•

UNIVErlSITY OF NORTH CAROLINADepartment of Statistics

Chapel Hill, N. C.

Mathematical Sciences DirectorateAir Force Office of Scientific Research

Washington 25, D. C.

AFOSR Report No.

ON THE DEFINITION OF A CERTAIN CLASS OF AUTOMATA

by

M. P. Schutzenberger

January, 1961

Contract No. AF 49(638)-213

In this note we verify that a largepart of Kleene's Theory of regular~vents still applies to a class of au­tomata with an infinite set of states.

Qualified requestors may obtain copies of this r~port from theASTIA Document Service Center, Arlington Hall Station, Arlington12, Virginia. Department of Defense contractors must be estab­lished for ASTIA services, or have their "need-to-know" certi­fied by the cognizant military agency of their project or contract.

Institute of StatisticsMimeograph Series No. 273

ON THZ DEFINITION OF A CERTAIN CLASS OF AUTOMATA

M. P. Schutzenberger

I. 1. Introduction. This note is concerned with the definiton of

a class of one way one tape automata,tOt, that includes that of finite

automata. (1) In a schematic manner an automaton from the class;Accon­

sists of an ~ way, ~ tape, finite automaton(2)controlling a very

elementary computational process in which enter a finite number of finite

but not bounded integral parameters. Three main restrictions are

imposed:

i) There is no feed back from the parameters to the finite

state machine and only addition and subtraction are used;

ii) For each input letter only a bounded number of additions

or subtractions are performed;

iii) Drastic conditions are imposed on the rules by which, from

the computed values of the parameters it is decided to accept or not an

input word.

As expected, the definitions insure that a good part of Kleene's

theory remains valid for the class it .In order to make the paper self contained, some concepts of the theory

of finite automata are recalled in 1.2; the definitions are discussed

at heuristic level in section I.3, I.4, I.5; a formal definition is given

in 1.6 and in I.7 this definition is reduced to a simple standard form.

The second part is devoted to the discussion of examples and counter-

examples in support of the definitions. In the last part Kleene's

theory is applied to the class J4:and an intuitive interpretation of the

set of words accepted by an automaton of this class is given.

1. This research was supported by the United states Air Force through theAir Force Office of Scientific Research of the Air Research and Development Com­mand, under Contract No.AF 49(638)-213. Reproduction in whole or in part is per­mitted for any purpose of the United States Government.

2

1.2 Finite Automata

He recall the definition of a finite, one wa;)' , one tape

liutomaton (0:, F).

Let X = tx1 and Z =1er } be t'-10 fin! te sets;

called the input alpha'Lel:. and E is the set of the

X is usually

state of the

automaton. To each x g X is associated a rnappint;, of L into i tseli'

",hich \le denote simpl;)T by LTX. An initial state 0"1 and a subset z:'

have been distinguished in Z.

If presented with an input word f = x. x. ",xi in the letters~l ~,.,c n

xi ~ X, we compute successively the states O'lxi ,ulxi x. (=(erlxi )x. ),1 1 ~2 1 ~2

Jlx. xi xi ••• , erlxi xi ",xi ' .•• ,erlf = Jlx. xi ",xi ",xi~l 2 3 12m ~l 2 m nand we say that Q accepts the word f (in sjmbols f ~ Fa) if the

state vlf belongs to the distinguished subset E' E.

The definition given here is in fact that of a right automaton

",hich reads the input word f from left to right and a perfectly

s~!unetric definition could be given for a left automaton. However,

the family Ro of those subsets of words which can be considered as

the set of the 'Words accepted by some finite automaton ,haS t.he·,

special property that to··eacbrig)it finite automaton (Q,F)l.theret

corresponds a left finite automaton ex'for which Fc(l = ~ The main

property of Ho is contained in Kleene's Theorem which can be presented

in the following manner:

1. ~o qont~ins a.lJ". the fini;te &J.l~ c;f J;

2. If F' ,F"g{if6'r. tl;1eIlcF"~l.,1I, F' F"", Fi<~FtI(F ,rf=f'f" wi1;Jlf'Gf.': and

fIlW"] )and F,*=~n(With :F,n= F,n-~'land-F,l~F') all belong to~.

3· If F\ 111, then the complement F-F' ~f F I also belongs to (A,o .

4. (}Lo is the smallest family of subsets of F that satisfies (1),(2).

(3).

