed n2• Thus, the set IIp is stabilized. li = lim IIp.

TJ->00

Assurne li = {x1, ••• , x 5}. For such a word xiyi and every word xiakyi (i = 1, ••. , s; j = 1, •.• , r, k = 1, ••• , m) it is necessary to perform no more than n experiments Exiy i or Exiakyi, respectively. The total number of words xiyi iss· r ::5 n2r, while the total number of words x iakyi is s • m • r ::5 n2mr. Therefore, the length of the experiment ~ does not exceed

n [(n2r + n2mr) l (E) + (n2 + n2m) L (Y) + L (II) (m + 1) r + n2r],

where L(II) is the sum of the lengths of the words xE II, while n2r is the upper bound for the number of entries of ak in all words xi akyi •

It remains for us to estimate L(IT).

From the method of constru cting the set li it follows that along with each word x E n all initial pieces of x are entered in li.

If the word xi is defined on the step p, then x i = x i ak, where l < i1, and for the states -+ 1 u P the outputs on all words of the form xyi are known from the preceding experiment, where j = 1, ••• , r, and the length of x is less than the length of xi. Otherwise, we would have taken


the largest initial piece of x, which is the word xz E II, to be such that l < 1 (in view of A E II one can find such a piece). From the construction of the experiment e it follows that the outputs of all words x1 Ok Yi are defined for the state Ü p, and all vectors Ü p xz ak can be expressed linearly in terms ofÜpx1, ü"'px2, ••• , Üpxz.

Otherwise, on the step p there would be added to n p-1 one of the words x h ak , where h ::s l, 1 ::s k ::s m, which is shorter than xi. But then the length of xi may not be greater than n - 1, since all words x of length ::s n - 1 form a complete system for any state Ü of the general linear Moore automaton 121, while if xi 2:: n, then all words Xf, f < i, would form a complete

~ ~

system for up ~hich would contradict the condition: UpXi cannot be expressed linearly in terms of the vectors u P Xf• Thus, the length of each word xi E II is no greater than n- 1, while their number is ::s n2• From this L (II) < n3 .t Thus, we obtain

l (e) < n3 (m + 1) [l (E) + L (Y) +nJ +n3r.

For the optimal choice of E and Y we obtain

A paradoxical result is obtained: the bound l (~) for given optimal E and Y turns out to be less than the bound l (~) in Theorem 7 for optimal E, Y, and n. However, this can be explained by the fact that during the process of constructing e in Theorem 8 we actually do not construct the entire complete (for the general linear automaton 121) step n, while in return the construction algorithm becomes more complicated.

Theorem 9. Assurne that for a complete general linear automaton 121ETn we know E and ll = {x 1 , ••• , xt}. Then a prime branching experiment e exists having the length

which allows the structure of the generallinear automaton 121 tobe defined.

Pro of. Let us place F1 = x1, y1 = A, Y 1 = {y1}, ~~ = EF1• Assurne that the experiment op = EF1EF2 ••• EFp has been constructed (i.e., F1, F2, ••• , Fp have been defined), and assume YP ={y1, .•. , Yq }.

~

Assume 121 has made the transition from the initial state v0 under the action of o pE to ~

the state up for which from a previous experiment the output has been defined on a certain set of words (perhaps empty) of the form xiyi and xi akyi (i = 1, ••• , t; j = 1, ••• , l; k = 1, ••• , m).

We shall order all words of this form as in the proof of Theorem 8. ~ ~~

For each word y and state v we use w(v, y) to denote the vector having the coordinates ~ ~

1{! (vxiy) (i = 1, ••• , t). If in the state u p the outputs are defined on all words XiYi, xiakYi, then we see w~t_2er the vectors ;(;, akYi) for k = 1, ••• , m; j = 1, ••. , q are expressed linearly in terms of w(up, Yi) (j = 1, ••• , q).

a) If this is so, then we assume Y = Yq, and considering Y to be a ref~rence set we define the matrix of transitions A(ak) in a basis which is first isolated from {upxi} by means of Y (as in the proof of Theorem 8).

tBy finer reasoning one may obtain the upper bound for L(ll) equal to n3/2.

212 A. A. MUCHNIK

b) If this is not so, then we take the least h suchthat for a certain k the vector ;(;p. a«Yh)

cannot be expressed linearly in terms of {;(U:,, Yi)} (j = 1, ••• , t). Assurne kt is the least of such k for j 1• We add the word a«1Yh to Yp, having placed

In this case we place Fp+i =x1u1Yq+1• -+

c) However, if in the state up the outputs were not defined from the experiment ~PE on all words XfYi and xiukYi (xi E II, ak E ~. Yi.E Yp), then we take the first of such words in our ordering and place F P +1 equal to this word, while Yp +1 = Y P • If after this we turn out to be in situation a), then the transition and output matrices of the generallinear automaton ~ are defined.

Let us prove that the process breaks off. We use Y to denote the set of all Yi defined by our procedure. From the construction it follows that:

-+ -+ -+ 1) for any state u the vectors w(u, yi) (j = 1, 2, ••• ) are linearly independent;

2) along with each word Yi, the sets Yp, containing the word Yi , als~ c2ntain ail of its "tails" (i.e., its final pieces). The i-th coordinate of the vector w(u, Yi) is equal to

'ljJ (~XiYi) = Cij.

Since the rank of the system of vectors {lrxJ (i = 1, ••. , t) is equal to n, it follows that the rank of the matrix II c iJ' II is likewise equal to n, and since all of its columns are the vec-

-+ -+ tors w(u, Yj ), it follows that their number is equal to n. Thus, in Y there are no more than n words. Since each of the words xiy, xiuky, where xiEII, akE~, yEY, maybe taken as Fp no more than n times, it follows that on some step the outputs on all words

-+ can be defined in the state up from the experiment ~PE or ~PEFp+i , while since the set Y may not be expanded further, the procedure breaks off and the structure ~ is defined.

Computation shows that

l (e)< (m+ 1) t-11Y ·nl (E)+ (m+1) ·flY·nl (IT) + (m+ 1) tnl (Y) +mtn-flY,

where f../,Y = n, l (Y) ::::: [n(n- 1)]/2, whence

which is what we were required to prove.

If E and II are chosen optimally, i.e~,

n2· n3 l (E) < T, l (IT) < T, t<.n2 ,

then


Note that Lemmas 1-3 and Theorems 7-9 arevalid foragenerallinear automaton 2I E T~ belonging to the class of reduced strongly connected n-dimensional generallinear automata of the II type if instead of completeness of the generallinear automaton 2I we require weak completeness of 2f, while linear dependence of the vectors is replaced by n o r m a 1 i z e d linear dependence in the proofs.

One may also reject the requirement of completeness of a general linear automaton (or weak completeness), while in Lemmas 1-2 and Theorems 7-8 one may stipulate the set Y, which is the reference set for the supplement (weak supplement) of the general linear automaton 2I E Tn (T;,) in place of Y.

Assurne now that we know only the experiment E which allows the terminal state of the investigated general linear automaton 2I E T n to be established. The class of such general linear automata shall be denoted by T11 (E). Then, taking account of the fact that the set of all words having a length no greater than n is complete in any n-dimensional general linear automaton, we obtain the following corollary.

C or oll ar y. For any generallinear automaton 2I E Tn (E) there exists a prime brauehing experiment F which establishes the structure of 2I, its length being

l (F) <; (m + 1) n2 (tl (E) + 2nt),

where t = m11 • (In order to define the output on all basis words it is sufficient for us to define it on all words af length n.)

8. Let us also note a number of cases with a priori information on the investigated automaton. Let us consider the case of a generallinear automaton 2I in reduced form having nondegenerate operators cp"i' i.e,, operators whichyield a one-to-one transformation of the hyperfine r M •

In the corresponding finite automata a permutation of states occurs due to the action of the input Ievels; therefore, S. Ginsburg called such automata permutation automata [16]. The class of such general linear automata shall be denoted by S 11• The intersection of the classes T 11 and 811 shall be denoted by s:1 • Modifying the Yu. M. Borodyanskii method [3] (our proof is based on geometric notions), we prove the following theorem.

Theorem 10. The length of the shortest universal uniform experiment U(n) which establishes the terminal state of any general linear automaton 2IES~ doe s not exceed (2 11 +1 - 2) (m + 1) 11 • The length of the shortest uniform experiment which all·ows any general linear a u t o m a t o n 2I ES~ t o b e d i s t in g u i s h e d f r o m an y o t her g e n er a 1 li n e a r a u t o -maton )()ES~, does not exceed (2 11 + 1 - 2)(m + 1) 211 •

Pr o o f. Let us give the proof for a Mealy general linear automaton.

The essence of the Yu. M. Borodyanskii method consists in the abridged notation of certain words in the input alphabet ~ in the form of word codes in the expanded alphabet ~ U {ffi}.

Each code word of the form <\ffiÖ2(J)()3(J) ••• ÖkffiÖk+!, where ö,, Ö2, ... , ök+1 are words in the alphabet ~, is decoded from the word ((öiö2)2 ... ök)2 ök+l in the alphabet ~ (i.e., the symbol w implies that it is necessary to repeat the preceding part of the word).

Assurne that in the beginning the general linear automaton 2I (r, ~, Z, cp, 'ljJ) may be in any state- any point of the hyperplaner. Then there exists an input a;1 suchthat the outcome of the experiment a;1 partitions r into parallel hyperplanes {r,}, r, = r (a;1), r (r,) < n.

Due to the action of the input 0';1 thesehyperplanes go over into the parallel hyperplanes

214 A. A. MUCHNIK

{f1 (cr;1)} , where r[r 1] = r[r 1 (ai )] in view of the nondegeneracy of the operators (j)a, crE~. . 1

Let us assume that E 1 =CJ;1 , and the code G1=o;1 • Assurne that wehave constructed the ex-periment Ek with the code ~. which partitions r into parallel hyperplanes {r (Ek)}, which the action of E k causes to go over into the hyperplanes {rk (Ek)}; under these conditions

Let us consider two cases.

a) rk (Ek) i!- r (Ek)· In this case the repeated application of Ek partitions the hyperplanes fk (Ek) into the hyperplanes fk +1 (Ek ), since all of f is partitioned into { f (Ek)}.

Under these conditions rk+i(Ek) is caused by the action of Ek to go over into rk+t (EkEk) =fktdEk+1). Let us assume EkH=EkEk, r[rk+dEkH)l=r[rkH(Ek)l-<n-k-1, while we take the code Gk+i to be Gkw.

b) fk(Ek) II r (Ek)· In view of the coincidence of dimensionality, each hyperplane rk (Ek) coincides with a certain hyperplane r (E k).

Then 1) either fk(Ek) is a point and the experiment U(n) = Eb since as a result of Ek the terminal state of rk (Ek) is defined as one of its points; or, 2) there exists a certain input

cri' that partitions rk(Ek) into rk+i(Ek) which due to the action of cr;· go over into rktdEkcr;·). In this case we assume Ek+i = Ekai, and the code GkH=Gkcri'. Here r [rktt(E~tcri·)l=r[f~t+i(Ek)l < r [_I\ (Ek)l < n- k, i.e., r [fktdEkH)l < n- k -1 ; 3) or for coincidence of the outputs of all points

fk(Ek) there mustexist on each input ai an input O'i*• which causes fk(Ek) go over into the hyperplane rk (Ekcri*) i!- rk (Eh). Otherwise, the output would be identical on each inputward x for all points rk (Ek), and since 21: is a reduced form it follows that r k(Ek) would be a point (but we have already considered this case).

Let us assume Ek+1 = E11cr;*Ekcri*, while Gk+1 = Gkcri*ro. Ek partitions the hyperplane r,. (Ekcri*) into fkt1 (Ekcri*) , since fk (Ekcri*) i!- r" (E~t), and therefore fk (E~tcri*) i!- r (E~t) ; after all,

r k(Ek) is a parallel shift of r (Ek). Therefore, the hyperplane {rkH (E~tcri*)} and the hyperplane {fktt(Ekcri*Ekcri*)= rkH(E~tt1)} have a dimensionality less than r[rk(Ek)] (i.e., not exceeding n- k- 1). Thu1~{, for a certain k ~ n it turns out that rk (Ek) is a point, i.e., the terminal state of the experiment Ek can be defined. Therefore, one of the co:les x1, x2, ... , xn, where xi = w, u;i or cr; iro (j = 1, 2, ••• , n), turnsout to be the co:le of the experiment E (x1x2 ••• xn), that determines the terminal state of W E Tn. Carrying out all the experiments E(x1x2 ••• xn) successively in any order for various allowed sets of values x1, x2 , ••• , Xn, we obtain the required experiment U (n) which is universal for Tn.

Let us estimate the length of U (n). We note only that in order to obtain the collection of experiments {E (x1x2 ••• xn)} suchthat for each W E Tn a certain experiment determines the terminal state, it is sufficient to assign each Xj just two values w and cr;iro, since the value cr;i is "absorbed" by cr;iro.

The total nurober of allowed sets x1, x2, ••• , Xn will thus be (m + 1)n.

In decoding the collection x1, x2, ••• , xn the maximallength of E (x1x2 ••• xn) will occur for Xj=O';iro (j = 1, 2, ••• , n). Under these conditions the length Zn of the experiment

E (XtX2 ••• Xn) = (((cr~10';2) 2 0';3) 2 ••• CJ;n)2

is equal to 2ln-l + 2, l1 = 2. From this we have


Therefore the length of U (n) does not exceed (211 +i - 2) (m + 1) 11 • The first statement of Theorem 10 has been proved.

Let us now consider the generallinear automaton m ES~=Tnnsn, i.e., a strongly connected general linear automaton m in reduced form with nondegenerate transition operators cp0 • For any other general linear automaton m Es~ let us consider the direct sum m E8 m, which will be a generallinear automaton from the class S211 (only strong connectedness is lost). The experiment u (2n), which is universal for s2n' allows the final state m EB m, tobe established and thereby allows m tobe distinguished from lB (and lB from 91). The length of U (2n) does not exceed

The theorem has been proved.

The basic idea running through the proofs of Theorems 1, 2, and 10 of this paper resides in the fact that the hyperplane r', which is parallel to the hyperplaner (r(r) > r(r')), where r is determined by the output corresponding to the inputs (J i' may be converted into the hyperplane I" (x) iJ.. r by the word x of length :s r (r) - r (r 1).

In the case of fields of real numbers D and complex numbers K one may consider the measure in the space

which is the direct product of the spaces of the transition operators <Jlai (i = 1, 2, ... ,m) and the output operators 1Jlai (i = 1, 2, ... , m). E ach alement <Jll: x 'ljll: for fixed V, r, 1:, and Z defines a generallinear automaton m (V, r, ~. Z), while each element r x (jll: x 1Jll: defines an initialgenerallinear automaton m. For "almost all 11 (in the sense ofthisnatural measure) initial generallinear automata m, coincidence of the trees D qi and D q2 of the automata in any n edges is sufficient to ensure equivalence of any two states q1 and q2• Moreover, for 11 almost all" general linear automata 21 any n vertices of the tree Dq will be linearly independent states. This derives from the fact that an algebraic surface defined by an equation in any n-dimensional space has measure zero in this space, while measure zero is invariant relative to the choice of the basis in the space.

From the indicated properties of 11 almost all" generallinear automata over the fields D and K it likewise follows that the structure of 11 almost any" generallinear automaton is determined by a multiple experiment (tree) having a length of the order of log m n, and any word of length n may serve as the input of a uniform experiment which determines the terminal state of a general linear automaton 2l. Moreover, any word of length n may form a basis (if one considers pieces of this word as basis words) in almost allgenerallinear automata. In order to distinguish "almost any11 generallinear automaton 2l E Tn from "almost alli1 generallinear automata iU E Tn it is sufficient to carry out a uniform experiment with any input word having the length 2n. A prime branching experiment of length mn2 which is constructed according to the method given in Theorem 7 allows simple (without sorting of general linear automata from Tn) establishment of the structure of II almost any11 general linear automaton m.

These statements are also valid for finite probabilistic automata.

Here it is of interest to note that analogous properties hold for finite deterministic automata if the term 11 almost all11 is understood in the sense of the tendency towards 1 of the fraction of (n, m, p)-automata having the given property out of all (n, m, p)-automata for n- oo [6].

It would be of interest to consider the case of generallinear automata over a field of rational numbers, having chosen a certain basis and considering the set lmn, " of n-dimen-

216 A. A. MUCHNIK

sional general linear automata with transition and output matrices whose elements are irreducible fractions p/q, where I p I + I q '1 < k.

Problem 7. How does the "majority" of generallinear automata from mn, 11. behave in experiments? It is required to estimate the length of the experiments.

Pr ob 1 e m 8 • It is required to consider the case of the general linear automaton '21 (V, M, ~. Z, cp, 'ljl) over finite fields P = GF(pk). How does the length of the experiments

depend on the dimensionality n, the numbers p and k, and the power M for "the majority" of automata '21 over P?

I express thanks to the editor of the paper Yu. Ya. Breitbart for a number of comments.

Literature Cited

1. K. Berge, Theory of Graphs and Its Applications [Russian translation], IL, Moscow (1962), Ch. 14, pp. 150-152.

2. Yu. M. Borodyanskii, "Experiments with finite Moore automata," Kibernetika, No. 6, pp. 18-27, Kiev (1965).

3. A. Hill, Introduction to the Theory of Finite Automata [Russian translation], Mir, Moscow (1966).

4. V. M. Glushkov, Synthesis of Digital Automata, Moscow, Fizmatgiz (1962). 5. N. E. Kobrinskii and B. A. Trakhtenbrot, Introduction to the Theory of Finite Automata

[in Russian], Fizmatgiz, Moscow (1961), Chap. II, V, and VII. 6. A. D. Korshunov, "On the degree of distinguishability of automata," in: Discrete Analysis,

No. 10, Nauka, Novosibirsk (1967), pp. 39-60. 7. M. S. Lifshits, "On linear physical systems connected with the external world by com

munication channels," Izvestiya Akad. Nauk SSSR, 27:993-1030 (1963). 8. M. S. Lüshits, "Open systemsandlinear automata," Izvestiya, Akad. NaukSSSR, 27:1215-

1228 (1963). 9. A. A. Muchnik, "Length of an experiment for determining the structure of a finite strongly

connected automaton," in: Systems Theory Research, Vol. 20, Consultants Bureau, New York (1971), p. 136.

10. Ch. Faisi, On the Distinguishability of Infinite Automata (see this volume, pp. 219-222). 11. M. L. Tsetlin, "On nonprimitive networks," in: Problemy Kibernetiki, Vol. 11, Fizmatgiz,

Moscow (1958), p. 31. 12. G. Bacon, "Minimal-state stochastic finite-state systems," Trans. IEEE, CT-11:307-308

(1964). 13. J. W. Carlyle, "Reduced forms for stochastic sequential machines," J. Math. Annal.

Appl., 7:167-175 (1963). 14. J. W. Carlyle, "On the external probability structure of finite-state channels," Inf.

Contr., ~:385-397 (1964). 15. S. Even, Comments on the Minimization of Stochastic Machines, Res. Rep., SRRC-RR-

64-50 (1964). 16. S. Ginsburg, "On the length of the smallest unüorm experiment," ACM Journal, 5(3):266-

280 (1968). 17. T. H. Hibbard, "Least upper bounds an minimal terminal-state experiments," ACM

Journal, 8(4):601-612 (1961). 18. E. F. Moore, Gedanken-Experiments on Sequential Machines. Automata Studies,

Princeton University Press (1956), pp. 129-153. 19. A. Paz, "Some aspects of probabilistic automata," Inf. Contr., 9(1):26-60 (1966). 20. M. 0. Rabin, "Probabilistic automata," Inf. Contr., Vol. 6, No. 3 (1963). 21. M. 0. Rabin and D. Scott, "Finite automata and their decision problems," IBM Res. Dev.,

3(2):114-125 (1959).


22. G. N. Raney, Sequential Functions, ACM Journ., 5(2):177-180 (1958). 23. M. P. Schutzenberger, "On the definition of a family of automata," Inf. Contr., 4:245-270

(1961). 24. M. P. Schutzenberger, "Finite counting automata, 11 Inf. Contr., 5:91-107 (1962).

DISTINGUISHABILITY OF INFINITE AUTOMATAt

Cho Faisi Moscow

1. Let 121 = {Q, X, Y, cpm, 'llm} be a strongly connected synchronous automaton (i.eo, the length of an output word is equal to the length of an input word), where Q is the set of states of the automaton, X the input alphabet, Y the output alphabet, cpm the transition function, and ')l~

the output functiono The automaton ~ is either finite or infinite. Hence for any automaton m={P, X, Y, cp53, 1/Jm}, whose input and output alphabets coincide with the corresponding alphabets of ~. we have:

a) Either for any state q belonging to Q there exists an equivalent state p belonging to P,

VqEQ3pEPVaöX: 1/l~(q, a)=1Pm(p, a);

( a ö X signifies that a is a word of finite length in the alphabet X),

b) or no state q belonging to Q has an equivalent state in P:

Pr oof o Suppose that for a qoEQ there exists an equivalent state p0 EP:

121 is strongly connected; hence for any qj E Q there exists a word bi ö X that carries 121 from q0 into qi, and p0 is equivalent to q0; hence

But

Va: 1/J~ (qo, bp) = 1Jlm (po, bia).

'lJI}l (q0 , bia) = tjll}l (qo, bi) tpl}l (cp\lt (qo, bj), a) = 1/J~ (q0 , bi) 1Jl21 (qh a),

tJlm (p0, bp) = 1/J)B (po, bi) 1PIB (qJ\B (Po, bj), a) = 1/Jm (po, bi) 1/Jm (pi<i)' a),

where piU> = qJm (p0 , bi) is a state of m. By our condition we have 1Jlm (qo, bi) = 1Jlm (po, bi); hence

VaöX: 1/Jm(qh a)=1Pm(PiU>' a)

the state p i(i) is equivalent to the state qi o

toriginal article submitted May 23, 1968o

219

220 CH. FAISI

2. Let ~ and m be infinite synchronous strongly connected automata, and ~ cannot be distinguished from m by any finite experiment, i.e.,

Vq E QVa ö X3p E P: 'iJm (q, a) = 'ljl58 (p, a).

In this case it can happen that no q E Q has an equivalent p E P.

Example. X=Y={0,1},Q={qi,i=O, ±1, ±2, ... },P={pi,j=O, ±1, ±2, ... },

cpm (q;, O) = q;+ cp18 (pj, O) = pj-j,

cpm (q;, 1) = q;+~> cp58 (pj, 1) = Pi+b

{ 1 for i = 0,

'iJm (q;, 0) == 'iJm (qi, 1) c= 0 for i =F 0,

{ 1 n (n --1); · · 1 2

'iJ)ß (Ph 0) = 'iJ58 (Pi• 1) = for j = ± 2 ' n = ' ' 0 for other j.

The transition diagrams of these automata (the outputs for a given state are given in the circles) are:

21 cannot be distinguished from m by any finite experiment; suppose we are given an initial state qi and an experiment a. In the operation of the automaton 21 over this word there participates a finite nurober of states belonging to Q, more precisely, a "section" of states [qr, q 5], r < s, since 21 can go over from one state to a neighboring (according to it~ nurober) state. Suppose that these are the states from qi _ m to qi + n (n, m :::: O).

a) i-m> 0 or i + n < 0. Then all these states yield zero. The transition function of m is the same as the transition function of 21; hence if we take p i as the initial state, the operation of m over a will invol ve the "second" from Pj _ m to Pj + n• Let us take

. (rn-t-n-t-1) (m-t-n-1-2) , , 1 J= 2 -t-mr .

Then the states in operation will be those with numbers from

(m-t-n-t-1) (m-t-n-t-2) +m+ 1-m > (m-i-.n-t-1) (m-t-n-t-2) 2 ' 2

to

(m--j-n-t-i)(m--j-n--j-2) + .L 1 -l- <. (mf-1.-l-n)(m-+-n-\~2) + -1-Z+ =(rn+n-t-2)(m--j-n-t-3) 2 ,m, ,n 2 m n 2 ,

i.e., these states likewise yield zeros alone.

b) i-m ::::; 0 ::::; i + n, i.e., -n::::; i ::::; m. As the initial state let us take P<n+mH~ <n+m+Z> +i·

The states in operationwill be those with numbers from (m-t-n-t-1) (m-\-n-\-2} . (rn-t-n-i-1) (m+n-i-2) ,

2 +i-m?- 2 • -(n 1-m)>

> (m-t-n-]-i);m+n-t-2) (m+n+ :1) = (m-t-n-+;1) (m-;-n)

DISTINGUISHABILITY OF INFINITE AUTOMATA 221

to (m-i-n-J-i)(m+n-J-2) + ._ /(rn·i n-j-1)(m+n+2) -1-( -+- )<

2 ~ n~ 2 ' m n

<(m-j-n-j- 1)t~+n+2) + (m -;-n 1_ 2) = (m-i-n-j-2)2(m-J-n+3)

Among these states, only P <m-:n+IHm+n-12) yields unity, whereas all the other states yield zero. 2

Therefore the result willlikewise coincide with the output of ~. But, for example, for q0 E Q

there does not exist an equivalent state belonging to P: 'IJlm (g0 , 16 ) = 106 - 1, whereas 'IJliB (Ph iö)

terminates with 1, where for j ~ 0, 1, 2, 3 we have ö = I i 1 (! ~ 1-i) - j + 1, since

and for j = 0, 1, 2, 3 we have

3. Theorem. Let ~ be a strongly connected infinite synchronaus automaton that cannot be distinguished by any infinite experiment (infinite sequence of input letters) from a synchronaus automaton lJ.)

with a countable number of states (with the same input and output alphabets). Then for any state of ~ there exists an equivalent state of Q3

Pro o f. Suppose this is not the case. But then (according to Sec. 1)

VgEQVpEP3aöX: 'IJlm(g, a):;i='ljJiB(P, a).

Let us number the states of ~ and )!:{: P = { p1, p2, p3, ••• } , Q = { q1, q2, q3, ••• } • Then we construct an infinite experiment x such that

By a i,i we shall denote a word in X such that 'IJlm (q;, a;, i) =I= 'IJliB (Ph x) and write x1 = a1, 1• By our condition we have 'IJlm (q~> al, 1) :;i= 'IJliB (p1, a1. t)·

If 'IJlm (g~> a1, 1) :;i= 'IJliB (p2, a1, 1), then x2 = x1; if 'ljl21 {q" a1, 1) =- IPiB (p2, a1, t), we shall denote IPm (q~o at, t) = q;., IPiB (P2• a1, t) = P.i •• and .,-2 = x1a;2, i2·

Suppose that x 11 has been constructed. If 'IJlm (gt, xn) :;i= 'ljl~ (Pn+t, xn), then Xn+t = :r,.; but if 'IJlm(g~> Xn)='IJlm(Pn+~> Xn), we shall denote IJlm(q~> Xn)=g;n+l' q~(Pn+t, xn)=Pin+l and write

Let us show that 'ljllll (q1 , xn) :;i= 'IJlm (pk, xn) for any k = 1, 2, ••• , n. For k = 1 this is true.