3

In this note we propose to verify that tha left-right symmetry and

the statements 1) and 2) in Kleene's theorem are still valid for a somE;·,'\

what larger class f4cin which E is replaced by a finite dimensional spac8

VN

with integral coordinates and where the operations are restricted to

ordinary arithmetic and, even more to the addition of integers.

1.3 Finite dimensionality and elementary nature of the operations.

We shall constantly assume that all the information abo~the first

m letters of any input

for further computation

word f =xi xi xi ..• x. that is available12m ~n

is summarized by a vector v(fm) belonging to VN,the N-dimensional vector space ~ii'h~in:teg1'a:l'·'~bO:Pdina:te~f'.'whereN ~sl.a :1'1nite

fixed constant characteristic of the automaton.

Thus, by definition, we are given independently of the input word

f:

x = x and of the N coordinatesim+l

This we summarize by saying that for

A vector v(e)€ VN from which we start the computationsj1)

2) For each letter x. € X and each number j = 1,2, ... ,N a~l

function ¢.(Vjx) which expresses the value of the j-th coordinateJ

v,(f 1) of v(f 1) as a function ofJ m+ m+

of v = v(f )(with v(f ) = v(e».m 0

each x € X we are given a mapping ¢x: VN~ VN·

3) A decision rule by which from the knowledge of v(f)(= v(fn »

we declare f to be accepted or not.

In themselves, these assumptions do not imply any restriction whatso-

ever upon the set of words F accepted by our automaton.Q;

This is shown by the display of a simple computational rule such

that for any two words f, f' € F, v(f) = v(f') only if f = f'; or, in

4

bctt~r terms by the display of a simple isomorphic representation of

the free monoid F. This, of course, is quite well known and we take

an example for which a proof if readily available. (4)

Example 1.3.1. Let us suppose, first, that the input alphabet

contains two letters x and y only. Let v(e) = (1,1) and inductively

for any word f with v(f) =(Vl (f),v2 (f))., let

v(f·x) = (vl (f), vl (f) + V2(f)); v(fy) = (vl (f)+v2(:t;,J1''''2(t,,»);(as usual "fz"

denotes the words made up of "f" followed by the letter z. It can be

shown that if f r f' then v(f) ~ v(f'). Thus, by selecting any subset

VI of the space V2 and by deciding that f is" accepted" when v(f) € VI,

we could obtain any F'e: F as the set of word accepted by our automaton.

The process easily generalises to any finite alphabet xl ,x2' •.• x .n

With Xl we associate the function v(f) ~ v(fx) described above; with

any x.(i = 2,3, ••• n-l)we associate the function V(f)~(fyi-lx) and1

finally with x we associate the function v(f) ~ v(fyn). Since anyn

word in the letters x and y can be factorized in a unique manner as a

n' n-lproduct of the words x, y x(l ~ n' ~ n-2), and y we again obtain

a one to one correspondence between the set of words in the Xi and a

subset of V2 and this proves our contention.

Thus, restrictions have to be imposed either on the operations

involved in computing v(fx) from v(f) and x, and/or on the decision rule

that is on the final operations by which f is accepted or not.

In order to formulate the restrictions we introduce the following

definition:

5

Definition 1: A mapping ¢:VN~ VN is said to be elementary if

every coordinate of ¢(v) can be obtained from those of v € VN by a

finite number of addition, subtraction, multiplication and reduction modulo

p (cfbelow). This mapping is said to be very elementary if it satisfies

the supplementary condition that only multiplications with a left factor

bounded by an absolute constant are allowed. It is positive if it maps

into itself the subset v~os of all vectors with non-negative coordinates

and if only addition, multiplication by a non-negative number and "re-

duction modulo p" are allowed.

In this definition, by "reduction modulo p" we mean the usual mapping

y of Z (the set of all integers) onto the interval (0, p-l) defined byp

Y n =- n (mod p).p -

The reader may observe that in the above example the algorithm was

a very elementary and positive one, since only addition and multiplication

by °or by 1 were .used.

Definition 2. A rule for deciding from v(f) if f is accepted or not

will be said elementary if it is given by the union VI of a finite number

of linear subspaces of VN together with the prescription that f is accepted

if and only if v(f) ¢ VI. It will be said positive if VI belongs to the

subspace v~os of the vector with non negative coordinates.

It is frankly admitted that no argument is given for deciding to say

that f is accepted when v(f) ¢ VI rather than when v(f) € VI. However these

two possible defiIlt10nsantail somewhat different structures as it is shown

by the example I.7.4 below.