Let k > 1. If 'IJlm (g~> xh-t) :;i= 'IJliB (Pk, xk-t), then 'IJlm (gt. :rn) :;i= 'IJll{l (PR, :.rn) (since X 11 is a continuation of xk-i• also 'IJliB (pk, xn) will be a continuation of 'IJl>S (pk, x"_1), and 'IJlm (gt, xn, will be a continuation of 'ljl~l (gt, .rh-t) ; if their initial parts do not coincide, they will differ from one another).

222 CH. FAISI

If lflm (q~> xk+1) = 'ljl\8 (pk, xk_1), then

'lJlm (qll xn) = 'lJlm (q1, xk-1) 'lJlm (q;", a;", i") · · ·,

'lJl}ß (ph, Xn) = 'lJl\8 (Ph, X~t-1) 'lJl!B (pik, a;k• ik) · · ·

But 'lJlm (q;", a;", r,,) =I= 'lJl}ß (Pi"' a;", i").

Let us write x = lim Xn • Hence n-+oo

i.e., ~I can be distinguished from lE by an experiment x. Wehave obtained a contradiction; this proves our assertion.

4. fu the proof we used the fact that the states of Q3 can be numbered, i.e., that it has a countable number of states. If Q3 is also a strongly connected automaton, this condition will be satisfied.

It is not known whether there exists for any state of ~ an equivalent state of m, if ~ cannot be distingUished by any finite experiment from m, and Q3 cannot be distinguished by any finite experiment from ~.

If the automata are not synchrbnouss the assertion of Theorem 3 is not necessarily true. From the nondistinguishability by any infinite experiment does not yet follow the nondistinguishability by finite experiments.

Example. I§:={Q, X, Y, fJlQ:, 'lJl~.i}, 'Il={P, X, Y, fJl:Il, 'lJl:Il}, Q ={qi, i = 0, ±1, ±2, ••• }, P = {pi, j = 0, ± 1, ±2, .•• }, X= { 0, 1}, Y = { 1}. (§; and '!l have the same transition as ~ and Q3 (this follows from Sec. 2)

{ A for i= + p,

'lJl~.i (q;' O) = 'lJlQ; (q;' 1) = 1 otherwise;

'lJl:Il(Pio O)='lJl:Il(Pi> 1)= 1 for i=+2k, k=O, 1, 2, ..• , { A for i = + p, p > 2,

11 otherwise,

where p is a prime, and A is the empty word. Any state of (§; is distinguishable from any state of '!l, and any two states of (§; (or '!l) are distinguishable from one another, but they yield the same output (consisting of infinitely many 11 ones11 ) for any initial states and infinite sequences.

Literature Cited

1. A. A. Muchnik, 11Generallinear automata, 11 this volume, pp. 179-217.

PROGRAMMING

ON ALGORITHM SCHEMATA WHICH ARE DEFINED ON SITUATIONS t

R. I. Podlovchenko

Erevan

The notion of a memory consisting of individual cells and the notion of a memory state consisting of states of cells play a fundamental role in programming theory. At the same time, in certain branches of this theory, information on memory structure turns out to be redundant, and the memory state is perceived as a unified element. In this case memory may not be introduced at all, and the set of memory states may be considered as a set of abstract "situations."

The notion of a situation, which forms the basis for all the constructions in the present paper, is introduced as an indeterminate object and may certainly have interpretations differing from the notion of a memory state; however, all of the constructions are carried out in such a way that if a situation is understood to be a memory state, then we have both a ready device consisting of basic programming notions and a number of results which may be applied in the programming field.

Mappings of two types are constructed over a set of situations: actions and predicates on situations. If a Situation is interpreted as a memory state, then an actionwill constitute an operator over a memory, and a predicate on Situations will constitute a predicate over a memory.

The notion of a schema advanced in the paper is a generalization of the notion of a graphscheme algorithm [2]. The construction of a schema is raised on a finite graph which may have several outputs for one input and is called a net; the vertices of the net are equipped with transformation functions (in this case a vertex is juxtaposed with an action) or recognition functions (and then the vertex is juxtaposed with a predicate on situations); both the links between the vertices of the nets and the objects juxtaposed with the vertices may be changed along with a change of situation; in particular, one and the same vertex of a schema (it is called a vertex of the mixed type) may be a transformer in certain situations and a recognizer in others.

In the majority of schemata defined on Situations, an algorithm for executing the schema is introduced - an algorithm I which is applicable to any schema 2 in any situation ~ • The process of applying the algorithm I to the schema 2 in the situation ~ is called t h e procedure for executing the schema 2 in the situation ~ and can be reduced

t Original article submitted July 23, 1968.

225

226 R. I. PODLOVCHENKO

to the parallel construction of two sequences: a sequence of Situations and a sequence of vertices of the schema 2, which are tagged in these situations. The process of executing the schema 2 in the situation ~ may be infinite, finite without result, and finite with result; in the latter case the situation ~ is assigned to the definition domain of the schema 2.

The algorithmic properties of a schema are reflected by various of the characteristics. Among them the so-called value of the schema on the Situation ~ occupies a special place; this characteristic carries information on the "history" of the execution of the schema which has been begun in the situation ~, and it is used in introducing equivalence relations between schemata. For equivalence of two schemata it is required that their definition domains coincide and, in addition that the values of these schemata on each situation from their definition domain (weak equivalence) or on each possible situation (strong equivalence) coincide.

Among schemata we distinguish the following:

s tat i on ar y in cp - the function juxtaposed with the vertex of the schema does not depend on the situation considered, and this applies to all vertices;

weakly nonstationary in cp - for nonstationarity the schema at the sametime does not have vertices of the mixed type;

str ongly nonstationary in cp - the schema does contain vertices of the mixed type;

stationary in cp - the arcs connecting the vertices of the schema remain unchanged (along with their markers) in all situations.

Stationary in cp and lf! sirrtultaneously schemata are classified according to the structure of the net on which the schema is based; thus there appear:

connected schemata- thenetoftheschemais connected;

directionally connected schemata - the net ofthe schema is suchthat a path connecting the input with one of its outputs passes through every vertex of the schema;

1 in e a r s c h e m a t a - the net of the schema has one output and is such that it allows the construction of an elementary path passing through all vertices of the schema.

The classification of schemata given in this paper yields a series of classes which are imbedded in one another. Theorems 1-7 establish sufficient attributes for a transition from a schema of one class to an equivalent schema of another (imbedded) class.

Examples illustrating the basic definitions and equivalent transformations of schemata are given in a separate section at the end of the paper. This emphasizes the particular character ofthe geometric interpretation of the structure of a schema and preserves the integrity of the exposition of the material of the paper.

§ 1. Auxiliary Notions

1.1. Let us give a series of definitions applying to the theory of groups (see [1]).

Assurne 2r ={a} is an arbitrary set.

Weshall say that the gr aph

has been stipulated if a) the nonempty set a=2t: is given;

ALGORITHM SCHEMATA DEFINED ON SITUATIONS 227

b) each element of cx E a is juxtaposed with its set TI (cx).:;; a (in particular, n (cx) may be empty).

The elements of the set a are called vertices of the graph r. 1.2. An ordered pair (a, a 1), where cxEa, cx'ETJ(cx) shall be called an arc of the

gr aph; concerning the arc (a, a 1) we say that it issues from the vertex a and enters the vertex a 1 ; a and a 1 are called the b e g i n n i n g a n d end o f t h e a r c ( a , a 1 ) •

A path in the graph r is called a sequence of its arcs -x.1, -x.2, ••• , -x.k, suchthat the end of each preceding arc coincides with the beginning of the next one. If 'X.i = (ai, ai + 1),

i = 1, 2, ••• , k, then the path, by definition, passes successively through the vertices

where a 1 is the beginning and a k+1 is the end of the path. A path is e 1 e m e n t a r y if no vertex in it is encountered twice.

A c o n t o ur is called a finite path in which the beginning and end coincide.

1.3. The edge of a graph r is called a set consisting of two of its vertices a 1

and a 2 (the so-called boundary vertices ofthe edge) suchthat either cx1 E II (cx2) or cx 2 E II (cx!).

A chain is a sequence of edges b.1, b.2, ••• , b.k, in which one of the boundary vertices of each edge b.i is also a boundary vertex for b.i_1, while the other is a boundary vertex for b.i+1•

A c y c 1 e is called a finite chain beginning and ending at one and the same v ertex of the graph.

1.4. The graph

is called a n e t and shall be written in the form

if the following are true in it:

1) the vertex cxo E a, called the in p u t o f th e n e t (it is such that exactly one arc of the graph issues from it and not a single arc of the graph enters it), is isolated.

2) The nonempty set aw s a is isolated whose elements are called outputs of the ne t and have the following property: not a single arc of the graph issues from them;

3) at least one arc of the graph issues from each graph vertex which does not coincide with a0 and does not belong to the set aw (such a vertex is called internal).

1.5. A graph is called c o nn e c t e d if any two of its vertices may be connected with a chain.

A net is called dir e c ti o n a 11 y c on n e c t e d if, regardless of its vertex, there exists a path beginning at the input of the net which ends at one of its outputs and passes through this vertex.

A directionally connected net in which there are no samples is called a t r e e •

A net with one input is called l in e a r if there exists an elementary path beginning at the input of the net and ending at its output, while passing through all vertices of the net.


§ 2. BasicNotions

2.1. Let us consider an arbitrary set 8 = g}, whose elements are called situations.

2.2. An a c t i o n is called a partial mapping of the set 8 in itself.

An action with an empty definition domain is called e m p t y and is denoted by A .

Assurne '1l = {A} is the set of all possible actions.

2.3. Let us introduce the set U = { u } .

A p red i c a t e o n s i tu a t i o n s is called a function which is defined on a certain sub-set of the set 8 and maps this subset in the set U.

Assurne ffi = {p} is the set of all possible predicates on situations.

2.4. We shall call a s c h em a a complex consisting of three mappings:

1. The ma.pping a (~) which juxtaposes each situation GE 8 with its net; all juxtaposed nets, by stipulation, have the following in common: the set of vertices a, the inputs a 0, and the set of outputs aw, and they differ from one another, perhaps, in the sets II (a), a E a.

The vertices of the nets which are different from its inputs and outputs will be called intern a I ; the net juxtaposed with the situation ~ shall be written in the form

where II (a, ~) is the set of vertices juxtaposed with the vertex a in the situation ~ •

2. The mapping cp (a, ~) which juxtaposes either an action A or a predicate p on Situations with each pair (a, ~ ), where a is an internal vertex of the schema, while ~ is an arbitrary situation.

3. The mapping 1/J(a, 0 which is defined for all vertices of the schema with the exception of its output and juxtaposes the pair (a, 0 with an arbitrary mapping of the set U in the set II (a, ~ ), if cp (a, ~) is a predicate, and the mapping of the set U in some particular element of the set II (a, ~), if cp(a, ~) is an action.

If the vertex a' E II (a, G) is the image of the element u E U in the mapping

'\(; (a, ~): u--? rr (a, ~).

then the arc (a, a') is called a tagged element u.

For the schema we use the notation

The vertex at which the arc issuing in the situation ~ fr_om the vertex a 0 arrives shall be called the fi r s t arc and denoted by a 1 W.

2.5. The procedure of executing the schema 2 in a stipulated situation ~ shall be understood to mean the procedure of joint construction of two sequences:

a) the sequence of situations

(1)

b) the sequence of vertices of the schema 2, which are tagged in the situation (1)

(2)


Let us define this procedure by induction.

T h e f i r s t s t e p • Let us assume that ~ 1 = ~, and let us tag the vertex a 1 (~ 1).

Assurne that l steps (l 2: 1) have been taken, ~z is the situation constructed on the l-th step, and az is the vertex tagged on the l-th step (and, therefore, in the situation ~ z).

On the l + 1-st step let us do the following:

I. Let us consider the vertex az • If a 1 is the output of the schema c(azEaw), then the process of executing the schema c

in the situation ~ is assumed tobe completed with a result on the l-th step, and the situation ~l is considered tobe terminal.

If the vertex az differs from the outputs of the schema c (az E aw), then we go over to point II.

II. Let us consider the function cp at the vertex az andin the situation ~z; its value is either a certain action A or a certain predicate p on Situations.

1. Assurne cp(az, ~z)=A and the vertex a*EII(a1, ~z) is the image ofthe set U in the mapping 1jJ (az, ~z); let us distinguish between the following cases:

a) the action Ais applicable to the situation ~l; then we assume that

~/+1 = A (~z), az+! = a*,

after which the execution of the l + 1-st step is assumed to have been completed, and we go over to point I to execute the next step;

b) the action A is inapplicable to the situation ~l; in this case the process of executing the schema c in the situation ~ is assumed to have been completed withou t r es ult on the l-th step.

2. Assurne cp (az, ~z) = p; let us consider the following cases:

a) the predicate p is defined in the Situation ;z, p (~z) = u*, and the mapping 1jJ (a1, ~1 ) juxtaposes the element u* with the vertex a* VI (az, ~z); then we assume that

~Z+1 = ~~, az+t = a*,

consider the execution of the l + 1-st step to have been completed, and go over to point I to execute the next step;

b) the predicate p is not defined in the situation ~l ; then the process of executing the schema 2 and the situation ~ is assumed to have been completed without result on the l-th step.

The process of executing the schema 2 in the situation ~, as is evident from its description, may be either finite or infinite. If the process has been completed with a result and consequently leads to a terminal situation ~', then the schema 2 is assumed to be a p p 1 i c ab 1 e to the original situation ~' while the situation ~ is denoted by 2 (~).

2.6. Assurne :Ba is a set consisting of all such situations to which the schema 2 is applicable.

Let us introduce the following characteristics of the schema.

The schema 2 is juxtaposed with the action A., which has the following properties:


1) the definition domain of the action Aa is the set Ba;

2) regardless of the situation ~ E Ba,

A,m=2(~);

we shall speak of A a as of an a c t i o n r e a 1 i z e d b y t h i s s c h e m a 2.

Assurne aw = {a(!l, a< 2>, ... , aU•>}, k ~ 1. We shall say that the schema 2 real i z es the partitioning of the set Ba into k disjoint subsets Eau', i = 1, 2, .•• , k, which are defined as follows: the Situation ~ E :::a is assigned to the set Ear;,, if the procedure of executing the schema 2 in the Situation~ is completed by tagging the output a(il,

Assurne the schema 2 is executed in the situation ~, and this leads to the construction of the sequences (1) and (2).

Let us consider the sequence

its elements are actions or predicates which are juxtaposed with the tagged vertices of the schema 2, the juxtaposition being in those situations in which these vertices were tagged during the execution of the schema 2. From this it follows that: if the procedure of executing the schema 2 in the situation ~ is finite with a result, then since the function <p is not defined at a single vertex of the set a w (outputs of the schema), the sequence L (2, ~) has a length one less than the sequence (1).

The value of the schema 2 in the situation ~ shall becalledasequence

L(2, ~)=ejj(ct;l' ~h), cp(ct; 2 , ~;2)," ••• , cp(a;t, ~;t), ...

of the sequence L (2, ~) , which is such that it contains all actions entered in L (2, ~) and only those.

§ 3. Relations between Schemata. Classification

of Schemata

3.1. Let us introduce a series of relations between schemata.

The schema 21 is called the expansion of the schema 22 if

and for any situation ~ E Ea2

The schema 21 and 22 are called weakly equi valent if

and for any Situation ~ E Ea1

The schemata 21 and 22 , are, by definition, s t r ong ly e q u i valent if, regardless of the situation ~ E E,

ALGORITHM SCHEMATA DE FINED ON SITUATIONS 231

and, moreover,

Weak equivalence of schemata derives from their strong equivalence; if each of the schemata 21 and 22 is an expansion of the other, then the schemata 21 and 22 are weakly equivalent.

The equivalence relation (both weak and strong) is reflective, symmetric, and transitive. The expansion relation is reflective and transitive.

3.2. The schemata

and

are isomorph i c if between the vertices of the sets a' and a" one may establish a one-toone relationship (}', such that:

1) a ~ and a ~ are corresponding vertices;

2) the sets a:v and a;;, go over into each other;

3) regardless of ~ES and a E a', the following statements are valid:

a) the sets II(a, ~) and II" ((Ja, ~) consists of vertices that are pairwise corresponding to each other;

b) cp' (a, ~) = cp" (aa, ~);

c) the mappings

tp' (a, ~): U ---?H' (a, ~)

and

tp" (aa, ~): U ----7 11" (aa, ~)

juxtapose each element u EU with corresponding (to each other) vertices of the sets II'(a, 0 and II"((J'a, 0.

The isomorphism relation of schemata is reflective, symmetric, transitive, and involves the relation of strong equivalence of schemata. It is obvious that isomorphic schemata realize one and the same operator and one and the same partitioning of the definition domain of the schemata.

3.3. Assurne

is an arbitrary schema.

We use II * (a, ~) to denote the image of the set U in the mapping


The schema 2 is called reduced in 1/J if for all a and ~

TI* (a, s) =TI (a, s).

If the schema 2 is not reduced in 1/J, then it may be juxtaposed with the schema

* 2 =[a0 , aw, {TI*(a, s), cp(a, s), 'ljl(a, s), c.:Ea, sEE}J,

which a) will be reduced in 1/J, and b) is strongly equivalent to the schema 2·

Hereafter we shall consider only schemata which are reduced in 1/J without making special mention of this each time.

3.4. We prefer classification of the schemataproper to classification of the vertices of a schema.

An internal vertex a of the schema 2 is:

s tat i on ar y in cp ü the value of the function cp (a, ~) is independent of the Situation ~, and nonstationary in cp otherwise;

s tat io n ar y in 1/J if the mappings 1/J(a, ~) do not change with a change in ~, and nonstationary in 1/J otherwise.

An interval vertex a is, by definition, finite-valued ü the function cp(a, 0 takes a finite nurober of values on the set E. An internal vertex a of the schema 2 is:

an A-vertex if the values of cp (a, ~) in all situations s E E are only actions;

a p-vertex if the values of cp (a, ~) in all situations s E E are only predicates;

a vertex of the mixedtype ü the values of cp (a, ~) in certain situations are actions and in other situations are predicates.

3.5. Let us distinguish among the following classes of schemata.

The schema 2 is called:

s t a t i o n a r y in cp ( i n lfi ) if all internal vertices of 2 are stationary in cp (corre-spondingly in 1/J), and nonstationary in cp (in 1/J) otherwise;

type;

stationary if 2 is stationary in both cp and 1/J.

Among schemata which are not stationary in cp we distinguish among:

w e a k 1 y non s tat i o n a r y schemata which do not contain the vertices of the mixed

s tr ongly no ns t a ti onary schemata which contain such vertices.

Assurne

2 = [c:t0 , aw, {TI (a, s), cp (a, 6), 'ljJ (a, s), a E a, 6 E E}J

is a schema which is stationary in 1/J; consequently, the mapping Il (a, ~), regardless of the vertex a, is independent of ~ if all situations 6E E are juxtaposed with one and the same net

Si)=[a0 , aw, {TI~(a), c.:Ea}].

In § 1, we define such properties of a net as connectedness, directional connectedness, linearity, and the property of the net making the net a tree.


If the net S z has one of the enumerated properties, then we shall agree to confer this property on the schema 2 itself. Thus, in a set of schemata which are stationary in l{J there appear classes of connected, directionally connected, linear schemata, and tree-schemata.

3.6. A schema 2 which is stationary in l{J (constructed on the net S 2 ) shall be stipulated by agreement using the accepted method of depicting a net. For such a stipulation:

1) The vertices of the net are depicted by points of a plane; the input is marked in standard fashion with the index 0, and the index w is used to denote the outputs of the net.

2) The arcs of the net are depicted by directional pieces of curves which connect the vertices of the net.

3) The arcs issuing from the A-vertices do not carry tags.

4) The arcs issuing from the p-vertices are tagged by elements of the set U.

5) If 2 is stationary in cp, then the vertices of the net are supplied with the designations of those actions and predicates which are juxtaposed with these vertices; for nonstationarity of 2 in cp the mapping cp (a, 0 must be stipulated separately.

Let us present e x a m p 1 e s of schemata of the simplest type which are stationary in l{J •

1. A

Here A is a certain action.

2.

Here U = { 0, 1}, and p is a certain predicate on situations.

§ 4. Operations on Schemata

4.1. In all the subsequent considerations:

1) all schemata are assumed tobe reduced in 1/J;

2) if a new schema is synthesized from several schemata, then the original schemta are assumed to have no common vertices.

In order for the first proposition to be valid, it is sufficient to go over from the original schema of transformations considered in subsection 3.3 to an equivalent schema; the transition from a schema to a schema isomorphic to it (and therefore equivalent to it) makes the second proposition valid also.

Besides these propositionsg we assume that the following rule is operative: for transformations and synthesis of schemata, the vertices of the schemata are always taken to be inseparable from the functions juxtaposed with them (the arcs are inseparable from their tags); if the added vertex is an A-vertex, then nothing is said of the tags of the arc issuing from it.

402. The schema


is given whose outputs are assumed tobe ordered aw={a<il, a< 2>, ... , aU<>}, and the schemata

in an amount equal to the number of outputs of the schema 2 (i = 1, 2, ... , k).

* Let us construct the schema 2 , which we shall call the 2 - c o mp o s i t i o n o f th e s ehern ata 21, 22 , ••• , 2". Let us begin by giving a substantive description of the procedure for constructing it.

1. We combine all of the vertices of the schemata 2, 21' 22 , ••• , 2" into the set.

2. The input of the future schema is designated to be the vertex 0! 0, and the outputs are designated tobe all outputs of the schemata 21, 22, ••• , 2".

3. In each situation ~ E E we execute the following:

a) we consider the set containing all arcs of the schemata 2, 21 , ••• , 2";

b) each arc arriving at the output a(il, i = 1, 2, ... , k, of the schema 2, is directed to the first vertex af.h~) of the schema 2i' with no alteration of the tag of the arc itself;

c) we discard the outputs a(il of the schema 2 and the inputs of all schema 2;, i =

1, 2, ... , k.

Now the 2 -composition of the schema 21, 22 , ••• , 21! will be defined formally; it will be the schema

in. which:

1) the input aö is the input a 0 of the schema 2;

2) the outputs constitute the set

at,= U a~>; i=!, 2, ...• lt

3) the set of internal vertices is obtained by a union of the sets of the internal vertices of the schemata 2, 21 , 22 , ..• , 2";

4) for all internal vertices a of the schema 2* and all situations ~ we have

* {cp(a,~). if cp (a,~)= rp<i)(a,~), if

aEa,

aEa<i)' i=1, 2, ... , k;

* 5) regardless of the vertex a of the schema 2 and the situation ~ E E,

a) if aEa(i), then TI*(a, ~)=TI<'>(a, ~), \jl*(a, ~)=ljl(il(a, ~);

b) however, if aEa , then ll*(a, ~) is obtained from II(a, 6) by replacement of each vertex aCO in the latter by the vertex a~i) (6), i = l, 2, ... , k (here a~i) (6) is the first vertex of the schema 2 i and the situation 0; analogously, the mapping \jl* ( a, 6) is obtained from the mapping

ljJ (a, 6): u --7 n (a, 6),


provided only that in the set n (a, 0 each vertex a(il is replaced by ayl(O, i = 1, 2, •.• , k.

4.3. The vertex a of the schema 2 is called the preoutput of the schema c in the situation ~ if the set TI (a, ~) contains just one output of the schema z.

W e shall say that

2 is a schema of the type A, if:

1) all of its preoutputs are A-vertices which are stationary in l{J;

2) not one output has two preoutputs (i.e., for any preoutputs a' and a 11 the inequality a' ;>! a" entails the equation Il(a') n n (a") 1·);

2 is a schema of the type p, if:

1) all of its preoutputs are p-vertices which are stationary in l{J;

2) all arcs issuing from each preoutput Iead only to outputs of the schema;

3) not one output of the schema has two preoutputs.

Let us introduce the following operations:

1) Substitution of the schema 2 1 of the type A into the schema Z2 in place of its A-vertex a *;

2) Substitution of the Schema ( 1 of the type p into the Schema c"in place of its p-vertex a *.

The result of executing each of these operations is a new schema which is denoted by 2* in both cases.

* Let us describe the substance of the procedure of synthesizing a schema 2 .

1. Let us combine all vertices of the schemata 21 and 22 into one set.

2. The input of the featured schema is assumed tobe the input a~ of the schema c\• while its outputs are assumed tobe all outputs of the schema 2 2 •

3. In each situation ~ ( 2 we do the following:

a) we combine all arcs of the schemata 21 and c2 into one set;

b) each arc arriving at the vertex a* is directed to the first vertex al (~) of the schema ? 1• without changing the tags of this arc;

c) we eliminate the input of the schema 21 and all of the outputs together with the arcs arriving at them;

d) if from the vertex a* there issues an arc (a*"', a), then from each preoutput of the schema 21 webring out an arc to the vertex a and tag it with the same elements of the set U which were used to tag the arc (a *, a); after this the arc (a *, a) is discarded; the described procedure is carried out until the set of arcs issuing from a* becomes empty;

e) we discard the vertex a* .

* The resultative schema 2 may also be stipulated formally. If

236

then

R. I. PODLOVCHENKO

22 = [a~, a;;,, {IT' (a, ~), <p" (a, ~), 1jl" (a, ~), a E a", ~ E B}],

2* cc= [a~, a~., {Il* (a, ;), <p* (a, ~), 1Jl* (a, ~), a E a*, ~ = B}J,

* 3) the set of internal vertices of the schema 2 · is obtained by combining the sets of internal vertices of the schemata 21 and 22 and eliminating the vertex a*;

* 4) for all internal vertices 01 of the schema 2 and all situations ~ we have

*(a ~)={<p'(a,;), if <p ' <p"(a, ;), if

aEa',

5) regardless of the vertex 01 of the schema / and the situation ; E E:

a) if 01 is not a preoutput of the schema 21, then

{ fl'(a,;),

rl*(a,;)== H"(a,s),

{a~(~)}Ufl"(a, ;){a*},

if a E a', if a E a"and a* E II" (a, ;),

if aEa", a*Eil"(a, ;);

1) 1jl' ( a, s)' if aEa';

2) ljl" (a, 6), if a E a" anda* E 11" (a, ;),

~'* (a, ;) = 3) however •. if a E a"and a* E I1" (a, ;) , then the mapping 1jl* (a, ;) is obtained from the mapping ljl" (a, 6): U ~ Il" (a, €) by replacement of the Vertex a* by the vertex a~ m in IT" ( a, s);

b) if 01 is a preoutput of the schema 21 , then

ll* (a, 6) = IT' (a, 6), ljl* (a, 6) = ljl" (a, 6).

4.4. The operationof splicing the outputs a' and a" ofthe schema 2, which * Ieads to the schema 2 , is defined only according to intent.

* In constructing the schema 2 we do the following:

1. We take the set of vertices of the schema 2.

* 2. The input of the schema 2 is assumed tobe the input of the schema 2, while the * outputs of the schema 2 are assumed to be all outputs of the schema 2, with the exception

of 01 11 •

3. In each situation ; E E:

a) we scan the arcs of the schema 2;

b) each arc arriving at the output 01 11 is directed to the output 011 ;

c) we discard the vertex a". * It can easily be verüied that the resultative schema 2 will be strongly equivalent to

the schema 2 , although it will also realize a new partitioning of the set Ez into subsets.


§ 5. Nonstationary Schemata and Their Transformations

5.1. Let us specify the sets D c;; ~ and R c;; ffi of actions and predicates on situations.

The schema 2=[a0 , aw,{IT(a, ~), lf(a, ~),'ljl(a, ~), aEa, ~EB}J is called a schema over D and R if

cp(a,~)EDUR,

regardless of the situation ~ from the internal vertex a of the schema 2.

2 will be called a s c h e m a o ver R if, regardless of the situation ~ and the interval vertex a of the schema 2,

cp(a,~)ER.

It can easily be seen that the operator realized by a schema over R is unitary.

5.2. Assuming D and R tobe nonempty sets, let us consider the class of all possible schemata over D and R; we establish a series of results.

Theorem 1. A strongly nonstationary schema 2 may be transformed into a weakly nonstationary s chema which is equivalentt to it if regardless of the partitioning of 8 into two subsets one can find a schema over R which realizes this partitioning.

Pro o f. Assurne 2 is a strongly nonstationary schema; this means that among its internal states one can find at least one vertex of a mixed type. Obviously, for proof of the theorem it is sufficient to advance a method of transforming the schema 2 into a schema 21 ,

equivalent to it in which the number of vertices of a mixedtype is one less than it is in the schema 2· Let us present such a method.

Assurne a is a vertex of the mixed type in the schema 2, and 8~~ is the set of all such situations (and only those) in which actions are juxtaposed with the vertex a; therefore, the set

combines in itself all such situations (and only them) in which predicates are juxtaposed with the vertex a .

-In accordance with the proposition ofthe theorem one can find a schema 2 over R which

realizes the partitioning of 8 into 8~ and B~ . The schema 2, just as any schema over R, will be weakly nonstationary.

Without loss of generality of the considerations, let us assume that 2 has a total of two outputs a' and a", and

(If this is not so and other outputs are present in the schema 2, then they are juxtaposed with empty sets of situations; in this case it is sufficient to carry out the operations of placing these outputs with, for example, the output a' in the final number, and we arrive at a schema equivalent to the original one and having the property postulated above.) Let us stipulate the following schemata which are stationary in 1/J :

t Here and below equivalence shall be understood to mean strong equivalence.


e' J''

, ( , t = { cp (cx, 6), if (jl y ' .,) A otherwise;

J'w

" ( " !:) _ { cp (cx, 6), if z n o----.-..o.".----'1-.o (jl y ' "' - . J'o' J'" y.;; p otherw1se;

here A and p represent an action and predicate which are arbitrary but specified. Both schemata, as is easily seen, are weakly nonstationary.

Let us compose the 2 -composition of the schemata c' and c": ", , "

2 = (2, 2' 2 ),

"' while assuming that in the schema 2 the first output is the output a'. The schema 2 will be wealdy nonstationary.

Finally, let us synthesize the schema 21, while performing the following operations.

1. We combine all vertices of the schemata 2 and 2'" into one set.

2. The input of the schema 21 is designated tobe the input of the schema 2, while the outputs are designated tobe all outputs of the schema 2.

3. In each situation ~ we do the following:

a) we combine all arcs of the schemata 2 and 2"' into one set;

b) each arc arriving at the vertex a of the schema 2, is directed to the first vertex of the Schema 2"', without altering the tags of this arc;

c) assume that in the situation ~ the vertex a is juxtaposed with an operator; therefore, · "' a single arc (a, a') issues from it; then from the preoutput y' of the schema 2

webring out the arc to the vertex a', while from the preoutput y" webring out an arc to the same vertex while tagging the latter arc with all elements of the set ut ; after this the arc (a, a') is discarded;

d) assume that the vertex a is juxtaposed with a predicate; then from the preoutput y' we bril"Jg out the arc to the same vertex; then we proceed as follows: if the arc (a, a') issues from the vertex a, then from the preoutput y" webring out an arc to the vertex a' and label it with the same elements of the set U that were used to label the arc (a~ a'); after this the arc (a, a') is discarded; the described procedure is repeated until the set of arcs issuing from a becomes empty;

e) we discard the vertex a, and the input and outputs of the schema z"'. The results of our operation will be a schema in which the number of the vertices of the

mixedtype is actually one less than it is in the schema 2. The theorem has been proved.

5.3. Theorem 2. A schema 2 which is nonlinear in cp may be transformed into a schema which is stationary in cp and is equivalent to it i f :

a) a ll i n t e r n a 1 ver t i c e s o f t h e s c h e m a 2 a r e f in i t e - v a 1 u e d ;

b) regardless of the finite partitioning of the set 8, one can find a schema over R which is stationary in cp and realizes this partitioning.

t Since the schema e1 is reduced in 1f!, it follows that it is superfluous to tag the single arc issuing from the A-vertex.


Pro o f. Since condition b) makes the statement of Theorem 1 valid, it is sufficient to consider the case in which 2 is a weakly nonstationary schema. Therefore, in the schema 2 each internal vertex is either an A-vertex or a p-vertex, and at least one of them is nonstationary in cp.

The theoremwill be proved if a method is given for constructing a schema equivalent to the schema 2 and containing vertices which are nonstationary in cp and are one less in number than those contained in the schema 2. Let us propose such a method.

Assurne a is a vertex which is nonstationary in cp and belongs to the schema 2. For a let us construct the schema 2', ha ving the following properties:

' 1) 2 is stationary in cp; '

2) 2 represents a schema of the type A if a is an A-vertex, and a schema of the type p if a is a p-vertex.

By virtue of the second property the schema 2' may be substituted into the schema 2

instead of the vertex a; assume that the result of the schema is 2". The schema 2' can be " constructed so that it is calculated to make 2 equivalent to the schema 2. The fact that

in the schema 2" the nurober of vertices which are nonstationary in cp will be one less than

in the schema 2 derives from stationarity in cp of the schema 2'·

' As has already been noted, in designing the schema 2 , it is necessary to distinguish whether or not the vertex a is an A-vertex or a p-vertex. We shall carry out all the necessary constructions only in the first case on the basis of the faet that the second case is completely analogaus to the first.

Thus, a is an A-vertex. Let us use the proposition concerning the finite-valuedness of a, and let us write out all values of the function cp (a, ~), ~ E 3,

Let us use 3i to denote the set of all those Situations (and only those) for which

cp (a, ~)=Ai, i = 1, 2, ... , m.

It can easily be seen that

lJ Si=B. i=1, 2, ... , m

Therefore, according to proposition b) one can find a scheme c over R which is stationary in cp and realizes the partitioning of the set 8 into S; , i = 1, 2, ••• , m. Without loss of gen-

erality, we shall assume (see subsection 5.2) that the schema Z has m outputs aC 1 ', aC 2 ', ••• ,

aCm), and that 3a<iJ = S;, i = 1, 2, .•. , m.

For each i E { 1, 2, ... , m} we construct the stationary (in cp and in lf!) schema c i:

rP yj,!i

(i.e., for all SE 3 cp (y(il, ~) = A;).


-Making use of the fact that we have already ordered the outputs of the schema 2 , we

construct the Z -composition of the Schemata 21, 22 , •.• , 2m. And this will be the schema /.


5.4. Theorem 3. A schema 2 which is stationary in cp may be transformed into a stationary schema equivalent to it if

a) e a c h p r e d i c a t e j u x t a p o s e d w i t h a p- ver t e x o f t h e s c h e m a i s finite-dimensioned;

b) regardless ofthe finite partitioning of the set 8, one can find a stationary schema over R which realizes this partitioning.

Pr o o f.. In just the same way as in the proof of ~heorems 1 and 2, Iet us indicate the method of transforming the schema 2 into a schema 2 , which is equivalent to it and has a smaller number than 2, of vertices which are nonstationary in 1/J. This will prove the theorem.

Let us consider the vertex a of the schema 2; which is nonstationary in 1/J ; this vertex will be an A-vertex or a p-vertex, and it will be stationary in cp for both cases,

Assurne a is an A-vertex; then in each situation ~ exactly one arc issues from it. Let us use 3 (a, a'), a' E a, to denote the set of all Situations (and only them) suchthat I1 (a, ~) = a'.

From among the sets 8 (a, a'), a' E a, let us isolate all nonempty words:

It is obvious that their number m exceeds 1, andin aggregate they have the properties

3 (a, CX;) n 3 (a, CXj) = </> for CX; =J= CXj,

U S (a, a;) = 3. i=1,2, ... ,m

Consequently, by assumption b) one can find a stationary schema over R (we shall denote

it by 2 ), which realizes the partitioning of the set 8 into 8 (a, rx;), i = 1, 2, ... , m. Without loss of generality, it may be assumed that z has the vertices a(1), a< 2 ), ••• , a (rnl as its outputs and

Sa(i)=S(a,rx;), i=1, 2, ... , m.

- ' Having the Schemata 2 and 2, available, we synthesize the schema 2. Forthis purpose:

1. We combine a vertices of the schemata 2 and Z into one set. ' 2. The input and outputs of the schema 2 are designated as the input and output of the

schema 2.

3. In each situation ~ we execute the following operations:

a) we combine the arcs of the schemata 2 and 2 into one set;

b) we discard the arc issuing from the vertex a;

c) webring out an arc from a to the first vertex of the schema 2; -

d) for each i = 1p 2, ••• , mall arcs arriving at the output a(il of the schema 2, are directed to the vertex a i of the schema 2;

e) we eliminate the input and outputs of the schema z.


The resultative schema 2 is our desired schema for the case in which a is an Avertex.

Assurne a is a p-vertex, and that the predicate p juxtaposed with a takes the values

[we use proposition (a)].

Let us specify a certain ui (1 ~ i ~ t); we use 8; (a, a') to designate the set of all those situations ~ (and only them) for which the following statement is valid: in the Situation ~ the arc (a, a') carrying the tag ui issues from the vertex a. Among the sets 3; (a, a'), a' e; a, let us enumerate all nonempty ones:

It is clear that

3; (a, au) n 3; (a, CX;k) = <[>, if j =/= k,

U Si (a, aii) = 8. j=1, 2, ... , mi

-Consequently, according to proposition b) one can find a stationary schema 2; over R which realizes the partitioning of 8 into 8; (a, aij), j = 1, 2, ••• , m. It can be assumed that

y;j, 1';2, ... , '\'im; are all outputs of the schema 2;, and

Bv .. =B;(a, tXij), jc=i, 2, ... , m;. l]

Assurne that for each ui we have found the schema 2; over R having the property de-' scribed above. Let us synthesize from the schemata 2 and 2;, i = 1, 2, ••• , t, the schema 2,

which gives the proof of the theorem. For this purpose: - - -

1) we combine all vertices of the schemata 2 and 21, 22 , ••• , 21 into one set;

' 2) the input and outputs of the schema 2 are called the input and outputs of the schema 2;

3) in each situation ~ we do the following: - - -

a) we combine the arcs of the schemata 2 and 21 , 22 , .•• , 21 into one set;

b) we discard all arcs issuing from the vertex a;

c) for each i = 1, 2, ••• , t we bring out an arc tagged by the element ui from the vertex a and direct it to the first vertex of the schema 2.;

' d) for each i = 1, 2, ..• , t and each j = 1, 2, .•• , m we execute the following operation:

all arcs arriving at the output y ii of the schema 2; are directed to the vertex aii ;

e) we discard all inputs and outputs of the schemata 2i' i = 1, 2, ••• , t.


§ 6. Stationary Schemata

6 .1. Let us show that for a nonempty set R one can construct a stationary schema over R whose definition domain is empty.


Actually, let us consider a schema 2 A

p

Yo Y,

in which all internal vertices (y1, y 2, y 3) are juxtaposed with one and the same predicate p, while the sets U' c;; U and U" c;; U (each of them combines those elements of the set U which tag one and the same arc of the schema) satisfy the conditions

U'UU"=U, U'nU"=cp.

It can easily be checked that when this schema is executed in any situation ~ the sequence of vertices we have mentioned has one of the two possible forms:

1) 'Yt, 'Y2' 'Y1' 'Y2, ••• , 'Yt, 'Y2, ••• '

2) 'Y 1' 'Y 3' 'Y 1' 'Y 3' ••• , 'Y 1' 'Y 3' • • • ;

the validity of our statement derives from this.

Note that the schema 2 A is directionally connected.

6.2. For stationary schemataover nonempty sets E and R the following theorems are valid.

Theorem 4. An arbitrary stationary schema 2 may be transformed into its equivalent connected schema.

Pr o o f • Assurne

is a stationary schema. Let us consider the set a* c;; a, which consists of all such vertices a of the schema 2 (and only them) for which one can construct a chain beginning at the vertex a 0 and ending at a. Let us distinguish between the cases:

1) a* contains just one output of the schema 2;

2) a* does not contain a single output of the schema 2.

In the first case one may choose the schema

* * - * 2 =[a0 , aw, {H(a), <p(a), 'ljJ(a), aEa }],

as the desired connected schema, where a:, = a* n au:; z* is obviously equivalent to the schema 2.

In the second case the definition domain of the schema 2 will be empty, and therefore the connected schema 2A will be weakly equivalent to the schema 2.

6.3. Theorem 5. For e ach c onne cted s chema <:! one m ay cons tr uct

a directionally connected schema which is an expansion of the schema 2.


Pr oof. Let us construct the set a* s a, while including in it all such vertices a E a, through which at least one path passes which connects the input of the schema 2 with one of its outputs.

If the set a* is empty, t then the schema 2 realizes an empty action, and therefore the schema 2,, which is weakly equivalent to it may be considered as an expansion of the schema 2 0

Assurne a* is not empty; let us construct the required schema 2*;

. * 1) the set a"' is considered to be the set of vertices of 2 ;

2) the input of the schema 2 * is called the input of the schema 2; the outputs of 2 *

are called those outputs of the schema 2, which are contained in a*, andin addition the vertex a~, where a0 E a;

3) we preserve all of the arcs of the schema 2 (with their tags) which issue from the vertices of the set a* and arrive at the vertices of this same set;

4) each arc issuing from the vertex a E a* and arriving at a vertex which is not contained in a *, shall be directed to a complementary output a ~ of the schema 2 *, without changing the tag of this arco

The theorem has been provedo

6o4o Theorem 6. Each connected schema 2 may be transformed

into a directionally connected schema c* which is weakly equivalent t 0 i t 0

For the proof it is sufficient to carry out the following constructions:

1) combine the set of vertices a * with the vertices of the schema 2A;

2) designate the input of the schema 2 as the input of the schema 2, the outputs of

the schema c* as all those outputs of the schema 2, which are contained in the set a *, and the output y w of the schema 2 A;

3) combine all arcs of 2, which issue from the vertices a* and arrive at the vertices of the same set and the arcs of the schema 2 A into one set;

4) direct each arc issuing from a vertex a E a* and arriving at a vertex which is not contained in a*, to the first vertex of the schema C?A;

5) discard the input y 0 of the schema 2A [along with the arc (y 0, y 1)]o

6o5o Theorem 7o Each directionally connected schema 2 having one output may be transformed into a linear schema which is equivalent to it, provided only that the set R contains an everywheredefinite predicate which does not take even one value uo EU.

t This case will occur ü 2 is a schema

11"

here .t\,_ and A2 are actions, and p is a predicate on Situations; U' s u and U":; u satisfy the conditions

U' U U"=cU, U' n U"=<j>.


Proof. Assurne

Z=(a0 , aw, {II(a), cp(a), 'ljl(a), aEa}]

is a directionally connected schema with one input, and

is the net of this schema.

If p is a certain e1ementary path in the net S, then each vertex be1onging to the path p

and differing from its beginning and end is cal1ed in t e r n a 1 •

The nurober of arcs forming the path p is called its 1 eng t h •

Two paths of the set S, bydefinition, do not cross ifnotasing1einternal vertex of one path is an internal vertex of the other path.

Two paths having identical beginnings and identical ends are cal1ed c o n t i g u o u s •

Assurne K is the set of e1ementary paths of the net S which are disjoint. The e1ementary path p which does not cross any other path of the set K and is suchthat the beginning and ends of p be1ong to certain paths from K is cal1ed b a s e d o n K •

Let us show that for the net S one can construct a sequence of e1ementary paths

K = po, P1, P2' o o o' Pm

which is suchthat

a) no paths p i and Pj, i ;z! j, cross;

b) for any i = 1, 2, ••• , m, the path p i is based on the set

{po, Ph · • ·, Pi-1 };

c) each vertex of the net S be1ongs to at least one path of the set K.

Let us begin the construction of the sequence K with the selection of the path Po·

As Po we shall take any elementary path on the net S, which connects the input a 0 with the output a w.

Let us use a 0 to denote the set of vertices belonging to the path Po·

If a0 = a, then the path Po exhausts the desired sequence K.

Let us assume that a0 =1= a, and let us note that the vertices of the set a 0 are naturally ordered in the path p 0•

Let us use Kf' to denote the set of all elementary paths of length greater than unity which are based on the path p 0• It is evident that Kj is not empty if a 0 =1= a 0

Let us introduce partial order in the set Kj . We shall say that p' Ex: p r e c e d e s p" Ex: (p' < p"), if one of the following two conditions holds:

1) the beginning of the path p' has a lower nurober than the beginning of the path p" ;

2) the numbers of the beginnings of the paths p' and p" coincide, but the nurober of the end of the path p' is lower than the nurober of the end of the path p".


Let us consider the maximum sequence of paths from Kf P1, Pz, · · ·, Pt, (3)

which satisfies the conditions:

1) P1 -< Pz -< · · · -< Pt;

2) Pt is a minimal element of the partially ordered set K~ ;

3) for all i == 1, 2, .•• , l the path Pi does not cross any of the paths p 1, p2, ••• , Pi -t·

It is evident that in the general case the sequence (3) may b e constructed ambiguously, but its length l remains unchanged.

Let us attach the following in order at the right to the sequence (3):

all paths contiguous with Pt;

all paths contiguous with p2;

all paths contiguous with p z

(the order established among the paths which are contiguous with a certain Pi, 1 :s i :s l may be arbitrary); the sequence of paths obtained shall be denoted by Kl.

Finally, to the sequence K0 consisting of the path p 0 we attach the sequence KI at the right, and use Kt to denote the resultative sequence. It is obvious that it satisfies the two requirements imposed on the sequence K.

Let us use a 1 to denote the set obtained by the union of all vertices through which paths from Kt pass.

If a1 = a, then the sequence Kt is the one desired.

If a1 =1= a, then we shall construct a sequence of paths K2 in a manner completely analogous to the manner in which the sequence Kt was constructed. For this purpose it is first necessary to order the vertices of the set a 1• Note that a1 is derived from ao by adding internal vertices of all paths of the sequence Kl. The vertices of the set ao have already been ordered; let us assume that they are precursors of all the remaining vertices of the set a 1 we also order these remaining ones.

If the vertices belang to the same path, then we preserve the order of their sequence in the path.

Of two vertices belanging to different paths we call the precursor that one which is contained in the path having the lower nurober.

Thus, the set a 1 is ordered, and one may go over to constructing the set K; and ordering its elements.

The procedure described terminates as soon as for a certain i 2: 1 the set of vertices ai coincides with a.

Having available the sequence K, let us carry out the following transformation of the schema 2·

Assurne p is a predicate on situations having a definition domain 3, which does not take the value u0•


1. To the set of vertices a we add the vertices

each of which is juxtaposed with a predicate p.

2. The input and output of the future schema remain a 0 and

Fig. 1 3. For each i = 0, 1, ••• , m we do the following:

a) all arcs arriving at the end of the path p i are directed to the vertex ßi without any change of their tags;

b) from the vertex ßi webring out two arcs: one, which is tagged by the element u0,

is directed to the first internal vertex of the path p i + 1 (for i = m it is directed to the vertex aw ) ; the second, which is tagged by all of the remaining elements in the set U, is directed to the end of the path Pi. It is easy to verify the fact that the designed schema is linear. Actually, assume the path Pi passes through the vertices

Then the sequence of vertices

ao = aoo, ao!l ao2, · · ·, ao, to-1, ßo,

au, a12' ... , a1,tt-1, ßll

which contains all the vertices of the schema, constitutes an elementary path. The equivalence of the original and resultative schema is obvious. The theorem has been proved.

§ 7. Examples

7 .1. As has already been assumed in subsection 3.6, the vertices of a graph are depictoo by points on a plane and the arcs are depicted by directional curves connecting the vertices of the graph.

Figure 1 depicts the graph r = {II (a) a E a }, where

a = {all a2, aa, a4, as, a6}, fi(at)={a2, aa, a4}, II(a2)={aa, as},

n (aa) = {a2, as},

rr (a,) = { a4}, n (as) = </>' rr (a6) = {aa}·

The graph r has nine arcs; here the arcs (a1, a 2), (a1, a 3), and (a1, a 4) issue from the vertex a 1, the arc (a4, a 4) issues from the vertex a 4, and no arc issues from the vertex a 5; at least one arc of the graph enters all vertices except a 1 and a6•

The sequence of arcs

may serve as an example of a path in the graph r; this path passes through the vertices a 6, a 3,

a 2, a 3, a 5 and contains the contour (a3, a 2), (a2, a 3). Assurne (a1, a 2), (a2, a 3), (a 3, a 5) is elementary.

IZo


Fig. 2

Fig. 3

Fig. 4

There are eight edges in the graph r: the vertices a 2 and a 3 are connected by one edge A = {a2,a3}.

The graph r is connected, since any two of its vertices may be connected by a chain. For example, the vertices a 6 and a 4 are connected by the chain

{cxa, cx3}, {cx3, cxi}, {cx~> a 4}, {a4, cx4},

while the vertices a 6 and a 6 are connected by the chain

If in the graph r the vertex a6 (not a single arc of the graph enters it, and exactly one arc issues from it) is called the input, while the vertex a 5 (not a single arc of the graph issues from it) is called the out p u t , then we obtain the net

the vertices a 1, a2, a 3, and a4 are internal.

The net S will not be directionally connected; actually no such path exists which wculd begin at the input, end at the output, and pass through the vertex a 1•

We present examples of a directionally connected net (Fig. 2), a net-tree (Fig. 3), and a linear net (Fig. 4); each of the nets is stipulated geometrically; only the input and output of the net are provided with designations. In Fig. 4, one and the samelinear net is given in two depictions; the second of them clarifies the appearance of the term "linear."

7 .2. Let us present an example of a schema on Situations.

Assurne the mapping u (~) is such that it partitions the set of all situations 8 into four subsets

';:;'(!) ~{2) ~(3) ):::'(4)

...... ' ....... '~ ' ....... '

which are disjoint, and juxtaposes all situations of an individual set 8(i> with one and the same (but its own for each set) net Si, i = 1, 2, 3, 4.

Let us recall that the set of vertices of a schema, and likewise its input and output, do not depend on the situation considered. Depicting each vertex of a schema by a point on a plane, we shall agree to preserve their mutual arrangement on the plane unchanged, regardless of the net associated with the schema which we might be describing. Then it is sufficient to give the designation of the vertices themselves on just one of the nets.

Assurne that in our case

where a 0 is the input, and aw is the output of the schema, while Fig. 5 provides descriptions of all nets Sp i = 1, 2, 3, 4.


a b

Fig. 5

The mapping cp(CL, ~) is constructed as follows: each of the sets 3m and ~<3> is partitioned into two nonintersecting subsets

and then the entire set E will constitute the sum of six disjoint subsets

E6= 8(4>

Let us assume that for each internal vertex CL of the schema 2 the following holds: the image of the pair (CL, 0 in the mapping cp(CL, 0 is determined solely by the choice of the vertex CL and by which subset B; the situation ~ belongs to (i.e., the image of the pair (CL, ~) does not change while the situation ~ traverses the subset 3;).

Consequently, the mapping cp(CL, 0 may be described thus: on each of the subsets 3;, 1 ~ i ~ 6, it is required to consider the net juxtaposed with the situations of this subset, and each internal vertex of the net is tagged with the action or predicate juxtaposed with it. Such a description of cp (CL, 0 is given in Fig. 6a; the symbols A1, A2, Aa denote certain actions, while the symbols p, q, r denote certain predicates on situations.

Let us now postulate that the set U consists of two elements 0 and 1, and let us make use of the already available partitioning of 8 into the subsets 8;, i == 2, 2, ... , 6 for stipulating the mappings lf! (CL, ~).

Let us suppose that lf! (CL, ~) is determined solely by the choice of the vertex a and by those of the sets 8; to which the considered situation ~ belongs. Then it is sufficient to use the elements 0 and 1 to tag the arcs of the nets juxtaposed with the set 8;. in order to describe the mappings 1/J(CL, 0 acting under these conditions in accordance with the definition given in 2.4. Such tagging of the arcs is displayed in Fig. 6a.

Thus, Fig. 6a contains complete information on the schema 2· Note that the first vertex of the schema 2 is CL 3 if the situation ~ belongs to the set 3<3> and CL1 in all remaining cases.

7 .3. In executing the schema 2 in a stipulated situation ~ it is necessary in the first place to establish which of the sets 8; the situation ~ belongs to.

Assume, for example, ~ E 3 1; then the following cases are possible:

1) the process of executing the schema 2 is completed without result on the first step, since ~ E 3p, where 8p is the definition domain ofthe predicate p; the value of the schema 2 in this case will be an empty sequence;

ALGORITHM SCHEMATA DEFINED ON SITUATIONS

b

r

V~ r

~3,

Fig. 6

2) the process of executing the schema 2 is completed without result on the second step; this occurs when s E Bp and one of the two following conditions holds:

a) p (s) = 1 and s E SA2;

b) p (s) = 0 and s E BA3;

249

here BA2 and BA 3 denote the definitiondomains of the actions A2 and A3; the value of the schema 2 in the case a) will be the action A2, while in the case b) it will be the action A3;

3) the process of executing the schema 2 is terminated with a result on the third step, the sequences (1) and (2) having the following form:

a) s, s. Az (S); (1) } for P(s)=1; a~> az, aw; (2)

b) s, s. Aa (s); (1) } for p (s) = o; a 1 , a 3 , au;; (2)

in case a) the terminal situation is A2 (~ ), and the value of the schema consists of the action A2;

in case b) the situation A3 (~) is the terminal one, and the value of the schema consists of the action A3•


However, if the original situation 6 E Ba and the Superposition

is meaningful, then the process of executing the schema 2 in the situation ~ leads to the construction ofthe sequences

here the terminal situationswill be A3(A2(A1 (~ ))), while the value of the schemawill be the sequence of actions

The process of executing the schema 2 may also be infinite. Actually, suppose

a) Bp1 n B 3 =1= 4>, where Bp1 is the set of all those situations on which the predicate p takes the value 1;

b) the action A2 maps the set Bpt n B3 in itself. Then for execution of the schema 2

(1)

(2)

in the situation 6 E Bpt n B3 weshall inspect the vertices a 1 and a 2 endlessly, going over from a 1 to a 2 and returning from a 2 to a 1• The value of the schema 2 in this case will be the infinite sequence

In order to describe the definition domain Be of the schema 2 and the action Ae

realized by the schema, it is necessary to concretize the choice of the actions A1, A2, A3 and the predicates p, q, r. One thing is obvious here: since the schema 2 has one output a w' the partitioning of Be realized by it consists of just this set B2 .

7 .4. Let us use the classification of schema introduced in§ 3, and let us define the type of schema 2·

First of all let us note that the schema 2 is not reduced in l/J. Actually, in situations 6 E B< 1> the arcs (a2, a 3) and (a3, a 2), which are not tagged by the elements of the set U ~ issue

from the vertices a 2 and a 3, while in situations 6 E B<2> the untagged arc (a2, O!w) issues from the vertex 0!2·

Figure 6b gives the schema 2~> reduced in lf!, which differs from the schema 2 solely by the elimination of the arcs enumerated above. The schema 21 is obviously equivalent (strongly) to the schema 2·

Let us go over to the characteristic of the vertices of the schema 2t· Of the three internal vertices of the schema 21 only the vertex a 3 is stationary in cp: in all situations ~ E B it is juxtaposed with the action A3 (i.e., a 3 is an A-vertex); the vertices a 1 and a 2 are of the mixed type. All three vertices a 1, a 2, a 3 are finite-valued and nonstationary in l/J.

Thus, the schema 21 is nonstationary in l/J and strongly nonstationary in cp, which allows the transformations described in Theorems 1-3 to be demonstrated on it.

Note that the schema 21 is stipuled by six schemata I-VI, each of which is reduced in 1/J, stationary in cp and 1f!, and desribes the schema 21 on one of the sets Bi, i = 1, 2, ... , 6.

In describing schemata which are reduced in l/J let us agree to leave arcs issuing from Avertices and from the input without tags, as has already been donein Fig. 6b.


7 .5. Postulating the schema 2t tobe a schema over the sets D and R, we shall to the degree necessary impose on R the requirements which ensure fulfillment of the presumptions of Theorems 1-3.

Let us first carry out the transition from the schema 2t, which is strongly nonstationary in cp, to its equivalent schema 2z, which is weakly nonstationary in cp. (It is obvious that in order to describe the latter we shall require no more than six schemata which are stationary in cp and 1{! and are relegated to the sets 2;, i = 1, 2, ... , 6 .)

For this purpose let us consider the mapping cp (a, ~) in vertices of the mixedtype

In order for the vertices a 1 and a 2 tobe replaced by vertices of the type A and p, it is necessary to realize partitioning of 3 into sets 31 U 3z U 83 and 3,, U 35 U 36 in one case, and partitioning of 3 into S" U :=:s and 21 1J 22 lJ 33 1J 36 in the other case.

Let us postulate that the predicates

belong to the set R, and let us construct weakly stationary in cp (and stationary in 1{!) schemata ' "

2 and 2. "..------------- .. \ I I

: I 0./

l ;Sz : a'o----J->C i

: ;a, o I at I I

\ Ia ' '------- ----- __ )

,,~- ------------,

: \ 0.1 I

a" I J'2 o----+--< : y,

1 : at I

: 73 : "---------- ___ .. J

~ E 311J 83, ~ E 82, ~ E 3, 1J 3s 1J 36;

~E2.U3s,

~ E 81 U 82 U 83 U 36 .


Here A and B are arbitrarily chosen actions, while A. and T are arbitrarily chosen predicates.

Making use of an intentionallanguage, it may be said that the schema 2 executes the

functions of the vertex a 1, while the schema 2 executes the functions of the vertex a 2• . .. The desired schema 22 can be synthesized from the schemata 21, 2 and 2 accord-

ing to the rules expounded in subsection 5.2. In order to trace more easily how this synthesis

is achieved, rectangles are drawn around those vertices and their connecting arcs in the . " schemata 2 and 2 which must enter into the composition of the schema 22; these same vertices and arcs, which are already incorporated in the schema 22 , are enclosed in rectangles there too (see Fig. 7). Considering the schemata I-VI describing 22 , and comparing them with the schemata I-VI describing 21 , it can easily be established that the rectangles take the place of the vertices a 1 and a 2•

'1.6. The schema 22 is weakly stationary in cp. Let us transform it into the schema 23

which is equivalent to it and stationary in cp.

Forthis purpose we must consider all vertices of the schema 22 which are nonstationary in cp. In order to reduce their number we place

then in the schema 22 only two vertices ß2 and ßs will be nonstationary in cp. If we take

then this reduces the set of values taken by the function cp2 (we denote the mapping of cp in the schema 22. this way) at the vertices ß.! and ß3• Thus,

tions

{ A2, ~E8,, 'P2 (ß2, ~) = A t -E >< •

f, 'o ....... 4,

{ q, ~ E 82, 'P2 (ßg, ~) = p, ~ E 82.

We shall postulate that the set R contains the predicates f.l. 3 and f.l. 4 defined by the equa-

{ 0,

lld~) = 1,

{ 0, 11• (~) = 1'

~E B,, SE 84;

Then the stationary (in cp and l/J) schemata 2"' and lv will execute the same functions as do the vertices ß2 and ßs•

At

!z

Az

~

p 0.1 ez

'I at eJ

ALGORITHM SCHEMATA DEFIN'ED ON SITUATIONS 253

I a 0

«w

f./1 tto

II

II! ;tt,

/II

/Y,Y

VI

Fig. 7 Fig. 8

IV Substituting the schema 2 (type A) for the A-vertex ß 2, and the schema 2 (type p)

for the p-vertex ß 3 in the schema 22 , we shall obtain the schema 23 which is stationary in cp (Fig. 8). 23 can be described by means of stationary (in cp and 1f;) schemata of four types; the rectangles in them denote the places occupied in the schema 22 by the vertices ß 2 and ß 3•

7.7. In the schema 23 four vertices are stationary in 1f;

and the remaining ones are nonstationary in 1f; •

From the A-vertices o2, o3, y2, and a 3 one arc issues in each situation ~; this arc is directed as follows:

{ ö for

from Ö2 andÖ3 to a: '\'t

from y2 to ß~ { a for

1'2 (1.3

~ E St U S2 U 83, HS4USs, ~ E Sa;

~ESt U 82, ~ E 83, HS4USs, ~E Ss;

(4}

(5)


from a3 to { aw for

y1 for

s E E1 U E2 U Ea U E6,

SEE, U Es.

From the p-vertices e 2, e 3, and y 3 arcs issue in each situation ~ which are tagged by the elements 0 and 1; an arc having the tag 0 travels

from Ya to { Ya for aw for

an arc with the tag 1 tra vels

from 81 and e2 to 1 { y for

8 1 for

{ '\'3 for from '\'3 to ß1 for

SE Bt U 82 U 83,

S E 84 U Bs U 86;

SE 81 U 82 U BaU Ee, SE 84 U Bs;

SE 81 U 82 U Ba, ~E84 U 8s U 86;

An arc issues from the input a 0 to the vertex ß 1,

and to the vertex a3 , if s E E, U Es.

(6)

(7)

(8)

(9)

(10)

(11)

In order to realize the partitionings which we require of the set E into subsets (these partitionings are represented by the descriptions (4)-(11)), we postulate that the predicate

belongs to the set R.

Let us construct the following stationary schemata over R:

zrl

... 1-lz Z"' o---..-.:<

'Ji' u 'Ji' u • u ::' ~..~, .. 2 -J '"'6

s4 u s,


Fig. 9

Here each of the outputs of the schema is tagged by a subset consisting of all those Situations in which we arrive at this output if we execute the schema.

V It can easily be verified that the schema 2 realizes the parititioning specified in the

description (7) and (9), the schema 2 vr realizes the partitioning specified in (6), (8), (10), and

(11), the schema 2vn realizes the partitioning specified in (4), and, finally, the schema 2vm realizes the partitioning specified in the description (5).

The transformation of the schema 23 into a schema 24 which is equivalent to it and is stationary (in cp and 1/J) consists of a series of transformations which eliminate nonstationarity in 1/J of the individual vertices of the schema 2 3•

Let us consider, for example, how nonstationarity of the vertex o 2 is eliminated. The

schema 2 vn is taken which realizes the partitioning required in this case; the input of the

schema 2 VII is made to coincide with the actual vertex o2; that output of the schema 2 vu,

which is tagged by the set 2, U 82 U 23, is made to coincide with the vertex o1 (in the schema 23

the arc from 02 arrived at it in the Situations SC~, U 32 U 8a); the Output of the SChema 2 VII,

tagged with the set 34 U 8 5 , is made to coincide with the vertex a 3 (the arc from o2 in the VII

situations s E 3, U 25) arrived at it); finally, the remaining third output of the schema 2 having the tag 36, is made to coincide with y 1 (the arc from o2 in the situations s E 86) arrived at this vertex in the schema 2 ) .

3 V VIII

Figure 9 shows the resultative schema 24 The vertices of the schemata 2 -2 which are included in it are enclosed in rectangles; it is easy to discern the fact that arcs (one or two; the latter holds in those cases in which the schema is used twice while being included in the composition of 24 at only one spot) enter each such reetangle from vertices which were previously nonstationary in 1/J.

7 .8. The schema 24 is a directionally connected schema having one output. We shall assume that the predicate

f,\6 m == o


Fig. 10

Fig. 11

is contained in the set R over which the schema 2~ is considered (this predicate does not take the value 1). Then the premises of Theorem 7 will be fulfilled; i.e., the possibility develops of transforming the schema z, into a linear schema equivalent to it.

Let us realize such a transformation.

Assurne S4 is a net on which the schema Z4 has been constructed. For s4 we must construct the sequence

K = po, P~o · · ., Pm

of elementary paths which are disjoint and are suchthat for each i (1:::::; i:::::; m), the path Pi is based on the set of paths

Each vertex of the net S4 must belong to at least one of the paths of the sequence K.

ALGORITHM SCHEMATA DEFlliED ON SITUATIONS

The construction of the sequence K has been demonstrated in Fig. 10. As Po we have chosen the path which passes through the vertices

this path is denoted in Fig. 10 by the heavy lines.

257

The elementary paths which are based on Po are marked on Fig. 10 by dashed lines; these are

(Here and further on a path is stipulated by the sequence of vertices through which it passes.)

It is obvious that the paths Pi,_p2, p3, and p4 are disjoint and are connected by successor relations

We leave it to the reader to check the fact that the sequence Pi, p2, p3, and p4 is the maximal one among sequences of elementary paths having the properties noted.

The paths

fail to contain only three vertices of the net S4: ö3, ß6, ßiz· Through these vertices we draw the paths

The paths p5, p6, p7 are disjoint, are based on the set of paths { P0, Pi, p2, p3, P4}, and are in the relation

they are shown in Fig. 10 by the thin solid lines; all paths of length 1 are drawn in Fig. 10 using dotted lines.

The sequence K has been constructed and has the following form:

Let us use 25 to denote the linear schema which is equivalent to the schema 24 and is obtained from 24 by the transformations described in subsection 6.5; S5 denotes the net on which the schema 2_, is constructed.

The net S5 includes all vertices of the net s4 and has vertices added in quantity equal to the nurober of elements in the sequence K. Each additional vertex is juxtaposed with apredicate f.ls; the vertices which go over into S5 from the net S4 are juxtaposed with the same actions and predicates as in the schema 24· Therefore, if the net s5 is constructed and the arcs issuing from its p-vertices are equipped with the tags 0 and 1, then we obtain complete information on the schema 25·

The net S5 with the tagged arcs is depicted in Fig. 11.

In describing the net S5 we isolated that elementary path (we shall call it the principal one) which begins at the input a 0, ends at the output aw, and passes successively through all vertices of the net. All branchings from the principal path are given by arcs whose beginning and end belong to the principal path. Each such .arc in Fig. 11 is represented by two arrows:


1) one issues from the vertex which is the beginning of the arc considered, and it is tagged by the symbol of the end-vertex of the arc; 2) the other arrives at the vertex which is the end of the arc, and it is labelad by the symbol of the beginning vertex of the arc.

7 .9. Let us sum up. In order to demonstrate transformations of schemata used in the proof of Theorems 1-3 and 7 we chose a schema 2 with one output, which was strongly nonstationary in cp and nonstationary in lf! •

The fulfillment of a portion ofthe premises of the theorems mentioned was ensured by the choice of constructions of the schema Z1; in order to fulfill the remaining premises it was necessary to include the predicate J.t 1, J.t 2, ••• , J.t 6 in the set R (over which the schema 2 was considered).

The schema Z which was not reduced in lf! was first replaced by the schema 21, which was equivalent to it and was reduced in lf!; strong nonstationarity in cp and nonstationarity in lf!

carried over to 21 from the schema 2.

By equivalent transformations, the schema 21 was converted into the schema 22, which was weakly nonstationary in cp; the schema 22 was converted into the schema Z3, which was stationary in cp, but nonstationary in lf!; the schema 23 was converted into the schema 24 ,

which was stationary in cp and lf! ; finally 8 the schema 2 4 was converted into the linear schema

2 • 5

Literature Cited

1. c. Berge, Theory of Graphs and Its Application [Russian translation], IL, Moscow (1962). 2. L. A. Kaluzhnin, "On algorithmization of mathematical problems," in: Problemy Ki

bernetiki., Vol. 2, Fizmatgiz, Moscow (1959), pp. 51-68. 3. R. I. Podlovchenko, "On transformations of program schemata and their application in

programming," in: Problemy Kibernetiki, Vol. 7 [in Russian], Fizmatgiz, Moscow (1962), pp. 161-188.

4. H. Thiele, Wissenschaftstheoretische Untersuchungen in Algorithmischen Sprachen, I, VEB Deutscher Verlag der Wissenschaften, Berlin (1966).

CONTROL PROCESSES IN LIVING ORGANISMS

ON THE PROBLEM OF MODELING FOR AN

EVOLUTION AR Y PROCESS WITH REGARD TO

METHODS OF SELECTION. IIt

T. I. Bulgakova, 0. S. Kulagina, and A. A. Lyapunov

Moscow and Novosibirsk

The present work is an immediate continuation of our previous paper [1]. The basic problern studied isthat of determining a more detailed statistical mechanism of divergence form andin studying the formation of a genetically isolated "invariant" of a group in a limiting population under different mechanisms of choice affecting the population.

§ 1. Types of Choices

In the work [1) we described one kind of choice (a "one-sided" choice) and represented results of experiments for the population No. 6, belanging to zones of "stable instability" (see [2]).

In the present work we examine three types of choices, and respectively, three series of experiments. As initial populations we select the populations Nos. 6, 8, and 9 of [2). P~pulations Nos. 6 and 8 refer to the zone of "stable instability" for a lack of choice, i.e., for which, under identical conditions in distinct realizations the changes occur in the same fashion: we are to give random numbers for the groups and the collections with the greatest or least isolation of one from the other (for characterizing such populations we will use the word "unstable"). Population No. 8 in the experiments without choice has a tendency to disintegrate into isolated groups (such populations will be called "disintegrating"). In addition to the populations described in [2] we describe a population No. 10, which in the experiment without choice is merged into one group (such a population is called "contracting"). The population No. 10 has the following characteristics: the nurober of elements in the initial population N0 = 150; the nurober of genes in the ge.netic type, influencing the restrictions on the process, n = 18; the nurober of support trains Mo = 25; for each support train for q = 5 genotypes, the distance from support to spread is not greater than l = 4. Definitions of all terms can be found in [2].

In the first series of experiments the choice of organisms was that described in [1]. Exactly as in [1] the advantage in survivial is given to the genotypes which are nearer to an "ideal" (an "ideal" genotype is supposed to consist of a unit). For the formation of a descen-

dant the nurober a is calculated by the formula a = 1- - ßo r~:r + y, where r is the length

tOriginal article submitted August 22, 1968.

261

262 T. I. BULGAKOVA, 0. S. KULAGINA, AND A. A. LYAPUNOV

of the geonotype of this descendant with respect to the "ideal" genotype; rm is the maximal possible distance from the "ideal" genotype (this "ideal" genotype consists only of the unit, up to r m = n, where n is a number and (r m - r) is the number of units in it); ß 0 is a positive constant for every experiment; y is a variable, regulating the size of the population in such a way that the size is not overly dependent on its initial value. The correction y is calculated in the following way. By definition the interval of time indicates the size of the population N and its deviation from the initial value ~ = N - N 0• If ~N > o, then y = + y 0 (y 0 and o are fixed positive numbers); ü (-~> > o; then y = -y 0, ü 1 ~ ~ < o, y = o.

The value a is equal to the random number ~, chosen from the interval [0, 1] and subordinated to this interval by the rule of distribution with constant density. If ~ > a, then the descendant survives and determines in this way its location in the population. If ~ :::; a, then the descendant will not survive. A choice of this type is called a one-sided choice. We recall that in an experiment without choice for the determination of the survival of the descendant the random number ~ is equal to 1/2.

In the second series of experiments the choice was organized in the following way. We introduce two "ideal" genotypes, with one consisting only of zero, and the other, only of one. The random variable ~ is chosen, as in the first series of experiments, equal to a, while a is selected by the formula

if r > r0 ,

where r is the former distance of the genotype from the unit, 0 :::; r 0 :::; r m; r m = n, ßo and ß1 are fixed positive numbers for every experiment. Therefore this choice gives a preference for the survival of those genotypes which are close to one of the "ideal" genotypes, and the intermediate genotypes have a smaller probability of survivial. This choice will be called two-sided.

In the third series of experiments the preference is given to those genotypes having r 0

zeros and (r m - r 0) units and such that a is determined by the formula

if r> r0 •

Thus for r 0 close to rm/2, the preferred genotypes have approximately equal numbers of zeros and units. This choice is called central.

The fourth series of experimentswas carried out because in the basic paper [1] very limited experimental material is found, while because of this, most subsequent papers bear the character of preliminary information. At the same time, experiments on the components of the I-series of the present paper entirely confirm the results of [H The series of experiments II and III in some other arrangements also verify the existing phenomena described in [1].

§ 2. Description of the Experiments

In [1] and [2] the details were given of a model population and a model process of reproduction which we will not repeat. We recall only that one act of the process, corresponding to unit time, consists of selecting a pair of separate entities, capable of having descendants, and forming four offspring; after this we determine the survival of each of the offspring, as described in § 1, and the survival of the offspring distributed in the population. As in [1] and

SELECTION IN MODELLING AN EVOLUTIONARY PROCESS 263

[2], we will determine the process of evolution for an initial population with a known structure. The structure is characterized by the collection of groups in the population, and by their size (i.e., the collection of elements found in every group), and the order of linkage of the groups in the family (by the number of isolated groups, the number of groups, found in every family, and the size of the families).

Since the choice is carried out in such a way that preference is given to those elements having certain defined numbers of units in the genotype, it is generally found that the evolution of the population bears the following characteristics.

N

~ em 1. The mean number of units in the genotype of the population Eav= m=~ • where em

is the unit set in the genotype with number m, m = 1, 2, ... , N.

Ni

~emi 2. Themeannumberofunitsinthegenotypeofthei-thgroup, Ei= miNi , where Ni is

the number of elements in the i-th group.

3. The mean number of units in the genotype of the maximal group, for which Ni0 =maxNi, is denoted by Eio· i

If the characteristics described at the points 1, 2, 3, are calculated by a machine, then the characteristics described at the points 4 and 5 are calculated automatically.

4. The mean number of units in the genotypes of the k-th group is Ek (in the present experiments k denotes the assumed values 0, 1, 2, 3).

If k = 1, then Ek is given by the formula

and i1 is the number of isolated groups.

If k = 0, then the quantity Ek is not calculated; if k > 1, then Ek is calculated approximately by the mean of the number of units in the groups found in the k-th family.

5. The mean number of units in the genotypes of the maximal family for which Nko = max N", is denoted by Eho·

k

Every characteristic E is calculated with an accuracy to the first decimal place.

The results of experiments are presented with the aid of tables and graphs. In every experiment we present the distance between elements of the group p = 5. In each of the experiments we enumerate the group, calculate Eav• Ei, andconstructfour graphs: the firsttime for the original population is given by means of all 960 units of time, which corresponds to a change of 13 generations. Thus, as in [2], the population is traced through a duration of 2880 units of time, i.e., for 40 generations. The computation of the correction y for every experiment is carried out for every unit of time.

Each of the tables corresponds to the determinations of a series of experiments and characterizes the structure of the finite population, obtained from the initial population after 2880 time steps.

In Table 1 we present the results of experiments for the one-sided choice, in Table 2, for the two-sided, and in Table 3, for the central choice. For every type of choice we carry


u, ~ 48

40

48

J2 J2

24 24 ~ ., '· ,~ ., I.

'1 15 18

~~.: ; !'1! I • 'y

8 8 /'· . ../ : i i

.~. ,.,_ • .J·

0 (j 24 0 4 8 e

Fig. 1 Fig. 2

out the experiments with four different populations. In each series of experiments with one and the same population differing pairwise by means of the order of the formation of the pair of genotypes, given descendants, and values of the parameters of the choice ß, y.

We now explain some of the graphs of the tables.t In graph 3 for every population we introduce the number n of genes in the genotypes of the population; in graph 5, the numbers of finite populations. In graph 9 we introduce the number of groups found in each of the families. For miscellaneous families the numbers are separated by the sign n I ,n In graph 10 we introduce the number of elements in the family and in the isolated groups, as weÜ as the number of elements in the families contained in the brackets. In graph 14 we find the mean number of units Ek in the genotypes of thefamily (included in brackets) andin the isolated group. For this the order of the groups and families are the same as that of graph 10. In graphs 13, 15, 16, whenever we indicate the number of units, we introduce the mean number of units, calculated as on p. 264. For the remaining graphs of the tables an explanation is unnecessary.

Additional tables refer to the graphs of Ne = j(e), where e is the number of units in the genotypes, which can be taken as a discrete value between 0 and n. Ne is the number of critical points in the population, whose genotypes contain e units. The graphs are found not for every experiment, but only for certain characteristic cases. The remaining cases are qualitatively similar to the previous ones. In each figure we present four graphs: the first for the initial time (denoted by lines and points), the second for the population obtained after 960 times units (the graph denoted by dots), the third; obtained after 1920 time units (denoted by points), the fourth for the final population (denoted by means of lines),

By means of the graphs and tables it is possible to observe that in the case of a n onesidedn choice (Figs. 1 and 2) the principle part of the population is displaced by the aspects of the nidealn genotype. Forthis ncompressedn population (No. 10) an nidealn displacement dictates the choice of one family (see Fig. 2, experiments 18, 19). The "unstablen populations

t The enumeration of the experiments is that of [1].


TABLE 1

I x ~ 1 § '" ~ § <I) Ii ~ ; ·~ E E x x ()

5 +-' +-' '8 ·~ I '" .s .s '" z ..... ..... '" '" ..c: ""

lJ 5. ::l 2 2 () "' bO

"' "' 0:. ci +-' Q) +-' +-' ci 4-< c ...... "0 0 c 5 c :> 0 .s .s z 0 <I) Ii <I)

~ 0.. <I) ·~ '" 0.. z 2 +-' fo 4-< 2 0 2 ."w 4-< "' "' +-' "' ::l '" 0 .~ ~ 0 .::: .:Z .';::! c c "" <I) 0

..... <I)~ ~ .~ Cl) .g <I) ..... .... 0

~ ~e: <I) n)Z CO Cl) §w c .§ ~ Cl) bO . ~

~ ~z ::s·a .... "' ;::l·~

'" 4-< 4-< 4-< 4-< ::l::l 4-<>-. 4-<ol ::l +-' 4-<>-. .... w 8;.. +-'0 +-' +-' ....

"" ..... 0 0 0 0 gbl () o- 0 0.. o,.... () d 0;.:! 00.. <I) ;::l 8 • ::l . ·a • ::l .[ e ::l . E ·5 0.. 0.. ci ci ci 0 .... ....

0 '" 0 0 oo

0 '" 0,_. X 0

'" 0 +->d +-' zbb +-' d w 0.. d o..<n z z z z (j).,.. (J) z .... Zo.. (J) .... z .... ZbO

10 6 24 0.3 148 11 5 2 4/2 (130), (7) 130 121 18.4 (19.1); (15.5) 19.1 19.1 0.2 3, 2, 4 9,0; 15.0;

1, 1 13.3; 8.0; 12.0

11 6 24 0,3 151 10 4 1 6 (125), 2, 125 111 18.1 (18.9); 8.5; 18.9 18.9 0.1 20, 1, 3 16,0; 5.0;

11.0

12 9 30 0.2; 130 12 12 - - 1, 105, 4, - 105 25.1 16.0; 27.1; - 27.1 0.1 7. 1, 1, 9.0; 19.5;

4, 2, 1, 21.0; 17.0; 1, 2, 1 21.0; 13.0;

18.0; 14.0; 19.0; 20.0

13 9 30 0.2; 135 8 6 1 2 (105), 105 104 24,8 (25.9); 23.0; 25.9 25.9 0.1 226, 1, 2, 19.0; 10.0;

1, 3, 1 16.0; 16.0; 18.9

14 9 30 0.2; 136 13 2 1 11 (130), 130 87 24.7 (25 .4); 9 .0; 25.4 25.9 0.1 4, 2 13.0

15 8 28 0.2; 145 25 5 1 20 (130), 130 43 22,3 (22.5); 8,5; 22.5 21.9 0.1 8, 1' 2, 14.0; 19.0;

3, 1 10.0; 16.0

16 8 28 0.2; 144 14 1 1 13 (139), 5 139 62 23,2 (23.4); 17,8 23.4 24.4 0.1

17 8 28 0.2; 144 17 3 1 14 (137), 4, 137 62 22.9 (23.4); 13.0; 23.4 24.0 0.1 1, 2 19.0; 13.0

18 10 18 0.2; 159 22 - 1 22 (159) 159 81 13.5 (13.5) 13.5 14.3 0.1

19 10 18 0.2; 153 14 - 1 0.1

14 (153) 153 103 14.4 (14.4) 14.4 14.8

(Nos. 6 and 8) arise in the following manner: we select a family which including the basic part of the population, and strongly displaced in quality, which in turn dictates the choice of this part of the population divided into isolated groups, which are.generally not displaced (the "invariant" form) (see Fig. 1, experiments 10, 11, 15, 16, 17). It can be noted that the non-evolution of a group arises if its genotypes are found at a very small distance from each other and often are simply equal.


TABLE 2

I I c c .s 0

11) "' ..:Z ·~ x x (.) .~ lii c ·s Cl! Cl!

. ! z ..... & z ·~ .....

8 8 'fS "' "' '§ "' "' 0 ii .... 0 .... ;>.. .j-J'.-l >

0 .s 0 c .... "Cl 11) Cl! c.-. §Z .;:l'" .s z 0 4-o

~ 19 ~ ·~ 4-o 0. ~·§ 0. 0 ..... 8 >, ."I:Ll 4-o "' .... z '§ 'ö 4-o ..... .... .E ~ "' ::s Cl! 0 ~~ (1.)'.-1 •.-j c:: 0 .... c c ...

~ 0 0 'QJ 6 §.S §I:Ll '§ .,!i <l) 0 <l)

iOl ~ <ll"' <l) <l) • <l)

6 ·~ t)~ "' ... ...o. ... Cl! .... ... "' ·~ .... ::s::s ::s .... x \.f-1~ 4-r (lj ::s .... ~~ 4-oi:Ll ·~Cl! 8 • ..... ..... .

4-< "' ..... .... 0 tl o'"

~X I~& .... ·~ 00. ........ 0 0 00. 0 u ... . 8 (.) c

ö '§ <l) ::s '" ..... o::S • ::s bO ::s 2 ::s • ::s O.p, ... 0 0 ao 0 ... .... oc 0 Cl! 0 0 oo >< 0 Cl! .... z +-'Q .... zB Zo. .... c z~ zbh i:LlP.. c p,.CO. z z zbh oo.~ 00 z.~ 00 0~

20 6 24 Oo2; Oo2; 147 4 2 1 2 (111), 40, 6 111 110 8o4 (2o5); 2200; 2o5 2.5 0.1; 12 1800

21 6 24 0.2; 0.2; 16111 4 1 7 (153), 2, 153 121 21.4 (22.5); 9.0; 4.0; 2205 2201 0.1; 12 4, 1, 1 1401; 10.0

22 6 24 0.2; 0.2; 152 14 4 1 10 ( 134)' 1' 134 120 4.3 (2 0 5); 12 0 0; 205 205 0.1; 12 8, 6, 3 17.0; 17.0; 9.0

23 6 24 0.2; 0.2; 0, 1; 12

151 7- 2 5;2 (131); (20) 131 126 19.9 (21.1); (3.1) 21.1 21.1

24 6 24 0.2; 0.2; 149 15 1 2 10j4 (98), (50), 98 71 9.5 (2 .8); (21.3); 2.8 205 0.1; 12 1 1400

25 9 30 0.2; 0.2; 130 7 7 - - 25, 57, 29, - 57 5o9 7,0;2.5;7.0; - 2.5 0 01; 12 17' 1, 2, 3 9.0; 27.0; 21,0;

7.0 26 9 30 0.2; 002; 131 8 7 1 2 (78), 34, 3, 78 50 11.3 (3.1); 26,0; 9.0; 3.1 4.0

0.1; 12 7' 7' 1, 1 25.0; 900; 8.0; 6.0

27 9 30 0.2;0.2; 131 7 7 - - 89, 20, 7, - 89 21.5 28.4; 6.1; 9.8; - 28.4 0.1; 12 1,1,1,12 13.0; 22.0;

14.0; 4.0

28 9 30 002;0.2; 132 3 3 -0.1; 12

- 19, 112, 1 - 112 25.7 4.1; 29.4; 8.0 - 29.4

29 9 30 0.2;0.2; 139 4 4 - - 66, 71, 1, - 71 14.4 26.9;3,0; 12.0; - 3.0 0.1; 12 1 900

30 8 28 002; 002; 157 14- 1 14 (157) 157 111 3.5 (3.5) 3.5 2.7 0.1; 12

31 8 28 002;002; 143 12 5 1 7 (125), 12, 2, 125 107 2208 (24.8); 608; 11.0; 24.8 24.8 Oo1; 12 1, 2, 1 7 .0; 17 .0; 13.0

32 8 28 0.2; 0.2; 152 8 4 1 4 (138), 11' 138 124 6o3 (4.2); 26.0; 4.2 4o3 0.1; 15 1, 1, 1 20.0; 10.0;

22.0

33 8 280.2; Oo2; 150 16 4 2 2/10 (13), (122), 122 80 8,6 (25.8); (5o9); 5.9 600 0.1; 12 1' 11' 2, 1 13.0; 12.0; 6.0;

7.0

34 8 280,2; 0,2; 146 4 4 - - 138, 4, 1, 13 - 138 208 2,0; 25,0; 24,0; - 2.0 Oo1; 12 700

35 8 280.3; 0.2; 166 12 4 1 8 (111), 50, 111 94 17.3 (25.4); 1.0; 25.4 25.4 002; 15 1, 3, 1 7 09; 800; 7.0

36 8 28 Oo3; 002; 216 5 3 1 2 (189), 25, 189 188 23.1 (25.9); 2.0; 25,9 26.0 002; 15 1' 1 13.0; 18.0

37 828 0,2; Oo2; 152 2 2 - - 150, 2 - 150 300 2.7; 2405 - 2.7 0.1: 15

38 8 28 Oo2; 002; 145 4 4 - - 130, 12, 1, - 130 2.1 LO; 10.0; 1900; - 1,0 Oo1; 15 2 22.0

3910 18002; 0.2; 153 8- 2 2;6 (90), (63) 90 89 10,8 (15.8); (2.9) 15.8 1508 0.1; 9

I

SELECTION IN MODELLING AN EVOLUTIONARY PROCESS

<!) ü

] ü

..... 0

TABLE 3

<!) .... ~ "'"' ... ·~ ü c:: :::> :::> 1::1c [/).~

.5 .5

40 6 240.2; 0.1; 168 15 6 32/4/3 (18), (65), 12 (17), 39, 11,

65 5811.9 (10,9); (11.4); 11.411.4 (11,1); 12.5;

10, 2, 4, 2

41 6 24 0.2; 0.1; 149 13 11 1 2 (84), 12, 3, 84 12 7, 17, 19,

1' 1' 2, 1, 1' 1

42 6 240.2; 0.1; 158 9 5 1 4 (76), 3, 46, 76 12 29, 3, 1

43 6 240.2; 0.1; 151 27 51 22 (132), 11, 1,132 12 2, 1' 4

44 6 240.2; 0.1; 118 8 6 1 2 (67). 53, 11, 67 12 7,15,4,1

45 9 300.2; 0.1; 153 10 8 1 2 (30)' 28, 56, 30 15 4, 13, 2,

18, 1' 1

46 9 30 0.2; 0.1; 140 13 6 1 7 (100),20, 2, 100 15 2,11,2,3

47 9300.2; 0.1; 130 1515- - 25, 19, 39, -15 1, 2, 7' 1,

4, 23, 1, 1, 1' 4, 1, 1

15,0; 12. 0; 16.0; 13.0; 7,0

81 12.0 (11.5); 10.0; 11,411.5 12.0; 11, 7; 11.7; 12.9; 18,0; 10.0; 14.0; 17.0; 14.0;

8.0

46 11.9 (11.3); 16.0; 11.312.4 12.4; 11.1; 12.0;

11.0

38 11.7 (11.5); 14,0; 11.512.1 12.0; 14,0; 14.0;

11.0

65 11.7 (12.0); 11.7; 12.011.9 12.0; 13.1; 10.8;

9.0; 13.0

56 15.0 (14.7); 16.7; 15.0; 14.7 15.0 13.0; 16.0, 16.0; 13,1; 12.0; 7.0

84 14.0 (14.1); 13.0; 14.114.1 15.0; 13, 0; 15,0;

12.0; 16,0

39 15.316.0; 15.1; 16.0; - 16.0 22.0; 11.0; 12.0; 9.0; 12.0; 15.0;

12.0; 19.0; 16.0; 17.0; 14.0; 13.0

48 9300.2; 0.1; 137 1010- -15

27, 21, 38, 7' 35, 1, 2, 4, 1, 1

- 38 15.0 15.0; 14.0; 15.0; - 15.0 16.0; 15.9; 9.0; 15. 0; 15. 0; 12.0;

11.0

49 9300.2; 0.1; 13010 81 2 (64), 31, 19, 64 6314.8 (13.8); 15.1; 13.813.9 15 9, 2, 1' 1' 15.0; 16.0; 16.0;

1,2 16.0;17,0;22,0; 17.5

50 8280.2; 0.1; 153 22 51 17 (105), 38, 4, 105 3818.3 (18.7): 18.5; 18.718.5 12 4, 1, 1 24.0; 11 ,3; 13.0;

15.0 51 8280.2; 0.1; 161 13 6 1 7 (113), 37, 1,113 82 14.5 (14,1); 15,0; 14.114.1

15 1, 3, 5, 1 17.0; 8.0; 21,0; 13.0; 10.0

52 8280.2; 0.1; 147 1410 1 4 (64), 33, 20, 64 51 15.5 (16,3); 16,1; 16.316.3 15 9, 1, 3, 10, 13.5; 11.5; 25.0;

53 8 28 0.2; 0.1; 170 10 5 2 3/2 15

54 828 0.2; 0.1; 152 9 2 3 3/3 15

5510 18 0.2; 0.1; 163 26- 1 26 9

56101 8 0,2; 0.1; 155 33- 1 33 9

2, 3, 1, 1 18.7; 16.0; 8.0; 14.0; 10.0; 16.0

(48), (15), 48 102 14,9 (14.5); (15.2); 102, 1, 2, 15.0; 8.0; 18.0;

1' 1 10.0; 25.0

(48), (93), 93 88 14,8 (14.4); (14.7); 8, 2, 1 18.0; 18.0; 11.0

(163) 163 67 9.0 (9.0)

(155) 155 35 8. 7 (8.7)

14.5 15.0

14.7 14.7

9.0 8.5

8.7 9.0

267


40

32

24

72 I I I

64 I I I I I

56 I I I I

48 I

40

32

Fig. 3

!ZJ

Fig. 4

" /I I I • I

e

fJ

SELECTION IN MODELLING AN EVOLUTIONARY PROCESS

tle

58

48 I

l h

40 II II II II II

.32 II II II I I I I

24 1 I I I I I I I I I

!ff I I I

t5

I I I I I I I I I I I I I

\A I I.

I·. /"' I I · / \ t J : .~., i '· •. .. , /' '.: / ............ '·.... r.·. . ~ .' .· ·. :">.·~.· . .• :,1·- ~ '~

': :· ·· ..... ~· ··.: ·:t:.._,:::·.·.'·~ . .&..-...

o~~~~~~-~-.. L-~~~·~~/_·~~\~.~~·--~~~~~·--·~·~:~~~·--4 8 Q M M U U

40

' .32 '\ II I I

2ft I I I I I:.

f{j :.

e

Fig. 5

e

Fig. 6

269

In Fig. 1 the peak of the value e = 16 corresponds to the group of slow evolution, while the non-extreme peaks correspond to the values e = 11, 8-9, and 5 correspond to the nonevolutionary "invariant" forms.

Reducing the population (No. 9) for a one-sided choice into different cases without choice, we obtain a self-similar instability. In this way we distinguish a larger part (a group or a

270

56

32

24

!6

8

0 ft.

T. I. BULGAKOVA, 0. S. KULAGINA, AND A. A. LYAPUNOV

8 12 /6 ZO Zft. e

Fig. 7

family) which is strongly displaced towards the direction of the "ideal genotype," and moreover, which generates through a slow evolutionary process into a small non-evolutionary group (experiments 12, 13, 14).

For the two-sided choice in the various experiments, we obtain results of three types. In one case we obtain a situation reducing to the one-sided choice: the separating basic mass of the population is displaced to a genotype consisting only of units (Fig. 3, experiments 21, 23, 27, 28, 31, 36). In the other cases the basic mass of the population is displaced to a genotype consisting only of zeros (Fig. 4, experiments 22, 30, 32, 33, 34, 37, 38). In this case the basic part of the population is generally represented by one family, while sometimes (for the "reduction" of the population) it separates into roughly equal numbers of elements of isolated groups, with every displacement in one direction but with diff~rent orders (see Fig. 5, experiment 25).

Finally, we consider the case when the population divides into two approximately equal numbers of parts, which are displaced in different directions (see Fig. 6, experiments 20, 24, 26, 29, 35, 39).

We will generally speak of the greater part of the population. In the majority of cases we may speak of the basic isolated mass of the group, which is displaced into one or another small group or is not displaced at all (an "invariant" form).

It may be noted that a faster disintegration of the population into parts than noted, in which its parts undergo evolution, is dictated by the method choice.

For the central choice of the population we have a displacement dictated by the choice, i.e., Eav becomes approximately equal to (n- r 0). From the graphs (Fig. 7) it is possible to note one peak, obtained at the point (n- r 0). However, this does not mean that the population combines to form one group, since it may instead evolve into various different genotypes, consisting of (n- ro) units and r 0 zeros. In the general case the basic part of the population is divided into approximately equal numbers of isolated groups, divided into sufficiently remote pairs (experiments 40-54). Exceptions are the "contractible" populations, which in this case give one family (and do not give isolated groups) (experiments 55, 56).

In every figure we can see that the first and second graphs (i.e., the graphs of the initial population and the population after 960 units of time), as well as the second and third graphs, differ pairwise much more than the third and fourth graphs, although each graph is formed 960 unit time steps after the previous one. This means that the drift of the population slows down with increasing time. Clearly, the larger the variety in the genotypes, the faster will be the evolution of the population.


§ 3. General Conclusions

On the basis of the experiments performed, we have arrived at the following general conclusions.

1. In the problern on the divergence and evolution of biological organisms, an evolutionary process is obtained of such a character that some small genotype descendants will not be given, while large descendants have a behavior indicative of Markov processes, i.e., in essence the fluctuation of genotypes of the population. Thus it is not at all necessary to differentiate between special geographically or biologically isolated mechanisms in order to perform a natural taxonometric study of new organisms. It is highly probable that because of this the general rhythm of evolutionary processes is accelerated, and that it is necessary to consider the theme of this evolution.

2. Repeated experiments with the same initial population and probability characterization of the process of selection and with only different random numbers (chosen from one and the same distribution) Ieads to completely different realizations of the process of evolution up to that which in the case of the two-sided choice for one realization results in a drift of genotypes of the population to one side; for two realizations, intotwo sides; and for three, two genotypically isolated pairwise populations, from which one genotypically drifts into another, and pair into pair. This shows to what extent a random element may affect the qualitative characterization of the course of a random process. Hence, it is clear that it is inappropriate, in order to explain the qualitative and macroscopic details of the evolutionary process, to look for special determining properties of an essential nature. On the contrary, the representation of very important features of this process requires the knowledge of the character of the four kinetic evolutionaryprocesses, whichcan be determined by means of random properties as well as the properties of this prucess.

3. We call attention to a common result of our experiments, in which for an unknown structure the choice of the character of the genotype dictates the evolution of the population with the passage of time.

Thus the population No. 8 (see Fig. 4) goes through a process of the following form. If we observe its changes after 960 time steps, then it seems that it divides into two parts, which will be evol ving in different forms. But then the character of the drift is known, and the part of the population which is close to the genotype consisting of one unit is almost invariant (see also p. 270 on the retardation of the drift). This indicates the great role of the fluctuations in the progress of evolution of the process as well as the fact that we must deal with great care with the history of known genotypes of a population under known conditions of choice.

4. In the present experiments it was assumed that the establishment of the invariant forms results in an evolutionary impasse when the population is quickly evol ving under known sufficiently abruptly changing conditions of choice. This collection of invariant forms turns out tobe highly genetically homogeneous. The case of strong fluctuations and the evolution of invariant forms, if observed for various types of choices, is essentially seen to be of a peculiar character and is caused by only exceptionally posed experiments. Great interest has arisen in connection with the extent to which the appearance of spreading is natural, and the genotype structure of the resulting groups.

5. Our results have attracted the attention of investigators towards aspects of the important problems of the mathematical theory of evolution of populations in the following form. Thus special interest has arisen in the cybernetic zones in time, which are characterized in the following way. The fate of the genotype population in a small interval of time is entirely in


agreement with the description of elementary probability-combinatorical methods. Similarly, over long time intervals it is weil described by asymptotic methods. However, there exists an extension of the time interval in which neither combinatorical nor asymptotic methods are applicable to the problem. In this case it is necessary to combine analytical methods with sufficiently accurate estimates for the experiment. This zone of time has been referred to as the cybernetic zone by Yu. I. Zhuravleb,

Literature Cited

1. T. I. Bulgakova~ 0. S. Kulagina, and A. A. Lyapunov, "The Question of Modeling the Evolutionary Process with Selection Taken into Account. I., Systems Theory Research, Vol. 20» Consultants Bureau, New York (1971), p. 22.

2. 0. S. Kulagina and A. A. Lyapunov, non modeling the evolutionary process,n Problemy Kibernetiki, Vol. 16, Nauka, Moscow (1966).

ON THE DYNAMICS AND CONTROL OF THE AGE STRUCTURE OF A POPULA TIONt

L. R. Ginzburg

Leningrad

The problern of time dynamics of the size of natural populations occupies one of the most important places in biologicalliterature [1-9, 11]. Many investigators isolate varied mechanisms for regulating the size of natural populations, which operate at both the interpopulation and intrapopulation levels. For many reasons, some of which will become clear below, the study of the dynamics of the overall nurober of a population without allowance for its age structure is hardly satisfactory. Besides being of purely ecological interest, problems of the dynamics of the age composition of a population have practical significance in connection with a nurober of applied problems of controlling the dynamics of natural and artificial populations.

In 1hefi.rst section of the present paper the mathematical model of the dynamics of the age structure of a population in a stationary medium is considered, and one of the possible hypotheses explaining the fluctuations in population size with time is discussed. In the second section a generalization of this model for the case of nonstationary external conditions is considered. In the third section the constructed model is used as the basis for stating certain control problems.

Dynamics of the Population Size in a Stationary Medium

Various approaches exist which explain the periodic character of the variation of population size. The best known and most widely used mathematical concept is the Lotka-Volterra concept [10-13] whose meaning is that the fluctuation of population size in time is caused by predator -prey relations. The equations

a:1 = a 1N1 - ß1N1N2, }

d~2 =- a2N2 + ß2NIN2 (1)

serve as the foundation of the mathematical model ofthis theory; here N1 is the size ofthe prey population; N2 is the size of the predator population; a 1, a 2 are coefficients which characterize the intrinsic birth and death rates of both populations; ß1, ß2 are coefficients which describe the interpopulational interaction.

t Original article submitted June 4, 1968.

273

274 L. R. GINZBURG

This model has undergone various kinds of criticism and refinement [14, 15, 16]. In particular, A. N. Kolmogorov noted the low accuracy of the system (1) in view of the fact that the derivatives dN/dt and dNJdt have meaning only for fairly long time intervals.

Moreover, the experimental material obtained up to the present by biologists allows the statementtobe made that factors operating at an intrapopulationallevel have an important significance in the dynamics of population size [1-5, 8, 9].

Wehavemade an attempt at constructing a mathematical model which describes regular fluctuations of population size on the basis of a nonunüorm age distribution of the individuals in the population (this holds in the overwhelming majority of natural populations). This fact makes necessary a study of the dynamics of populations with allowance for the age composition. As will be shown below, consideration of the age composition of a population improves the accuracy of the model and reveals qualitatively new effects connected with an explanation of the fluctuating processes in the dynamics of population growth.

Let us go over to a construction of the mathematical model for the growth of a certain population with allowance for the age composition. Related problems were considered in [17, 18, 20, 21] in connection with the growth of human population and the age composition of the growing population.

In order to simplüy the mathematical reasoning we shall consider the growth of a unisex population in a stationary medium. All of what follows will be generalized for the case of a bisexual population in a nonstationary medium. In certain places those mathematical proofs that are not essential to an understanding of the biological meaning of the constructions of the model will be omitted.

Weshall call the function of two variables u(x, t) the age density (or simply the den s i t y) of the nurober of indi viduals of a gi ven age at time t if for any ages a and b the nurober of individuals having an age from the interval [a, b] is expressedas

b

ju(x,t)dx. a

Obviously, the function u(x, t), which is defined by the equation

Ü (x, t) = oo u (x, t)

.\ u (x, t) dx 0

may be treated as the density of the probability that an individual chosen at random has an age not exceeding x at time t.

Let us assume (again only for simplicity) that the population is totally isolated (i.e., there is no exchange of migrants with other populations of the same type).

The basic system of equations which is satisfied by the function u(x, t) has the form

au ' au d --t--=- (x)u at ax ' 00

u(O, t)= J b(x)udx, 0

u(x, O)=g(x).

(2)

DYNAMICS AND CONTROL OF POPULATION AGE 5TRUCTURE 275

Here d(x) is the mortality rate of individuals of age x; b(x) is the birth rate for parents of age x; g (x) is the initial numerical distribution by age.

The first equation of the system (2) describes the population loss at various ages as a result of natural mortality and as a result of natural aging of individuals with the passing of time. The second equation expresses the number of newborn at time t, summed over all ages of the parents. The third equation of the system (2) constitutes the initial condition. In deriving the equations the assumption of a linear dependence of birth and mortality on numerical size was used. This hypothesis corresponds to a situation in which there is no noticeable effect of limiting factors such as, for example, shortage of food and restricted living space. The fact that the coefficients b (x) and d (x) are independent of time is what designates the medium as stationary.

Let us now investigate the system of equations (2). The general solution of the first equation has the form

X

-I a<slds u(x, t)=Q(t-x)e o

where Q is an arbitrary function.

Having substituted (3) into the second and third equations of the system (2), we obtain

Let us introduce the notation

X

"" -I d<slds Q(t)= i b(x)e 0 Q(t-x)dx,

0

X

~ a<sld\i Q (- x) = g (x) e o

X l -I d(s)ds K (x) = b (x) e

1 b ,

~ d<~lds cp(t)=g(t)e 0 •

Equations (4) and (5) take the following form for the conditions (6):

Q(t)= I K(x)Q(t-x)dx,}

Q(-t)=cp(t).

(3)

(4)

(5)

(6)

(7)

We shall seek the solution of Eq. (7) in the form Q (t) = ezt; then we obtain the basic equation for z:

00

F(z)= J K(x)e_zxdx-1=0. 0

(8)

276 L. R. GINZBURG

It may be shown that the function F (z) is an integer function of the complex variable z having an order of growth which is higher than that of apower law. Consequently, it has an infinite countable set of zeros on the complex plane [19 ], and therefore one may attempt to find the solution of Eq. (7) in the form

00

Q(t)=~Ci/i1 , i=1 (9)

where Zi are the roots of the function F, and Ci are constant numbers. Let us rewrite Eq. (7) in the form

t 00

Q (t) = 1 K (x) Q (t-x) dx +) K (x) Q (t-x) dx. u t

In the second integral the argument of the function n is nonpositive, and therefore from the second condition (7) we obtain

00 00

J K(x)Q(t-x)dx= J K(x)rp(x-t)dx=f(t). t t

Now we apply the Laplace transform to Eq. (10) with allowance for (+); then,

Q (z) = /(z) • 1-K(z)

(10)

(+)

Here n, f, and K denote the Laplace transforms of the functions n, f, and K. According to the expansion theorem from oparational calculus, the original of the function fi is determined according to the equation

00

Q (t) = ~ res Q (z) /i1, i=1 zi

where res Q are the residues of the complex-variable function n of the corresponding poles, zi

while zi are the roots of the equation

K (z) = 1.

The latter, as can easily be seen, coincides with Eq. (8).

Thus, the coefficients Ci may be calculated explicitly from the stipulated functions cp (t) and K(x) as residues of the function of the complex variable (3) at the corresponding poles.

Let us note certain properties of the solution obtained. First of all let us note that among z i there is one and only one real number. Actually, by virtue of the positiveness of the kerne! K (x) the function F is monotonic on the real axis and takes all possible real values from -1 to oo • Consequently, it has one and only one real root.

Moreover, it can easily be shown that for all the remaining complex roots of the function F (z) the condition

(11)

is fulfilled, where A. is the sole real root of Eq. (8). Actually, from (8) it follows that

DYNAMICS AND CONTROL OF POPULATION AGE STRUCTURE 277

00 00

1 =I J K (x) e -zix dx I.:::;;: J K (x) e-(Re zi)x dx. (12) 0 0

By virtue of the monotonicity of F on the real axis, Re zi::::; i\., i.e., the statement (11) has been proved.

Thus, in Eq. (9) one can isolate the principal term eA.t, which has the principal effect on the dynamics of population size for larget t:

00

Q (t) = Cel.t + ~ C1/i1• (13) i=2

Allterms of the sum in Eq. (13) describe the time fluctuation of population size. The frequencies of the fluctuations are the imaginary parts of the roots zi of the function F(z). Thus, the solutions of the system of equations (2) may be oscillatory in character, which is particularly noticeable for small t. In the case i\ == 0, which corresponds to equality of the average birth and mortality rates, undamped fluctuations of the population size u(x, t) with time may be observed.

Let us consider the problern of the frequencies of the size fluctuations of a natural population with time. It is clear that high frequencies are not realized in practice, and therefore it is of interest to attempt to estimate the lowest frequencies corresponding to the most noticeable fluctuations having the maximum period. Of course, for a stipulated function K(x) the frequencies may be calculated numerically with any degree of accuracy, but we shall attempt to estimate them,having made the simplest assumptions concerning the function K(x).

a) Assurne that the reproductive age is concentrated in the interval [A, B], i.e.,

then Eq. (8) takes the form

where

K (x)=Ü, {x: x<A, x> B};

B

J K (x) e-ax cos wxdx = 1, A

B

J K (x) e-ax sin wx dx = 0, A

a = Re z, w = Im z.

(14)

(15)

(16)

From the second equation of the system (15) it follows that sin wx changes sign on the interval [A, B]; therefore, for the maximum period T of the fluctuations one may obtain the estimate

T< 2B; (17)

i.e., the maximum period of the fluctuations of population size is a quantity of the order of the lifetime of one generation.

b) Assurne that the reproductive age is concentrated at one point M (i.e., the birth rate has a o-shaped form). This case may provide a good description ofthe situation which holds for several species of fish for which total mortality of the parent individuals occurs immediately after spawning. Moreover~ let us assume that A. == 0; therefore,

278 L. R. GINZBURG

00

j K0ö(x-M)dx=L 0

Equation (18) has the following form for K(x) = K0o(x- M):

e-zJI = 1.

(18)

(19)

Taldng account of (18) and making use of the expansion of the difference of the exponent into an infinite product, we obtain

1 00

- Z Mz I1 [1 M2z. 2 J =' O ze + 4k2;t2 • h=1

In this case the entire spectrum of frequencies can be determined from the exact formula

2kn (t)k = ---xr (k= 1, 2, ... ),

while the possible periods can be determined respectively from the equation

M Tk=T·

Thus, in this case the most noticeable will be fluctuations having the period T 1 = M. In this particular case the final solution of the problern has an especially simple form:

X

oo - S a<~Jds u(x,t)=[~ Ckcos 2:: (t-x)]e o ,

h=1

where Ck are the coefficients of the expansion of the function cp (t) into a Fourier series in cosines.

Let us attempt to compare the fluctuation frequencies obtained approximately from the model constructed above with the actual fluctuation frequencies cited in the literature.

(20)

(21)

(22)

(23)

Let us consider, for example, the fluctuations of the population size of foxes in Canada on the Labrador Peninsula. According to the data given by Elton [6] the average period of population-size fluctuations has been approximately four years during the past 100 years. In order of magnitude this approximately corresponds to the period of fluctuations obtained theoretically [see (17)]. The fluctuation frequencies obtained in the preceding section likewise correspond to the data on the population fluctuation of field voles and Iemmings [3-6]. For these species the fluctuation period is equal to three tofour years. The population-size fluctuations of the salmon family with a period equal to the reproduction age (M) are weil known. The accuracy of the Observations is so low that there is no need to carry out exact calculations. The order of the compared quantities is evidence of the fact that the model given may be used for a rough prediction of the size of natural populations. Of course, the actual fluctuations are considerably more complex than those obtained from the model. This is natural, since the basic hypothesis on which the model is founded is stationarity of the medium, and this hypothesis is naturally violated under actual conditions.

In conclusion let us again compare the model advanced for the dynamics of population size in the present paper with the model developed by V. Volterra. However paradoxical this might be, it is nevertheless true that in certain cases the objections to a model of the predator-prey


type lies in the fact that it is too general; i.e., the fluctuation frequencies which derive from it may be the most varied. Actually, however, an amazing similarity of the frequencies of the observed fluctuations [3-6] can be detected for one and the same species under different conditions. What has been said in no way denigrates the role of the interaction model of the predatorprey type in the explanation of many ecological facts. However, the thought comes to mind that practical oscillations of population sizes may in many cases be explained by simpler notions (for example, those which are presented in the present paper).

The Dynamics of Population Size under Nonstationary

External Conditions

In order to introduce nonstationarity of the external conditions into the model considered, it is sufficient to suppose that the birth and death rates depend on time.

The system of equations describing the dynamics of the population size will be of the form

~-l-~--Dt 1 Dx -- d (x, t) u,

00

u(O, t)=) b(x, t)udx. 0

The general solution of Eq. (24) has the following form:

u (x, t) = Q (t-x) e-D(x, tJ,

where n is an arbitrary function, while

X

D(x, t)= \ d(~, t-x+~)d~. .! 0

(24)

(25)

(26)

(27)

Having substituted the general solution (26) into Eq. (25) along with the initial condition (2), we obtain

00

Q (t) =) b (x, t) e--D(x, t)Q (t-x) dx, 0

Q( -x) = g (x) eD(x, 0).

(28)

(29)

Let us note that since the reproductive age is positive, the integration in Eq. (28) may be assumed to extend from a certain A > 0 to oo, rather than from zero. Moreover, let us introduce the notation:

K (~, t) = b (x, t) e-D(x, t), }

<p (t) = g (t) eDU, 0 ). (30)

Inthenewnotationthe integral equation (28) and the initial condition (29) take th.e foiiowing form:

00

Q(t)= J K(x, t)Q(t-x)dx, (31) 0

Q(-t)=~(t). (32)

280 L. R. GINZBURG

The function n is known to us for negative values of the argument. It is necessary to recover it from Eq. (31) for positive values of the argument. Since x varies from A to oo in the integration in (31), it follows that the argument of the function n varies within the limits

- 00 < t- X< t- A.

If one considers the functions n on the interval 0 ~ t ~ A, then the argument n (t- x) is negative, and therefore the right side of Eq. (31) is known to us. Thus, for 0 ~ t ~ A we have

00

Q(t)= \ K(x, t)~(x-t)dx. _4

Now the function n is known to us on the interval -oo ~ t ~ A. Having repeated the same procedure for the interval A ~ t ~ 2A, we obtain the function n for -oo ~ t ~ 2A. Continuing this process ad infinitum, we find the function n for all values of the argument.

Let us determine the sequence of functions w i (t) by means of the recurrence formula

Wi+dt) = I K (x, t) wi(t- x) dx, ] --1

w0 (t) = ~ (t) for t < 0;

then the solution of Eq. (31) for the initial condition (32) will have the following form:

[ "'(t) for -oo<.t<.O,

(!)1 (t) for O<.t<.A, Q (t) = ... . ........ ,, (33)

(!)i (t) for ( i - 1) A < t < iA, ••••• 0 ••• 0

The solution of Eq. (31) may be carried out by another method also. Let us partition the integration interval in Eq. (31) into two intervals from 0 to t and from t to oo • Then by virtue of the initial condition (32) we obtain

t 00

Q(t)= J K(x, t)Q(t-x)dx+ J K(x, t)qJ(x-t)dx. 0 t

This equation may be solved by the method of successive approximations.

The following fact may likewise be established using the method of successive approximations.

Assurne that the nonstationary birth and mortality rates for all t are confined between certain stationary values, i.e.,

bdx) <.b (x, t)<.bz (x), }

d1 (x) > d (x, t) > dz (x).

Then the corresponding solutions of the three systems of equations having the coefficients b1(x), d1(x); b(x, t), d(x, t); b2(x), d2(x) satisfy the inequality

(34)

(35)


Making use of the knowledge of the analytic solution of the stationary problem, one can indicate the boundaries within which the solution of the nonstationary problern lies. The asymptotic estimate

X X

-A!X- S d1<slds - ~ d2(slds

C1e"I1e 0 <;u(x, t)<;C2eJc2te-'A2xe o

has an especially simple form. This estimate is especially useful in view of the fact that in many cases the exact values of the coefficients are unknown. In this case the possibility presents itself of giving a bilateral estimate of the solution.

The final exact solution of the problem is given by Eq. (26), where it is determined by Eq. (33) or Eq. (34).

Let us now consider the case of abisexual population with allowance for separate dynamics of the population sizes of the sexes. By analogy, the dynamics equations will have the following form:

ou ' ou d Tt -t- ax = - u (x, t) u,

av av iit+iii= -dv(x, t)u,

00

u(O, t)= J bu(x, t)udx, 0

00

u (0, t) = J bv (x, t) u dx, 0

u (x, 0) = gu (x),

U (x, 0) = gv (x).

Here u(x, t) is the population density of females; v(x, t) is the population density of males; du, dy, bu, bv are respectively mortality and birth rates of females and males.

Let us introduce the notation:

In this notation we obtain

X

Du(x, t)= ~du(~, t-x+~)d~, ö

"' Dv(x, t)=~ dv(~, t-x+~)d~,

0

Ku (x, t) = bu (x, t) e-D,.(x, t),

Kv (x, t) =bv (x, t) e-Du(''• t),

fjlu (t) =:= gu (t) eDu(t • 0)'

fJJv (t) = gv (t) eDv(t. 0).

u (x, t) = Qu (t- x) e-Du(x, t), }

u (x, t) = Qv (t- x) e-Dv(x, t),

where nu and nv can be determined from the equations

(36)

(37)

(38)

282 L. R. GINZBURG

00

Qu(t)= i Ku(x, t)Q(t-x)dx, 0

00

Qv (t) = J Kv (x, t) Qu (t-x) dx, 0

Qu ( -t) =(jlu (t),

Qv (- t) = Cjlv (t).

(39)

(40)

(41)

(42)

It is evident that Eq. (39), which describes the dynamics ofthefemale population size, can be isolated for the condition (41). This equation coincides completely with the corresponding equation for the unisex case. And this should be expected, since the dynamics o f a bisexual population is determined in the final analysis by the dynamics of the female population size.

Equation (39) is solved for condition (41) by the method described above, while the function nv is found simply by substituting nu into Eq. (40).

Finally, the dynamics of the populations of the sexes in a bisexual population can be described by Eqs. (38), where the functions nu and nv are determined from the rule described above.

On Controlling the Dynamics of the Sizes of Natural

and Artifi ci al Populations

In the previous sections it was shown how, knowing the birth rate b(x, t), the mortality rate d(x, t), and the initial age distribution g(x) of the population size, we may predict the population size as a function of time with allowance for the age composition. It is of interest to state the problern of controlling population size for a certain purpose. In the case of combating the populations of various species of pets such a purpose may be to maintain the population size at a certain fixed level, while in the case of the exploitation of a school of fish (a natural population) or a herd of large horned cattle (an artifical population) the purpose of the control may be to optimize a certain economic criterion.

The determination of the required characteristics of the population, such as birth rates, mortality rates, etc. (we are speaking, of course, of natural populations), causes great difficulties in solving problems of this kind. The problern ofthe means of control available to U3 is likewise important. Thus, the age composition of caught fish is regulated by the size of the net meshes, but the relationship between age and fish size is, generally speaking, statistical. This requires consideration of additional information associated with the corresponding statistical characteristics.

Frequently (as, for example, in the case of exploitation of a herd of large horned cattle) we have the possibility of externally adding a certain quantity of individuals to the population. This provides additional possibilities for control. In this case the system of equations takes on a somewhat different form:

fJu fJu ae+ax-= -d(x, t)u+w(x, t),

00

u(O, t)= J b(x, t)udx+w(O, t), 0

which, in general, does not increase the difficulty of solution. The function w describes the rate of artüicial influx or removal of individuals having the age x from the population at time t.


However, we shall for the time being neglect this possibility and consider problems of controlling isolated populations [w(x, t) = 0]. We shall assume that control constitutes an additional mortality rate J.l.(x, t) (the catching of fish, the slaughtering of cattle) in such a way that the mortality equation with control takes the form

iJu au m+ax= -(d(x, t)+f!(X, t)ju.

First of all let us consider the problern of determining the necessary ecological characteristics (i.e., using control-theory terminology, let us consider the problern of observability and identification). For this purpose let us recall the integral equation for the function n (t) of the number of newborn at timet (for simplicity we shall deal with the stationary characteristics b(x) and d(x)):

t 00

Q(t)= J K(x)Q(t-x)dx+ J K(x)ffJ(x-t)dx. 0 t

If we can measure the number of newborn in the population over a generation (i.e., for a change in t from 0 to S), then we obtain the integral equation

00

J K(x)ffJ(x-t)dx=l(t), t

from which, knowing K(x), one can determine the initial age composition cp (x) of the . .population, or, knowing cp(x), one can determine the function K(x) (i.e., a certa1n relationship between the birth and mortality rates). Then, knowing the birth rate b(x), which is usually known more accurately than the mortality rate, one may determine the mortality rate d(x) from the formula

d b (x) d (x) = Tx ln K (x) •

Actually, the initial distribution cp (x) may be obtained by observing not only the function n (t) but also the dynamics of the population size of any fixed age, and even the dynamics of the overall population size. The latter is especially interesting, since the overall population size

00

N (t) = j u (x, t) dx is the parameter that probably is most accessible to observation. 0

The second problem, naturally, is the problern of the attainability of a certain age structure G(x) by means of our control J.l.(x, t) (i.e., using control-theory terminology, the problern of controllability).

In many cases of practical interest we are actually concerned with the problern of the attainability of a certain optimal stationary age structure G(x), but for stationarity of a certain distribution it is required that the condition

00

G (0) = j b (x) G (x) dx 0

be fulfilled, or, recalling that in the stationary case X

- r[d(sl+!!(s)]d~

G (x) = G (0) e 0 (43)

284 L. R. GINZBURG

we obtain the condition

X

oo -\ ~-t<slds i K (x) e b. dx = 1 0

(44)

for the control. This equation is a constraint on the control J.l. (x) in the case of a stationary policy. We may state that all distributions G(x) representable in the form (43), where J.l.(x) satisfies the condition (44), are stationary and attainable.

With this we shall end our brief discussion of controllability, observability, and identification problems, which of course require special consideration in view of their importance to practical problems.

As an example let us consider the problern of optimizing the age structure of a herd of large horned cattle. The exploitation of such a herd is connected with obtaining two basic forms of production -milk and meat, whose intensified production produces an obvious contradiction between them. Therefore, the age structure of the herd should be optimized on the basis of a certain resultant criterion. Such a criterion may be, for example, the income obtained by the farm from the exploitation of the herd. Let us begin by dwelling on the problern of the optimal stationary structure of the herd. In this case the equations take the simple form:

du dx = -[du (x) + f..tu (x)] u, dv dx = - [dv (x) + f..tu (x)),

00

u (0) = i bu (x) u dx, (45) 0

00

V (0) = J bv (x) u (dx). 0

The income from the exploitation of the herd is made up of the income obtained from the sale of milk and meat less the expenditures required to maintain a herd of a stipulated size.

We shall not enter into detail in the present paper (i.e., weshall not write out the dependences of the milk productivity of the cattle and of the slaughtering rate on age, etc.), but it is completely clear that the optimality criterion will be a linear functional of the distributions u(x) and v(x), which may be treated as the independentvariables instead of J.l.u(x) and J.l.v(x). The constraints are stipulated by Eqs. (45) and the limitations on feed which are stipulated by inequalities of the form:

00

J Lu (x) u (x) dx < Luo; 0

00

J Lv (x) V (x) dx-< Lv0,

0

where L (x) is the amount of feed required for one individual of age x, and L0 is the overall amount of feed of the given kind.

(46)

Thus, after quantization the problern of optimizing the age structure of a stationary herd reduces to a conventional problern in linear programming. The initial experience in solving this kind of problern for one of the state farms of the Leningrad Region shows that the income may be increased by 5 to 7% compared with the actually existing income by realizing the optimal age structure.


The problern of the optimal process of transition from a stipulated nonoptimal age structure to the optimal stationary age structure is more complex. This problern can be sol ved numerically by one of the existing methods.

The problern of optimizing the age structure of a herd for planned growth of its size is of great practical significance. First of all, having calculated the growth index i\. 0 for P.u = 0 as the root of the equation

00 1 K (x) e-l.x dx = 1, 0

we may establish the upper bound of the population growth rate in the form

N (t) <Noel.ot.

For optimization of some economic criterion one may pose the additional condition

which will guarantee the required planned growth of the herd size.

It is not difficult to see that all of the problems indicated may be related with equal success to the dynamics of a school of fish or other natural populations.

As a second example of the application of the mathematical model considered, let us dwell on the problern of planning the intensity with which agricultural pests are combatted. In this case one can state the problern of maintaining the population size at a certain stationary level. In the general statement of the problern the population size N (t) can be expressed by the equation

X

oo - ~ ll <s. t -x+s> ds

N (t) = 1 Q (t-x) e-D(x, ne 0 dx. 0

Our problern is to choose the function P.(x, t) in such a way that N(t) = N0, the function Q (t) likewise being a complex functional of p,(x, t)- the solution of the corresponding integral equation with a kernel that depends on p,(x, t). Under these conditions some economic criterion may be optimized. The basic difficulty in solving the problern stated lies in the absence or inaccuracy of information on the natural birth and mortality rates for different ages.

Thus, in the present paper we have considered the mathematical model of the dynamics of the age composition of unisex and bisexual populations in stationary and nonstationary media. Basedon the model constructed, problems of the control of population-size dynamics by choosing the mortality rate as a function of age and time have been discussed.

Literature Cited

1. S. P. Naumov, "General regularities of governing the population size of a species and its dynamics, n in: Investigation of the Causes and Regularities of the Dynami es of the Population Size of the White Rabbitin Yakutia, Izd. AN SSSR (1960).

2. L. z. Kaidanov, "On the problern of the role of behavior as a factor in microevolution," in: Issledovaniya po Genetike, Vol. 3, Izd. LGU {1967).

3. T. V. Koshkina, "Population density and its significance in regulating the population size of the red field vole," Byull. MOIP, Otdel. Biol., Vol. 20, No. 1 (1965).

286 L. R. GIN ZBURG

4. T. V. Koshkina, "On periodic variations of the population size of field voles, 11 Byull. MOIP, otdel. Biol., Vol. 21, No. 3 (1966).

5. T. V, Koshkina, "Population control of rodents," Byull. MOIP, Vol. 22, No. 6 (1967). 6. C. S. Elton, Voles, Mice and Lemmings, Clarendon Press, Oxford (1942). 7. C. S. Elton and M. Nicholson, 11The ten-year cycle in numbers of lynx," J. Animal Ecol.,

Vol. 11 (1942). 8. V. C. Wynne-Edwards, Anima! Dispersion in Relation to Cosial Behavior, London (1962). 9. I. I. Christian, Endocrine Adaptive Mechanisms and the Physiological Regulation of Popula--

tion Growth, London (1963). 10. V. Volterra, Lecons sur la Theorie Mathematique de la Lutte Pour la Vie, Paris (1931). 11. R. N. Chapman, J. Anima! Ecol., London (1931). 12. U. D'Ancona, The Struggle for Existence, Leiden (1954). 13. A. Y. Lotka, Essays on Growth and Form, Clarendon Press, Oxford (1945). 14. I. A. Poletaev, "On the mathematical models of elementary processes and biogeocenoses,"

in: Problemy Kibernetiki, Vol. 16, Nauka, Moscow (1966). 15. T. I. Eman, "On certain mathematical models of biogeocenoses," in: Problemy Ki

bernetiki, Vol. 16, Nauka, Moscow (1966). 16. W. R. Utz and P. E. Waltman, 11 Periodicity and boundedness of solutions of the generalized

differential equation of growth," Bulletin of Mathematical Biophysics, Vol. 25 (1963). 17. R. A. Fisher, The Genetical Theory of Natural Selection, Clarendon Press, Oxford (1930). 18. P. A. P. Moran, The Statistical Processes of Evolutionary Theory, Clarendon Press,

Oxford (196 2). 19. B. Ya. Levin, The Distribution of the Roots of Integer Functions, Gostekhizdat (1956). 20. Boyarskii (ed.), A Demography Course, Moscow (1967). 21. T. Harris, Theory of Brauehing Random Processes, Mir (1966).

ONTHECONTROLOFCARDMCRHYTHMt

Yu. A. Vlasov and A. T. Kolotov

Novosibirsk

The importance of controlled variation of cardiac rhythm need hardly be stressed. A successful solution of this problern would make it possible to control a nurober of pathological conditions such as auricular and ventricular flutter, extrasystole, and high- or low-frequency rhythms.

Unfortunately, in spite of the fact that investigations in this direction continue for a relatively long time, practical solutions have been obtained in only a few mostsimple cases such as, for example, increasing the heart rate by means of independent cardiac Stimulation.

The present article is an attempt to analyze with the aid of a model the effectiveness of control intervention in reducing the rate of spontaneous heart contractions.

The problern of changing the cardiac rhythm can be approached from two fundamentally different directions. To the first belong methods of changing the spontaneous activity of the automatic cardiac nodes (by various pharmacological means or by acting directly on the nervous system). Subsequent evolution ofthe excitation process proceeds without further intervention. In contrast, the second approach presupposes active intervention into the excitation process. As tools of such intervention serve various electrical stimulation devices that are being intensively developed in recent times.

In our discussion we shall deal mainly with the second approach.

It is well known that any spontaneous or induced extra contraction of the heart (extrasystole) can be followed by a prolonged (compensating) pause as a result of the fact that the next pulse arriving from the rhythm source is blocked by refractive cells. The first attempt to use this mechanism for clinical reduction of the rate of heart contraction has been made as recently as in 1963-1964 [3]. Adecisiverole inthislagisplayedapparentlybythefactthatmost physicians associate extrasystole with a pathological condition, and this hindered the attempts to use extrasystole for slowing down the rhythm. Unfortunately, the method of paired cardiac stimulation especially developed for this purpose, inwhich regular application of a pair of stimuli to the heart produces extrasystole, does not provide reliable reduction of the rate of heart contraction. This circumstance forced experimenters to resort to various modifications of this method (such as varying the nurober and shape of the applied pulses, changing the locationof the stimulating electrodes, etc.). In the best case, the paired Stimulation method ensures an approximately twofold reduction of the rhythm rate.

tOriginal article submitted July 23, 1968.

287

288 YU. A. VLASOV AND A. T. KOLOTOV

At present, there is no comprehensive theory concerning the interaction among cellular elements of the heart which could serve as a basis for developing reliable methods of controlling the rate of heart contractions. In such a situation it is quite important to analyze systematically all the factors relating to this problem.

I. Problem Formulation

We shall base our discussion on the model described in [1].

Consider a connected net of cells T which has two poles: A, the net input, and B, the net output.

If to the input A we apply pulses so that the time interval ® between two conseutive pulses is Ionger than the refractivity period of the cells (® > n), the output pulses at B will be of the same periodicity. (If desirable, the pole A can be assumed to be capable of periodical selfexcitation.)

The discussed problern can now be formulated as follows: how can the rate of the output signals of the NetT be reduced without changing the rate of input signals? (It is assumed that the net structure is fixed and that only stimuli are allowed to act on the net.)

It should be noted that the pole B cannot be completely blocked, i.e., its excitation cannot be discontinued after a certain finite time interval.

In fact, otherwise the set of all cells of the net T would be divided into two disjoint subsets ID1 and m, where ~m is the set of all blocked cells and m is the set of all remaining cells. None of these sets is empty as the pole B belongs to the first set and the pole A to the second. In virtue of its connectivity the net T will necessarily contain a pair of adjoining cells ai and ai (directly connected one with the other) suchthat a; E m and ai E ID1. But in such a case the cell a i would be excited an infinite number of times. Consequently, after a certain finite time interval the cell ai will force the cell a i to fire as soon as the latter turns into a quiescent state, i.e., the cell ai cannot be blocked, which contradicts the condition of its choice.

Wehave answered the problern stated under the following assumption: for any section of the net one can find a stimulus suchthat its region of application coincides with the given section; in other words, every cell of the net can serve as its input. Let the sequence of stimuli applied to the net T be called t h e e x p e r i m e n t e ( T ) o n t h e n e t T •

It is clear that an arbitrary experiment e (T) can change the sequence of output signals.

II. Solution

We will show that the pause between two consecutive stimulations of the pole B is limited to a certain value independent of the experiment.

Let T* be an arbitrary connected subnet of the netT suchthat both the poles A and B are contained in it. For any experiment e(T) on the netT in which the maximumpause at the output of this net (i.e., the maximum interval between two consecutive stimulations of the pole B) is equal to er, we can indicate a certain experiment e * (T* ) on the net T* that gives a maximum pause er* 2: er. (It is assumed that the input and output of the netT* are the same poles A and B, and that A receives pulses with an initial periodicity ® .)

In fact, such an experiment e* (I'*) can be realized by retaining all stimuli that affect the subnet T* in the original experiment e (T) and replacing all stimuli from the direction T"-._ T* with T* equivalent stimuli. Hence, in particular, follows that any upper bound of attainable pauses at the output of the netT* is at the sametime the upper bound of the attainable pauses in T.

CONTROL OF CARDIAC RHYTHM 289

Let L = { a1, a2, ••• , az} , where a 1 = A and az = B, be the shortest chain of cells connecting both pol es, and Iet l be the length of this chain. Considering L as an autonomaus net, we shall apply to A pulses with a periodicity ®. Let the arbitrary instant t0 of stimulation of the pole B be taken as the origin, andlet t 0 = o. Any cell a (1 :::: i :::: l - 1) being at the instant t in an excited state will force the cell a i + 1 to fire at the next (t + 1) -st instant provided the cell ai + 1 has not been in a refractive phase at the instant t.

Thus if even a single cell a i (1 :::: i :::: l - 1) is excited in the interval [t1, t2], then at least one excitation of the cell ai + 1 will take place in the interval [t1 - (n- 1), t2 + 1], where n is the period of refractivity and 1 is the magnitude of the latent period of reaction.

Reasoning as above we arrive at the following conclusion: if the pole A (cell a 1) is excited only once in the time interval [t1, t2], then:

a2 will be excited at least once in the interval [t1 - (n - 1), t2 + 1],

a3 will be excited at least once in the interval [t1 - 2(n- 1), t2 + 2],

························ ·····················~ and, finally, the pole B (cell az) will be excited at least once in the time interval [t1 - (l- 1) (n -1), t2 + l -1].

It now remains to choose the appropriate values of t1 and t2• Since the pole A receives external pulses with a period ®, it must be excited at least once during any time interval of the form [t, t + ® + n- 1]. Thus, we take t2 = t1 + ® + n- 1. To find t1, note that if the interval [t1 - (l - 1) (n - 1), t2 + l - 1] does not contain the point t 0 = 0, then any excitation of the pole B within this interval will be distinct from the initial stimulation. We can thus assume t1 -(Z - 1)(n - 1) = 1.

Thus, t 1 = (l - 1) (n - 1) + 1 and t2 = t 1 + ® + n - 1.

Hence, the pole B must be excited at least once within the time interval [t1 - (l - 1) (n - 1), t 2 + l - 1] = [1, (Z - 1) (n- 1) + 1 + ® + n- 1 + l - 1] = [1, l n + ® ], i.e., the pole B will be stimulated a second time not later than the instant t = l n + ® •

Now, regarding the chain Las T* c T, we can make use of the previous remark. Thus, the quantity l n + e is the upper bound of the admissible paus es also for the original net T.

Thus, if a certain experiment e (T) on the netT reduces the rate of output signals, then the initial pause equal to @ can be lengthened by not more than ln, where l is the length of the shortest chain of cells that joins both poles, and n is the refractive period of the cells.

The established upper bound is an attainable one, i.e., it is possible to devise an experiment e (T) in which the maximumpause at the output of the net is lengthened by exactly ln as compared with the initialpause @. However, if @, l, and n are arbitrary, such an experiment, generally speaking, cannot guarantee constant pauses between the output pulses since some pauses are liable tobe lengthened at the expense of others. We shall thus consider the case in which the pole A is stimulated at a high rate (i.e., e is, roughly speaking, identical with n) and l is sufficiently long.

The experiment proposed below ensures significant reduction of the rate of output signal of the netT while at the sametime keeping a constant interval between consecutive excitations of the pole B. (Sacrificing the constancy of the intervals, it is possible by means of a slight modification to obtain the maximumpause ln + ® at the output.)

Let us associate the number 1 with the pole A. With each cell in the neighborhood of the pole A (i.e., with each cell directly connected with A) let us associate the number 2. Further,

290 YU. A. VLASOV AND A. T. KOLOTOV

if the number i - 1 has been used already, we shall assign the number i to all those unnumbered cells that belong to the neighborhood of at least one cell with the number i- 1. Thus, the cell a has the number i assigned to it if there is a chain that joins this cell with the pole A and contains exactly i cells, but there is no connecting chain containing less cells.

The set of cells to which the number i is assigned is called the i-th 1 ay er and denotedbyPi• Clearly, BEPt.

The totality of cells belonging to alllayers whose number is lower than k + 1 is called the R.

domain Rk, i.e., RR. = U P; (k = 1, 2, ... ). i=i

We will successively narrow down the region of application of stimuli exciting at first the domain Rz (completely), and then Rz_ 1, Rz_ 2, and so on down to the domain R1 whose only point is the pole A. Each time we shall delay as far as possible the application of the next stimulus, but under the condition, however, that at the instant of excitation of the stimulated domain R k there is around it a barrier of refractivity formed by the layer Pk + 1 as a result of the preceding stimulus. (For this the time interval between two consecutive stimuli should be equal to n- 1.) As soon as the domain of application of stimuli contracts to a single point, we, after waiting for a time equal to the period of refractivity, once again cover the entire domain R1• This procedure is then repeated.

It is evident that the proposed experiment guarantees at the output of net T a constant pause a = l(n -1), i.e., the pause is shorter than the maximumpause only by l, which is equal to the time of propagation of the excitation wave from the pole A to the pole B.

III. Interpretation

The concept of refraction used above is local in nature since it relates to an individual cell and not to the heart as a whole. This property makes it possible in principle to exploit the inhomogeneity of the different parts of the heart in respect to refractivity.

One of the foregoing remarks leads to the following conclusion: the atrioventricular A-B node cannot be blocked by any Stimulations, i.e., it is impossible to secure complete rest of the A-B node with the periodic activity of the sinus node remaining unchanged. Thus, either we allow some p.llse to pass from the sinus node to the A-B node, or pulses of spontaneous origin due to the blocking stimulation will break through to the A-B node in the course of interception of pulses from the sinus node.

It should be kept in mind that a spontaneous periodic excitation that suppresses the desired effect arises in the A-B node when the pause between two transmitted pulses exceeds some criticallength. This can be coinpared with the shift of the rhythm carrier leading to the center of the second-order automatism that takes place in the case of a complete transverse heart block caused by anatomical interruption of the conducting tract. In fact, in the considered situation we also deal with a block but of an entirely functional character.

As an experimental test of the results obtained with the described model we can conceive the following experiment which can be realized in two variants: 1) on a complete working organ and 2) on an isolated muscular strip from the heart wall.

1. On the external surface of the heart auricles let us markout a sufficiently large number of zones imbedded one into the other so that the maximal zone borders on the A-B node and the sinus node lies at the center of the minimal zone (the mostinner one); this is in practice the region of the ostium of the superior vena cava. Each marked zone is fitted with separate electrodes arranged around its perimeter. The first stimulus is applied at once to all

CONTROL OF CARDIAC RHYTHM 291

electrodes. After a certain time interval, somewhat shorter than the refractivity period, the second stimulus is applied to all electrodes except those around the maximal zone, etc. After a nurober of steps, equal to the nurober of zones marked out, the stimulus is applied to the electrodes of the minimal zone containing the sinus node. The entire procedure is then repeated.

2. In the second version we isolate from the heart wall a muscular strip so that it contains no cells capable of spontaneous excitation (cells of the conductive system). On the strip we select two poles to one of which we apply stimulating pulses of a constant frequency whose passage is recorded at the other pole. From here on the experiment is conducted as in the preceding version.

In the first version of the experiment we should expect spontaneous excitation of the center of the second-order automatism. In the second case, we should obtain a maximumpause between two consecutive excitations of the output pole of the strip. In both versions we neglect the thickness of the myocardium of the auricles as well as the thickness of the myocardial strip and regard both of them as flat muscular layers.

By interrupting in the model experiment the sequence of stimuli at an appropriate instant we can obtain practically any retardation from the minimum possible to the maximum obtainable, i.e., practically any frequency both above and below the frequency of spontaneous excitation in the sinus node. Thus, we now deal with a case in which the heart rhythm can be varied at will in any desired direction. We wish once more to stressthat the proposed concept of reducing the heart rhytlun is based on multiplying the unit delay by a factor which is a multiple of the nurober of cells in the shortest chain between the sinus node and the atrioventricular node (i.e., using the local property of refractivity).

Literature Cited

1. A. T. Kolotov, An Automatie Model of the Heart, in: Systems Theory Research, Vol, 20, Consultants Bureau, New York (1971), p. 210.

2. P. F, Cranefield, "The force of contraction of extrasystoles and the potentiation of force of the postextrasystolic contraction: a hystorical review," Bull. N. Y. Acad. Med., 41(5):419 (1965).

3. J. E. Lopez, A. Edelist, and L. N, Katz, "Reducing heart rate of the dog by electrical stimulation, 11 Circul. Research, 15:414 (1964),

BRIEF COMMUNICA TIONS

A NOTE ON DETERMINISTIC LINEAR LANGUAGEst

A. Ya. Dikovskii Novosibirsk

§ 1. Basic Concepts

Definition. A finite automaton with two tapes (in short, a 2-K-automaton) is specified by an ordered sextupole S = (K, -~ U {:j::j:}, q~> qo, e, ö), where: 1, 2) K and ~ arefinite sets (of states and input symbols), and +tE~ (a right boundary marker); 3) q1 EK (an initial state); 4) qo E K (a terminal state); 5) e E ~ U {+t} (an auxiliary symbol); 6) o is a mapping of the set (~ U {+t} U {e} x K x (~ U {+t} U {e})- {e} x K x {e} into the set of all subsets of KJ

D efin i ti o n. Any ordered triple a belanging to the set Q =(~*{V}~* {:j::j:}) X K X (~*{V}~* {+t}) § ), where V E ~ U {+t} U {e} U K (the indicator of the location of the reading head) is called a configuration of the 2-K-automaton S.

The mapping o induces the following "1-" relation on the set Q:

(1) (x1Vaxz, q, y1"Vby2) 1- (x1aVx2 , q', y1bVyz), if q' E Ö (a, q, b); (2) (xiVaxz, q, Y1Vy2) 1- (x1aVx2 , q', y1VYz), if q' Eö (a, q, e);

(3) (x1Vxz, q, y1Vby2) 1- (x1Vx2 , q', y1bVy2), if q'Eö(e, q, b)

for any q, q'EK; x1 , y 1 E~*; x2 , y2 E~*{:j::j:}; a, bE~U{:j::j:}.

Let a, ß E Q • We shall write a I=~, if there exist a.11 a.2, ••• , ak E Q suchthat a 1 = a, ak = ß, and ai 1- ai+1(1<i<k).

D efi ni t i o n. Astring pair (x, y) E ~* x ~* is all ow ed by a 2-K-automaton S if (Vx:j::j:, q1, Vy:j::j:) I= (x:j::j:V, q0 , y:j::j:V). The set L(S)={xyj(x, y) is allowed by a 2-K-automaton S}~ iscalleda language allowed by a 2-K-automaton S. Theset L 2 (S)={(x,y)j(x,y) is allowed by a 2-K-automaton s} is called an event representable in S.

Definition. A context-freegrammartt r =(V, V1,llp, P) is saidtobe linear (and the language gener ated by it is called a 1 in e a r I an g u a g e) if the scheme P contains onlyrulesoftheform A- aBb,A-aB,A --Bb,A-a,where A, BEV1 and a, bEV.

tOriginal article submitted June 27. 1968. t Sometimes the inclusion q' E ö (~, q, lJ) will be written in the form of an instruction (~, q, T))-- q'. § Let X and Y be sets of strings. xydf= {xy 1 x Ex, y E Y, xy being a concatenation of x and y} •

U U Uoo XO={A}, A being the empty string; Xi+l=XiX; X*=;~/i·

.,. A d! ~ d! A A d! A 0 .. 0 O O f th d u A=A, xa=ax, L={yi y E L}, 1.e., x 1s an mverswn o e wor x. tt The principal concepts of the theory of grammars are assumed known.

295

296 A. YA. DIKOVSKII

Theorem 1. A language L is linear if and only if there exists a 2-K-automaton S such that L = L (S) (see [3], Theorem 11).

Definition. A diagram of a 2-K-automaton S=(K,~U{#},qllq0 ,e,ö) is defined as a directed graph D s of the following form:

1) The vertices of Ds are states of K; 2) the vertex q is connected with the vertex q' by an

a) arc a ' q t: q' if q'Eö(a, q, b);

b) arc a ' q 7 g' if q'Eö(a, q, e);

c) A if g' Eö (e, q, b); arc q t: q',

3) The vertices of Ds are connected by arcs only by virtue of a)-b).

In Ds let us consider a path from the vertex q to the vertex q'

a(l) a(2) a(n) n(q, x, y, q')=q -~ q(1) -~ q(2)~ ... ~q(n-1) -~q' t

b( 1) b(2) b(n)

(it is possible that some vertices q(i) and qü) coincide with one another or with q or q', and that some a (i) and b (j) are equal to A) such that a (1) a (2) ... a (n) = x and b (1) b (2) ... b (n) = y. By W(q, x, y, q') weshall denote the set of all paths 7T(q, x, y, q'). Moreover, let I(D8 ) be the set of all pairs of strings (x, y) suchthat x, y E~* and suppose that in Ds there exists a path

n (q1, x:g:, y:J:t, q0). It is evident that L2(S) = I(D8).

Definition. A2-K-automaton S=(K,~U{#},q1 ,q0,e,ö) is saidtobe deterministic if the following conditions hold:

1) If '21-? q and '21 ~ q' are instructions of S, then q = q';

2) if Q1, Q2, and Q3 are sets of states encountered in the left-hand sides of instructions of S of the form (a, q, b)- q', (a, q, e) - q', (e, q, b) - q', then Q1, Q2, and Q3 will be pairwise disjoint.

Definition • A language is said to be 2 - K- d e t er mini s t i c if it is allowed by a deterministic 2-K-automaton.

and

W e have the following evident

Lemma 1. Let S be a 2-K-automaton and Ds its diagram. Then:

1) If

( , ")- ,a'(1) '(1)a'(2) '(2) '( 1)a'(m)" nq,x2,y2,q -qb'(1)q b'(2)q ... q m- b'(m)q

are paths in Ds, then

n (q, x 1x2, y1y2, q") = n (q, xi> y1, q') n (q', x2, Y2, q") =

a(1) (1)a(2) (2) ( 1)a(n) , a' (1) , (i) a' (2) , (2) , ( 1) a' (m) " =qb(1)q b(2)q ... q n- b(n)q b'(1)q b'(2)q ... q m- b'(m)q

will also be paths in Ds;

t In the following we shall drop the arr0ws.

DETERMINISTIC LINEAR LANGUAGES

2) i f S i s a d e t er mini s t i c 2 - K- a u t o m a t o n, an y non e m p t y s e t W(q, x, y, q') will be a one-element set.

C or o 11 ary. Any 2-K-deterministic language is a well-defined language. t

297

Let rJC be the class of all CF-languages, r;cn the class of all CF-languages allowed by deterministic automata with a storage memory, :zo the class of all well-defined linear CFlanguages, X the class of alllinear CF-languages, and xn the class of all 2-K-deterministic languages.

§ 2. Relations between the Classes X, Xn, X 0 and ;!CD

Letus considera linear language L={xc~yfxE~*, cE:~. yE(d~1U{A}), 2::=~,U{d}, d€~,, I~ I> 3} t

Lemma 2. Any 2-K-automaton S that allows a language L satisfies the condition 3k>0Vx1, x2 , x, y[x=x1x2 &((x1, yxcx2)EL2(S) V (xcx2 , yx1)EL2(S)) :=Jl (x2)<:k].

Pro o f. Suppose that there exists a 2-K -automaton S with a diagram Ds that allows a languate L suchthat one of the following conditions holds:

a) There exists a sequence of natural numbers k1 < k2 < k3 < ••• < k 11 < ••• , such that \fi > 03x1 , X2, x, y [x .·.~ x1x2 & l (x2) - k; & (x1 , yxc~2) E L2 (S)];

b) there exists a sequence of natural numbers l 1 < l 2 < ••• < lj < ••• , such that V j > 03x,, x2 , x, y [x ~ x1x2 & l (x2) = lj & (xcx2 , yx1) E L2 (S)]. Let us assume that Condition (b) holds (the simpler case (a) can be analyzed in a similar way). Weshall consider the set F={(xc~2 , yx1)}

of all pairs of strings yielded by this condition. It is easy to see that for a sufficiently large j 0 there exists a pair (xocx~, i;ax~> E F and a path n:0 (q1 , x0c;;~ =lf, yox~ =lf, q0) E W (qj, xOcx~ =lf, y0x~ =f\=, q0)

suchthat n°(q 1 ,x0c;;~=fl=·,y0x~=f\=,q0)=qgUlq(1)gWq(2) .. . q(y-1)g~~lq(y) ... q (x-1)gi~jq (x) ...

q(A--1) gWq(A) ... q(m-1)Z(~jq0 andthat a(y)=c, q(x-1)=q(A-), a(x)a(x+1) ... a(A)=I=A

and l (x~) = lJo (otherwise the lengths of strings x2 such that (xcx2 , yx1) E F, are bounded by the number of states of the 2-K-automaton S). Let us consider the following paths and strings: a(l) _d!_ - 1 a(Y.) :!!_ - 2 a(m) :!!_ -3 ( 1) q1b(1)q(1) ... q(x-1)-n, q(x-1)b(x)q(x) ... q(~.)-n, q(A-) ... q(m-1)b(m)qo-n, ay+ ...

df - df . df-a(x-1)=u1 , a(x) ... a(A)=z, a(A-+1) ... a(cp-1)b('ljl-1) ... b(A-+1)=v (where cp and 1f! are

specified by the relations a (cp)=b ('ljl)~:ff), b (A-) ... b (x) dt z' and b (x-1) ... b (1) ctt u2 • It is

evident that n:0 (q1 , :&c~~ #, yox~ #, q0) = n1;:;:2n3 and x0cXoy0 ~· xücu,zvz'u2 • Furthermore, for any Ji1Ji:2;:(2 Ji:2;:(3

t > 0 we have in Ds the path ~ (Lemma 1 of Sec. 1), and hence for any t > 0 t t1mes

we have x0cu1 (z) 1 ;; (z') 1 ~ E L. Let us note that vz'u2 = wyo and u~zw = Xo for an appropriate

stringw. Since l(z) > 0, there exists at0 suchthat l[(z/0 ]>l(x0). Let y0 Ed~j (the case

y 0 = A is trivial). For an appropriate string w we then have the relation Ü (z'/0u2 = wy0 , where

yo E d~i. On the other hand, since x0cu, (z) 10 Ü (z') 10 i0. E L, it follows that Ii; (z) 10 =:Co~ for an

t A CF-grammar r is said to be w e 11- d e f i ne d if for any string belonging to L (I') there exists a unique left-side derivation (i.e., a derivation in which the rule is applied at each step to the leftmost auxiliary symbol). A CF-language is said tobe well-defined if there exists a well-defined CF-grammar generating this language.

:j: I ~ I is the number of elements of the set ~ .

298 A. YA. DIKOVSKII

appropriate string w. But in this case we have wwy0 Ed~t, which cannot be the case, since

l (w~) > 0 and y0 E d~i. This completes the proof of the lemma.

Lemma 3. lf there exists a deterministic 2-K-automaton S that allows a language L, it is possible to construct from it effectively another, equivalent, deterministic 2-K-automaton S that satisfies the add i ti on al c o nd i ti o n 'r:/';V11 [xcxy EL ...... (xc, yx) EL2 (S)] .t

0 utl ine of Pro of. By A weshall denote the set of paths n(q,, Zt=#:, zz=#=, q0) in the diagram Ds of the automaton S that satisfy the condition: 3x, y, w [xcxy = z1i 2 & wz2 = y & l (w) > 0]. By B weshall denote the set of other paths in D5 leading from q1 to q0• By reasoning in the same way as in Lemma 2, we can show that the set X A is finite. On the basis of the diagram D5 it is easy to construct a diagram D~ in which any c-A-path (i.e., part of an A-path of the form qg(~?q' ... q0 , where either a (a) = c, or b (a) = c) is incident to any c-B-path only at the vertex q0• Indeed, D5 does not contain any fragment of the form

.. ;"A-path

".···~B-path c •••

. . /

(by the definition of an A-path), and no fragment of the form

.-C--- • ••q, c-A-path / ••

qoL c-B -path

where the path from q1 to c is traversed by a sufficiently long string x (since as a result of the deterministic property the paths from q to q0 are distinct and the traversal of one of them does not violate the "mirror image" with respect to c). If the fragment D5 has the form

the common section from q to q0 must be "split," by adding a limited nurober of new vertices. By reasoning in the same way, we can construct on the basis of the diagram D~ a diagram D~ in which any two c-B-paths are incident only to the vertex q0• By virtue of Lemma 2 the c-Bpaths D" do not contain cycles. In this case the c-B-path q (1)g{N q (2)~g? q (3) ... q (i)~[f/ ... q (j)g({? q0,

where (for definiteness) a(1)=c, a(i)=b(i)==#=, can be replaced by apath q(1)~\\>p(2)~gjp{3) ... p (l)~tg q0, where the vertices p (2), ... , p (l) are new vertices, a(l) = b(l) = :#:, ä (1) = c,

a (2) = a (3) = ... = a (l-1) = A and b (1) b (2) ... b (.l- 1) = b (1) b (2) ... b (j -1) a (i -1) ... a (3) a (2).

The diagram Ds obtained by this transformation is the diagram of the sought-for 2-K-auto-

t V';P (x) dt P (x) holds for all x, with the possible exception of finitely many.

DETERMINISTIC LINEAR LANGVAGES 299

maton S (since the condition Vy [xcxy E L-+-+ (xc, yx) E L2 (S)J is not satisfied by strings belonging to a finite set).

Theorem 2. There does not exist a deterministic 2-K-automaton that allows a language L.

P r o o f . Suppose that there exists a deterministic 2-K -automaton S that allows a language L. On its basis let us construct a deterministic 2-K-automaton S as in Lemma 3. Let

us consider a sufficiently long string

ycyEL, hence W(q1, yc=#=, y=#=, qo)=focp.

y E L.1d; ycyy E L. Then W (q 11 yc =#=, JIY =#=, q0) =I= cp • N ext,

Since in D8 any B-path terminates with an arc q #> q0 , #

this being the only arc originating at q, we have

It follows from Lemma 1 of Sec. 2 that the path

is unique in the set W (qh yc =#=, yy =#=, qo). Yet at the vertex q there originates a unique arc

q ~ q0 with y ~ A. Therefore, ycyyE, L, which contradicts our assumption. #

Let us go over to relations between the classes X, xn, X 0 and Qlt'n. We can present the following table (the symbols

:::::>, c::. CD,

standing at the intersection of a row of class X and of a column of class Y signify

X;?Y, XS:Y;

Xn Y~ 1; &.X-Y7cp ~ Y-X

respectively):

Relation 2 follows from the fact that the linear language

L' = {x I x = anbkanb1 or x = akbmalbm; k, l, m, n>;> 1}

is essentially undefined (i.e., there does not exist a well-defined CF-grammar generating it) (2]. The inclusions 1 and 3 follow from Theorem 2.t Relations 4, 5, and 6 follow from the

t The inclusion 1 follows also from the corollary of Lemma 1 and from the fact that the language L' is essentially undefined. Our attentionwas drawn to this fact and to relation 3 by A. V. Gladkii.

300 A. Y A. DIKOV SKll

fact that the language L" = {xx 1 x E ~*, ~ contains at least two symbols} belongs to class xn and does not belong to 6/Cn (the latter is proved in [1] and [4]).

The author expresses his deep gratitude to A. V. Gladkii for valuable remarks.

Literature Cited

1. S. Ginsburg and S. Greibach, "Deterministic context-free languages,n Information and Control, 9:6 (1966).

2. R. Parikh, "Language-generating devices,n RLE Quart. Progr. Rept., MIT, Cambridge, Mass, No. 60, pp. 199-212.

3. A. L. Rosenberg, nA machine realization of the linear context-free languages,n Information and Control, 10(2):175-188 (1967).

4. A. Ya. Dikovskii, nRelations between the class of all context-free languages and the class of deterministic context-free languages,n Algebra i Logika, 7(3):23-37 (1968).

NONRECURRENT CODES WITH MINIMAL DECODING COMPLEXITY t

A. A. Markov Gorki

Let U be a finite system of distinct words in the alphabet A, ~ a free semigroup over A, [U] a subsemigroup of ~, generated by the set U, and A. the empty word. By II X II we shall denote the nurober of elements of the set X, and by I x ~ the length of the word x.

To the system of words U we shall assign a finite rooted digraph r (U) with a set of vertices V formed by some suffixes of the words of U, the set of edges E, and a function cp with values ± 1 defined on E:

1°. /.. E V and is a root of r(U).

2°, If u = u'a (u, u' EU, a E ~. a "bA.), then u' E V, a E V and (/.., u'), (u', a) E E.

3°, If a E V and aa' E U(aa' E ~. a =f='A, a' =/='A), then a' E V and (a, ta') f E.

4°. If a E V and a = ua' (u EU, aa' E ~. a =/='A, a' =/='A), then a' E V and (a, a') E E.

5°. The graph r (U) contains only the vertices and edges that can be obtatned by Rules 1°-4°, If the pair (a, a') can be obtained both by 3° and by 4°, the set E will contain the pair of edges (a, a'), and we shall distinguish between them.

6°. For e E E we shall write

( ) = { + 1 if e has been obtained by 2°-3°, cp e -1 if e has been obtained by 4 o.

The graph r (U) is a slight modification of the construction proposed for the first time in [1] (see also [7]) for studying variable-length code systems that do not have the prefix property (socalled nonrecurrent codes). The purpose of the improved version proposed by us here is to enrich and sharpen the information content of the graph assigned to a system of words. In the same way as in [1], we can show that for a E ~ "" U ( a =1= /..) we have

(1) a E V~ there exists a relation u1, ... , u ka = ui, ... , u~, where Uü ui EU, I a I< I u; I·

In the following we shall consider only complete independent systems of words. Let us recall that a system of words U is said to be independent if any word in ~ can be represented as a union of words in U in at most a unique manner, and it is said to be complete if for any v E ~ "" U the system { U U v} is no Ionger independent [2]. The following property is char

acteristic for independent system of words (see [7], [1], and [5]).

tOriginal article submitted December 10, 1968.

301

302 A. A. MARKOV

(2) For any path A., v1, v2, ••• , vkt ••• in the graph r (U) we have v; E U if and only if i = 1. If an independent systero U is coroplete, it will satisfy (as is shown in [2]) the conditions

(3) (3uE[U]) (VaE~)(3a'E~)uaa'E[UJ,

(4) ~ IIA 11-IUI = 1, uEU

each of the conditions (3) and (4) iroplying not only that U is independent, but also that it is coroplete. If (3) holds for a given u = u0, we shall refer to u0-coropleteness of U.

Let I+(U) be the nurober of edges e E E with cp(e) = 1, and I-(U) the nurober of edges with cp(e) = -1.

As a quantitative roeasure of the inforroation about the systero U contained in the graph r (U) we shall take I(I' I U) = r+(U) + I-(U), i.e., the nurober of edges of the graph r (U). The quantity I (I' I U) represents a certain aspect of the coroplexity of decoding if the systero is taken as a code. More precisely, it characterizes in general the diversity of probleroatic Situations that can occur in sequential decoding of roessages. In particular, in accordance with intuition we have I (I' I U) = 0 if and only if r (U) = { A.}, i.e., in the case of recurrent (prefix) coding [3, 4, 6]. We shall henceforth assuroe that I(I'IU) ~ 0, thus confining ourselves to nonrecurrent codes.

The redundancy of information in r (U) will be defined as I (I' I U) - II U II· This is due to the following reasons. Any edge e E E contains together with cp(e) inforroation about one (and only one) word in U which will be denoted by JJ.(e). More precisely,

(5) { aa'

!l (e) = " ' a,

if e=(a, a')andcp(e)= +1,

if e=(a, a'), cp(e)= -1 and a=a"a'.

Let us show that if the systero U is coroplete, JJ. will roap E onto the entire systero U, and hence

I (r I U)-JJ UJJ> 0.

Moreover, in this case we have

(6) U =J.t(E+), where E+={eleEE, cp(e)= +1}.

Indeed, we know [6] that if r (U) ~ { A.}, then

(7) (3a E ~) (Va' E ~) aa' ~ [U] and the equation a = ua" cannot hold for any u EU. In the case of u0-coropleteness of U for any u EU and any n of the form (3), there exists an a E m, such that uounaa E [ U], where a is selected on the basis of (7). In this case we shall have for some m and a' the relation u0uma' = vt ... vh = v E [U], where u = a'a", I a' I< I vh I, uoum E [U]. Now we have a' EV by virtue of (1), whereas 3° and 6° yield e= (a', a") EE with cp(e) = + 1. Hence, we conclude that I (I' I U) :::: J+(U) :::: II U II· But for constructing U on the basis of r (U) we must know only II U II edges, i.e., a nurober I (I' I U) -II U II of edges is redundant for this purpose. In particular, I-((J) negative edges can be always regarded as redundant.

Our objective is now to study a class of complete independent systems of words such that I(I'fU) -II U II = 0.

If W is a system of words in the alphabat B and II W II = II A II, whereas T is a one-to-one mapping of A onto W, we shall write for any words a,, az E 21 the relation T(a1 a 2) = T (a1) T(a2)

and for the set of words U we shall write • (U) = {• (u) I u EU}. In [5], the system T(U) is called a composition of the systems U and W. By a * weshall denote the inversion of the word a, i.e.,if a=a1 ~-·ah (a;EA),then a*=ah ... a1. Hence, Ü*={u*JuEU}. LetKiibeacomplete

NONRECVRRENT CODES OF MINIMAL DECODING COMPLEXITY 303

system of words in the alphabet Bii = { 1, 2, ••• , i + j} consisting of words 1, 2, ••• , i of length 1 and of words (i + p)q of length 2 for all p = 1, 2, ••• , j; q = 1, 2, ••• , i + j. Weshallshow that if the system V is complete and I(r I V) = II U il , then the set V'\.._ {A,} will be a complete prefix system and for some i and j there exists a mapping T of the alphabet Bii onto V such that u = T (Kii) •

By virtue of (6) we have cp (e) = + 1 for any e E E , and the mapping f.1. in (5) will be a bijection. Any word uEU can be uniquely represented in the form vivi, where v;, viEV,

(v;. vi) E E. Let us show that none of the words {V""' A,} is the initial section of another word of this set. Let us assume the contrary: v;,vj E V and vi =via. In this case the vertex vi in r (U) must be a dead end; otherwise we would have for some u EU a relation via' = u of type 3° and f.J.(vi, a') = f.1. (vi, aa'), which is impossible. Let (v~<, v;)EE (such an edge mustexist in view of (1) if vi ~ A., with vk ~ A.) and v = f.J.(Vk, vi). Let us consider the system V'= {UUva}, where a has been selected for Von the basis of (7). Without loss of generality it can be assumed that I a I > I u I for any u EU. It is evident that r (U) is a subgroup of r (U'); hence if r (U') contains any vertices or edges other than V and E, the former must appear according to 2°-4° for u = va or u' = va. Let us consider these possibilities.

2°. u = u1a, u' ~ va, and hence va = u'a. This relation has a unique solution u' = v, a = a, since otherwise we would have f.1. (A., v) = f.J.(Vkf vi), or u'a' = v and f.J.(u', a') = f.J.(vk, vi). Thus according to 2° we must include in V' the vertices V and a, andinE' the edges (A., v) and (v, a) and only them.

3°. aa' = va, a, a' =I= A, a E V. We have aa" = v (if a = va", then cp (aa") = -1 in r (U), which is impossible). But f.l.(a, a") = f.J.(vk, vi), whence a = vk. Hence according to 3° we must add v;a E V' and the edge (vk, via) in E'.

Rule 4° is evidently not applicable, and all the added vertices are dead-end vertices; therefore r (U'), as well as r (U), have Property (1). Butthis is impossible, since U is complete by assumption. The obtained contradiction proves that the system V""' {A} is a prefix system.

Thus» u = (U n V) u (uu = V;Vj), V;, Vj E V. It follows directly from the completeness of u that V is a complete prefix system. Let a (v;) = i, a (V""' {I.})= B. Hence there exists a system W consisting of words of length not greater than 2 in the alphabet B and suchthat U = o-1(W). It follows from the latter formula that W is independent and that if U is u0-complete, then W will be a-1(uo)-complete. For concluding the proof we must convince ourselves that the only complete systems W with I w I< 2 (w E W) and i words of length 1 are Kii and Kti. Indeed, let W = { 1, 2, ••• , i, 1A1, 2A2, ••. , mAm}, where A; c A, m = II A II·

~IIAjll m

Byvirtueof(4)wehave ~m-e1 =~+ i m2 =i,whence ~IIAill=m(m-i). The i i=l

condition of independence of W can be formulated in terms of a binary relation R on B: jRk ~ (jk) E W. W is independent if and only if Rn n I X I=</>, for any n; here I= { 1, 2, ... , i}.

Suppose that n1 symbols of A are to the left with respect to I, whereas n2 symbols are to the right. Since by virtue of the independence of W there are no cycles passing through I, we have

~ IIAili -<.(m-i) i+ni+(m- i-n1) 2 • Hence we obtain n1 2: m- i. But n1 + n2 + i = m, n1, n22: O, J

and we have n1 = m- i, n2 = 0 (or conversely), and these cases correspond to W = Kii, or W = K~i. By taking a - 1 = T, we obtain U = T (K~i), which completes the proof.

Literature Cited

1. A. A. Markov, "Alphabet coding," Dokl. Akad. Nauk SSSR, 132(3):521-523 (1960).

304 A. A. MARKOV

2. A. A. Markov, "Completeness condition for nonuniform codes," Problemy Kibernetiki, 9:327-331, Fizmatgiz, Moscow (1963).

3. A. A. Markov, "Nonrecurrent coding," Problemy Kibernetiki, 8:169-186, Fizmatgiz, Moscow (1962).

4. B. Mandelbrot, "On recurrent noise limiting coding," Symposium on Information Networks, Polytechnic Institute of Brooklyn (1955).

5. M. Nivat, "Elements de la theoriegeneraledes codes," Automata Theory, Academic Press, New York- London (1966), pp. 278-294.

6. E. N. Gilbert and E. F. Moore, "Variable-length binary encodings," BSTJ, 38!4):933-967 (1959).

7. A. A. Sardinas and G. w. Patterson, "A necessary and sufficient condition for unique decomposition of coded messages," Conv. Rec., Trans. IRE, IT-8:104-108 (1953).

REALIZATION OF DISJUNCTIONS AND CONJUNCTIONS IN MONOTONIC BASESt

E. I. Nechiporuk

Leningrad

This note belongs to a series of papers devoted to finding "nonlinear" lower bounds for the complexity of Boolean functions [1-5].

In the note we show that in some monotonic bases it is possible to realize a disjunction and a conjunction of n arguments with a complexity of order n c, where C is an arbitrarily large constant.

We shall consider the realization of functions

by Superpositions in a basis consisting of one monotonic function

l m

qJz,m=V &xi,j, wberel>2,m:;>2, i=1 i=1

Each of these bases generates all the monotonic functions apart from the constants 0 and 1. It is easy to see that in the basis { cpz m, 0, 1} the functions Dn and K11 are realized with a complexityt of order n. Yet we have' the following theorems:

log m 1 L (D ) \.J ll log 1 +

(j.!z, m n r'\ · • Theorem 1.

Theorem 2. log l 1

L (K ) \.J nlog m t-QJz,m n r\

1°. We shall represent a Superposition by a tree such that: Each internal node is an occurrence of a base element cp z, m; each terminal node is an. occurrence of the argument; each edge will be numbered by a pair (i, j) indicating the input of a base element. The occurrences of a superpositionwill be split into tiers (see Fig. 1). The number of the tier of an occurrence is the number of edges connecting the occurrence with an output element. The number of the last tier (it contains only arguments) will be denoted by P and called the de pth of a superposition. If for any p, 0 s p s P- 2, the p-th tier contains. (lm)P base elements, the superposition is said to be c o mp 1 e t e.

toriginal article submitted October 31, 1969. t The complexity of a Superposition is defined as the number of base elements in it. By L'" (!)

'f'l,m

we denote the minimum number of base elements cp z m sufficient for realizing a Boolean func-, tion f.

305

306 E. I. NECHIPORUK

Zeroth tier

First tier

Second tier

Third tier

Fourth tier

Fig. 1

2) Let us consider the subformulas

2°. Pr o o f o f T h e o r e m 1 • The upper bound is reached on complete Superpositions of depth ]logz n[.

The lower bound. Weshall transform the given Superposition that realizes the disjunction Dm without increasing its complexity, into an equivalent complete Superposition.

1) If a given Superposition has "superfl.uous" letters other than the letters x1, x2, ••• , Xn, we shall replace each such letter by one of the letters x1, x2, ••• , X11 •

l m V & A-.

'• J i=1 i=l (1)

in the order of decreasing tiers.t For each i we shall select from the subformulas Ai,1,

Ai,2, ••• , Ai,m the most economical one and replace all the subformulas Ai,i by this subformula. The function realized in this way can only increase; but it does not, since the "superfl.uous" letters have been eliminated.

After all these replacements have been effected, each subformula will realize a disjunction.

3) For each p, 1:::; p:::; P, we shall assign to a subformulai of the p-th tier an array of pairs

corresponding to the edges connecting the given Slbformula with an output element. This array will be called the r e c o r d of the subformula. Subformulas which ha ve the same arrays (i1, i2, ••• , ip) in the record, are said tobe monoconjunctive. It is evident that monoconjunctive subformulas located in the p-th tier consists of mP subformulas.

If a Superposition has letters xk (1 :::; k :::; n) in the p-th tier and subformulas A which are not arguments in the (p + 1)-st tier, we shall replace in the p-th tier each of the mP monoconjunctive letters xk by the subformula A, andin the (p + 1)-st tier each of the mp+1 monoconjunctive subformulas A by the letter xk.

After all these replacements, the Superposition will be uniform with a depth not less than ]logz n[. Hence,

3°. Proof of Theorem.2. The upper bound isreachedoncompletesuperpositions of depth }logm n[.

tAt first we shall consider all the subformulas of the (p- 1)-st tier, then all the subformulas of the (P- 2)-nd tier, etc., up to the zeroth tier.

t The subformula can be one of the letters x1, x2, ••• , xn.

REALIZATION OF DISJUNCTIONS AND CONJUNCTIONS 307

The lowe r bo und. Weshall transform the given Superposition that realizes the conjunction Km without increasing its complexity, into an equivalent complete Superposition.

1) We eliminate the "superfluous11 letters as in 2° above.

2) We consider the subformulas (1) in the order of decreasing tiers. In (1) we select an i such that the set of subformu1as

A;, ~> A;, 2, ••• , A;, m

is the most economical. In (1) we then replace for each k, k ,c i, the subformula Ak,i by Ai,1' Ak,2 by Ai,2, ... , Ak,m by Ai,m. The function realized in this way can only decrease, but it does not, since the "superfluous" letters have been eliminated.

After all these replacements each subformula realizes a conjunction.

3) As in 2° we assign to each subformu1a its record. Subformulas which have the same arrays (j1, b .,., jp) in their record are said tobe monodisjunctive. It is evident that monodisjunctive subformulas are identical, and that each group of monodisjunctive subformulas 1ocated in the p-th tier consists of zP subformulas.

If a Superposition has letters x k (1 :::; k :::; n) in the p-th tier and subformulas A which are not arguments in the (p + 1) -st tier, we shall replace in the p-th tier each of the l P monodisjunctive letters Xk by a subformula A, andin the (p + 1)-st tier each of the zP+l monodisjunctive subformulas A by the letter x k·

After all these replacements, the Superposition will be uniform with a depth not less than ] log mn[. Hence,

log l L (K ) -::;:, (Zm)]Iogmn[ -2 > nlog m +i

'Pl,m n ""' ~

Co r o 11 a r y • For log m =1= logt the bases cp z m and cps t are incommensurable (for log l log s ' ,

the definition see [6]).

Literature Cited

1. B. A. Subbotovskaya, "Realization of linear functions by formulas in the bases V,&,-," Dokl. Akad. Nauk SSSR, 136(3):553-555 (1961).

2. A. A. Markov, "Minimal gate-contact networks for monotonic symmetrical functions," Problemy Kibernetiki, Vol. 8, 117-121, Fizmatgiz, Moscow (1962).

3. R. E. Krichevskii, "A minimal circuit with make contacts for a Boolean function of n arguments," Diskretnyi Analiz, No. 5, pp. 89-92, Novosibirsk (1965).

4. E. I. Nechiporuk, "A Boolean function," Dokl. Akad. Nauk SSSR, 169(4):765-766 (1966). 5. E. I. Nechiporuk, "On a Boolean matrix, 11 Systems Theory Research, Vol. 21, Consultants

Bureau, p. 236. 6. B. A. Subbotovskaya, 11 Comparison of bases in the realization of functions of the algebra

of logic by formulas," Dokl. Akad. Nauk SSSR, 149(4):784-787 (1963).

CIRCUITS TO RAISE RELIABILITY t

M. M. Rokhlina Moscow

Let us consider circuits of functional elements that in a certain sense can be said to "vote." Voting consists of the following: in a circle of radius k the set (0, 0, ••• , 0) realized by the circuit of the function takes on the value 0, while in a circle of the same radius the set (1, 1, ••• , 1) takes the value 1; in this case we say that the circuit "corrects k errors." This type of circuit must be used in the synthesis of self-correcting circuits made of functional elements [1]. The circuits considered here are built of ~~' elements, realizing the function

~+1

h11 (xt, 0 0 0 , x[l+t) = V XtXz o 0 0 xi _1x;+1 .. 0 x~+to From the set of all circuits that correct k ,,= 1

errors we separate the subset of circuits in which the output of each element, different from the output of the whole circuit, is connected to one input of a certain element.

Note 1. The circuits just defined above are isomorphous to the equations in a base consisting of one function hp (x1, ••• , xJ.l + 1). All subsequent conclusions concerning the complexity of such circuits can be formulated as conclusions concerning the complexity of the corresponding equations if the complexity of the equation is defined as the number of symbols of the base function entering into it.

We will show that the minimum complexity of circuits with tree-like shapes correcting k errors is no less than klogdJJ.+!)o

As an application of this result for any constant c, note the incomplete base Be and the "effectively defined" sequence of functions f~ (x1, ••• , Xn), expressed in base Be such that the order of complexity of the equation when jg is realized in this base is no less than nc.

Let us introduce certain concepts and terminology.

A function is said to satisfy the condition (A11), fl > 2, if any J..t of the sets on which the function returns to one have a common unit component [2]. Obviously:

a function satisfying condition (A11 ), cannot take the value 1 o n o p p o s i n g s e t s • (1)

11+1 Clearly, function h11 (x~> 0 0 0, x11+1) = V x 1x 2, 0 0 0 x;- 1x;+lo 0 .x11+1 satisfies condition (A 11).

i=1

Weshall designate by l 0(a') the number of zeros in the set a, and by l 1(a) the number of ones in set 01. Obviously:

h11 {a)=0 if and only if Z0 (a)>20 (2)

t Original article submitted April 15, 1968. 309

310 M. M. ROKHLINA

Fig. 1 Fig. 2

Weshall call the function of !llgebraic logic g(x1, ••• , x 11 ) the k-function (k = 1, 2~ ... ), if it satisfies the following conditions:

if Z0 (rx) <. k, then g (a) = 1, if Zt(a) <. k, then g (a) = o

(the values of the function on the remaining sets are insignificant). Obviously, if g(x1, ••• , x 11 )

is the k-function, then it is also an l-function for any l less tha'n k. From this discussion and from [1] it follows that for each k there exists a k-function realized by a circuit of ~"' elements.

The complexity of the circuit S we shall define by the number of elements ~"' in it [symbol L(S)]. Let Lll(k) = minL(S), where the minimum is taken over all the circuits S realizing the k-function. From the definition of the k-function it follows that

if k' <. k", then L"' (k') <. L"' (k"). (3)

Circuit S is called nonrepetitive if to each circuit input is connected no more than the input from one element while different variables are ascribed to all the inputs. We designate by L~ (k) the minimum of the numbers L(S) over all the nonrepetitive circuits S realizing the kfunction.

It is true that

L"'(k) = L~ (k). (4)

In fact, the inequality Lll(k) ::5 L~(k) is obvious. On the other hand, let S be a circuit realizing the k-function and suchthat L(S) = Lll(k). If S isanonrepetitive circuit, then we separate the identified inputs of the elements forming the circuit inputs (Fig. 1). We obtain a nonrepetitive circuit S'. Clearly, L(S) = L(S'). And S' also realizes a certain k-function, so L(S') 2: L~(k)

and Lll (k) 2: L~ (k). Thus weh ave shown the validity of (4). From here it follows that the number of inputs for the minimum circuit realizing a certain k-function may not be fixed.

We shall designate the tree-type circuits connected to the inputs of the output elements ofcircuitSthe major subcircuits of circuit S (Fig.2).

Lemma 1. LetS be a nonrepetitive circuit and S 1 , ••• , Sp+i be its major subcircuits. The following two conditions are equivalent:

for any i and j such that i ~ j and 1 ::::; i, ::::; f.1. + 1, it is true that

K (S;)+K (S1)> k-1. UI)

Proof. We designate by F (or Fi, respectively, 1::::; i::::; f.1. + 1) the function realized by circuit S (Si, respectively).

CffiCUITS TO RAISE RELIABILITY 311

(I)- (II). Let K(S) 2: k. It is sufficient to prove one of the relationships of condition (II), such as K(Si) + K(S2) 2: k- 1. Assurne that it is not true, i.e.,

(5)

From the definition of K(Si) it follows that for each of the subcircuits Si (i = 1, 2) there exists one input set ai satisfying one .of the following conditions:

a) l 0 (a;) = K (Si)+ 1 and F, (a;) = 0,

b) l1 (a;) = K (S;) + 1 andF; (a;) = 1.

Further, if one set ai satisfies condition b), then by virtue of (1), the set ~~ opposite to it satisfies condition a). Thus, there exist two sets, a; and a~, satisfying condition a) for i = 1, 2.

Then F(a;, a;, 1, ... , l)=h~-'(0, 0, 1, ... , 1)=0. But from a) and (5) it follows that l0 (a;, a~, 1, ... , 1)<k, and this contradicts the fact that Fis a k-function.

(II)- (I). Let a be an arbitrary set and a i its part applied to the inputs of the major subcircuit Si (1::::: i::::: fJ. + 1). Let lo (a)<.k. We will show that F(ä) = 0. Assurne that F(a)<.k. Then from property (2) it follows that no less than twovalues ofifulfillF; (a;) =0.

Let F 1 (ad ~ 0 and F2 (a2) c.~ 0. Then from the definition of K(Si) and K(S2) it follows that l0 (a1) > K (S1)+ 1, and l0 (a2) .,_,. K (S2) + 1. Since circuit S is nonrepetitive, from the latter two

inequalities and by virtue of condition (II) it follows that l 0 (a) ~ l0 (a1) + l0 (ac)::> k -f· 1, and this contradicts the fact that Z0 (a) k.

Now let zJa)<.k. Let us show that F(a) = o. Assurne that F(ä) = 1. From property (2) it follows that then no less than fJ. values of i satisfy F, (a;). Assurne that these values of i are 1, 2, ... , fJ. • Considering only subcircuits Si and S2, as above, we contradict the fact that zJa) <.k •

Lemma 1 is proven.

We introduce the notation:

1:'1" is a set of sets of numbers (ki, ... , k!J + 1) satisfying condition (II) of Lemma 1.

Q" is a set of sets of numbers (k1, ... , k!J + 1) satisfying the conditions:

11+1

Lemma 2. L~(k)=1+ min ( ~ L~(kt)]. (k1, ..• , kf.l+I)E\Bk i=1

Pro of. First we show that

11+1

L~(k)=1+ min (~L~(k;)). (kt. .... k11+1lEilln i=1 (6)

LetS be a nonrepetitive circuit realizing a certain k-function and suchthat L(S) = L~(k). Further, let Si, ... , SJJ+l be its major subcircuits and ki= K(Si) (1::::: i::::: fJ. + 1). Then by virtue of lemma 1 (k11 ... , k~-'ct) E ~In and furthermore L (Si) 2: L~ (k). Therefore,

11+1 11+1 11+1

L~(k)=L(S)=1-+- ~ L(S;)>1+- ~ L~(k;)>1+min ~ a (k;). i=i i=1 lll~t i=1

312 M. M. ROKHLINA

~t+i

On the other hand, let (k~, ••• , k011 + 1) be a set on which we reach min ~ Lt (k1). Let S

su" i=i

be a nonrepretitive circuit whose major subcircuits S1, ••• , s11 + 1 (they are nonrepetitive) have the properties:

Si realizes a certain kr-function and L(Si) = L~ (k~).

Then by virtue of Lemma 1, circuit S realizes a certain k-function and

~t+1 ~t+1 ~+1

1 + min ~ Lt (k;) = 1 + ~ Lt (kY) = 1 + ~ L (S;) = L (S) > Lt (k). suh i=i i=1 i=i

Thus, (6) is proven. Further, it is obvious that ~h ~ Q3k. From this it follows that

The opposite inequality is easily derived from (3), and Lemma 2 is proven.

We define an auxiliary function f (k) as follows:

It is true that

f (0) = 0,

f (2s + 1) = (ft -1-1) f (s) + 1,

f ( 2s + 2) = ft f ( s+ 1) + f ( s) + 1 ,

f (k) > 0,

f(k+1)-/(k)>O.

s:>O,} s:>O.

The first is obvious; the second is easily proven by induction in k.

Lemma 3. It is true that (for t 2: 1, l- 2t 2: 0):

!-l (I (l) -· f (l- t)) > f (l- t)- f (l- 2t),

f.l (! (l + 1)- f (l + 1- t)) > f (l- t)- f (l- 2t).

Pro o f. First we consider the case l = 2t. Then (Az t) has the form '

f.l (! (2t) -- f (t)) > f (t).

(7)

(8)

(9)

(Az,r)

(Bz,t>

(~)

Expressing f (2t) in accordance with (7), we find an equivalent and obvious relationship (because fJ. 2: 2):

(f.l"'- f.l) f (t) + f.lf (t-1) > f (t).

In this case (B l t> takes the form '

f.l(f(2t+1)-j(t+1))>f(t). (B~)

We shall prove (B~) by induction in t.

Obviously (Bl) is true. Let (B~.) be true for all t' < t. We shall prove the validity of (B;).

a) Let t be even. Expressing f (2t + 1), f (t + 1), and f (t) in accordance with (7), we arrive at a relationship equivalent to (Bp:

CffiCUITS TO RAISE RELIABILITY

~ ( (~ + 1) f ( t)- (fl + 1) f ( +)) > ~~ ( ~ ) + f ( +- 1) + 1.

Relationship (10) follows from:

~ (t (t)- f ( +)) :? f ( ~ - 1) + 1.

Equation (11) follows from the already proven relationship (A; ), and (12) from (AI) 2 2

and the inequality f ( +-1) + 1-< f ( ~) [see (9)].

b) Let t be odd.

Equation (Bp is equivalent to

(( 1 ( t--1-1) (t-L1 )) (t--1) 1 ~ ~+ )f(t)-~f -2- -! -2--1 >(~t+ 1)/ -2-- + .

Equation (13) follows from:

~ (~ (t(t)- f ( t-;-1 -+-1))) >~! ( t-;-1)'

313

(10)

(11)

(12)

(13)

(14)

(15)

Equation (14) is derived from (BI-J), which is true by assumption, and (15) is derived 2-

from (AI-J) and (9). 2

Now consider the case when l- 2t > 0. Equations (Az,t) and (Bz,t) are proven by induction in l - 2t as (B{) was proven.

Equations (Az t) and (Bz t) follow from the relationships shown in Table 1 and assumed ' ' by induction tobe fulfilled.

l- even , t- even

l- odd, t- even l- even , t- odd

l- odd, t-odd

TABLE 1 Az, 1

Al/2, 1/2 • A(l/2) -1, 1;2

A(l-1)/2, 112

Al/2, (1+1)/2' A(lf:n-2, (1-1)/2 *

B(l-1)/2, \1-1)/2 **

* Here we also use the inequality

A</2, 1!2' B(l/2)-1, 112

B(l-1)/2, 1/2' A(l-1)/2, 112

B(l-1)/2, (l-1)/2•

Al/2, (t+1J/2

A(l+il/2, (t+!J/2'

B((l-1)/2)-1, (1-1)/2

fl (! (+- t t 1 ) -I ( ~ - t)) <rt (! ( +- tt1 ) + f (f-(t+1))) . * *Terms must be grouped, with some of them transferred from one patt of the in

eq uality to the other.

314

Lemma 4.

M. M. ROKHLINA

I (k) = min {I (k1) + ... +I (k11+t) + 1}. jl)k

(16)

Proof. For k = 1, 2, Eq. (16) is verified directly (sets W1 and W2 contain one set each).

Now let k 2::: 3. We can set up an arbitrary set of numbers from mk such that:

if k=2s+1, then k1=s-t, k2 =k3 = ... =k11+t=s+t; if k=2s+2, then k1=s-t, k2 =k3 = ... =k11+1=s+t+1.

In our case s 2::: 1 and s - t 2::: 0.

By virtue of (7), it is sufficient to show that for each s 2::: 1 and for each t 2::: 1 (for s- t 2::: 0) it is true that:

1 + (fl + 1) I (s)<.l (s- t) +fll (s+t) + 1

and

1 +flf(s+ 1) +I (s)<.f(s-t)+ flf (s+ t+ 1) +1,

which are equivalent respectively to

fl (! (s+ t)- I (s)) >I (s)- f (s-t),

JL (! (s+ t+ 1)- I (s+ 1)) >I (s)- I (s-t).

Assuming s + t = l, we obtain the relationship (Az t> and {Bz r) that were already proven. ' '

Theorem. L (k) =I (k).

Proof. Fork=1wehave Lll(1)=Lb(l)=f(1)=1. LetLil(k')=j(k')forall k'> k.

From Lemmas 2 and 4 and (4) it follows that:

11+1

L'; (k) = L11 (k) = 1 + min ~ L~ (ki) = 1 + min (f (k1) + ... +I (k11+t)) =I (k). lBn i= 1 lSn

Note 2. The function LJ.I(k) has an order k10g2 <11+0 •

First we show by induction in m that L 11 (2m-1) = <~-t+i)m -i. For m = 1 this equation u

is obvious. Let it be true for all m' < m that

From the theorem and (7) it follows that:

Further, let 2m -t - 1 < k < 2m - 1; then from (3) it follows that

<~-t+iJm-1 -1 <- L~" (k) < <~-t+fr-1 !! - u '

CIRCUITS TO RAISE RELIABILITY 315

or

Therefore the statement is proven.

Note 3 • Obviously all the results of this analysis are also valid for the complexity of realization of the k-function by circuits of e~t, elements realizing the function

Addendum

Consider the application of these results to the "effective" design of a complexly realized function. Consider the equations in a base consisting of one function hiJ (x1, •• , xll + 1).

The complexity of the equation 2f is the number of base functions entering into ~ For a function f, expressed by hiJ (x1, •.• , xll + 1) the notation of the main text is preserved (by virtue of note 1), i.e., Lll(j) == min Lll (W), where the minimum is taken over all equations 2(, realizing the given function f.

Let f ~ (x1, ••• , X 11 ) be a function that assumes the value 1 on sets a such that l o(a) ::5

[(n- 1) /,u], and is equal to 0 on the remaining sets.

The function f~ (x1, .•. , x11 ) can be defined differently as a symmetrical function with operating numbers n- [(n- 1) /,u], n- [(n- 1) /,u] + 1, ••• , n. It is a monotonic symmetrical function.

It is easy to show that f~ (x1, .•• , x11 ) satisfies condition (A~t). Accordingly, it is expressed by hiJ (x1, ••• , xll + 1). Furthermore, f ~ (x1, •.• , x11 ) is an [ (n - 1) I .U ]-function. Therefore, by virtue of note 1

and by virtue of note 2 for sufficiently large n

where c ll is a certain constant.

Thus, for any constant c, a base (incomplete) and a certain simply defined sequence of monotonic symmetrical functions ft (x1), •.• , fn (x1, ••• , x11 ) can be indi cated such that the complexity of realization of the function fn in this base has an order to less than nc.

Literature Cited

1. G. I. Kirienko, "Self-correcting circuits," in: Problemy Kibernetiki, Vol. 12, Nauka, Moscow (1964).

2. S. V. Yablonskii, G. P. Gavrilov, and V. B. Kudryavtsev, Functions of Algebraic Logic and the Post Class, Nauka, Moscow (1965).

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