6

1.4 Definition of the class~

Even with the extremely restrictive conditions that both

¢: VN~ VN and the decision rule are elementary, I am not able to

verify the not trivial parts of Kleene's Theorem. Thus, we propose:~

Definition 3. An automaton a will be said to belong to the class~

if for all x E X the mapping ¢x: VN~ VN is very elementary and if the

decision rule is elementary.

Definition 4. An automaton a e~will be said to belong to the

sUbclass~ if the initial vector v(e) has non negative coordinates, if'0

for all x ex, ¢ is very elementary and positive and if the decision·x

rule is elementary and positive.

If one accepts to consider that "addition" and "reduction mod p"

are operations taking one unit of time irrespective of the value of

the operands one can think Of~S of the class of those automata which

use only elementary operations and which work" in real time." Indeed,

in each computation (v(f); x) ~ v(fx) only a bounded number of units

of times is needed since because only multiplications with a bounded

factor appears, these can be replaced by bounded cascades of additions.

If we had also imposed the condition that the length of any vector

v(f) is bounded it is intuitive that we would have reduced ourselves to

the class of finite automata. It can be shown that because of our re-··

strictive definition of the decision rule it turns out that~ is equiva­

lent to the class of the finite automata. (c.f. 111.4.3 below).

1.5 Preliminary reduction.

We want to bring any elementary mapping ¢x:VN~ VN into a simpler

form by getting rid of the "reduction modulo p" operations. ThiS, of

7

course, is done at the cost of increasing the dimensionality.

By hypothesis, there exists only a finite number of integers p which

are used for performing reductions in all the computations ¢ (v(f» andx

we denote their product by q.

Let now)for any f g F) V'(f) be the direct sum of v(f) and of a N-v8ctor

Vi (f) whose coordinates are those of v(f) reduced modulo q.

Thus, by definition v"(fx) is obtained by computing first

v(fx) = ¢x(v(f)) and, then, by deducing Y'(fx) from v(fx) by reduction

modulo q. However, as we shall show, the same result can be obtained in the

following different way:

i) we obtain Y(fx) as a polynomial function of v(f) with coeficients

depending upon v'(f) and x.

ii) we obtain Y'(fx) as a function of x and of yl (f) only.

Since "for';: 'any of' ~Ft~oo;r.di"'nate.S o'f ytfl)a:re bounded by. q,; the ma,pplng..

described in ii) is just a mapping from a finite set to itself and does

not bother us any more since it can be realized by a conventional finite

state automaton.

Now to the proof~

By hypothesis the computation of any coordinate of Vi (fx) involves only

iterations of the following operations:

Computation of the sum or product of two partial results a and b, reduction

modulo p of some partial result c. We use induction and we assume that

every partial result is either a polynomial in the coordinates of v(f)

with coefficients depending upon x and the coordinates of v'(f) or a

function of x and of the coordinates of v'(f) which consequently is

bounded in absolute value.

8

Thus, trivially, this is true of a+b and ab if it is true of a and

of b.o

' If c is a pOlynomial in the coordinates of v(f), c(mod p) is a

polynomial in the values (mod p) of the coordinates of v(f) and consequent-

ly, since p is a divisor of q, c (mod p) is a function of the coordinates

of vl(f); finally if c is a function of vl(f) the same is true of c(mod p).

We now apply our requirement that only mUltiplications by factors less

than K are allowed. Admittedly we do this by giving to the definitions

their most restrictive interpretation and consequently we offer the present

derivation as being a heuristic introduction to the formalized definition

1 1 given below.

Nonetheless it may be argued that the instruction to carry out a

mUltiplication by a factor can only be under the control of x and Vi (f)

because we only allow arithmetic operations. Accordingly, if a coordinate

v.(f) happens for some f e F to exceed K in absolute value, then it must1

appear in anyone of the polynomials only under at least one of the two

conditions.

i) as a linear term (with a coefficient depending upon (x, vl(f), i»

ii) in a monomial of degree two or more in the other coordinates of v(f)

but with the provision that this monomial has a numerical coefficient

k(x, vl(f),i) which is zero for all the fwhich are such that IV.(f)1 > K.1

A liberal interpretation of the definitions would only lead to the

colclusion that when a product vi(f)vi,(f) appears in some polynomial the

construction of the algorithm is such that)for all f) Vi(f) and Vi(f)

are not both r > K. However, if we admit that the ,sequential time order

in which the operations have to be performed is part of the algorithm,

9

this interpretation would result in taking, at least provisionally, un-

bounded numbers as multipliers.

Now, reverting to our above assumptions i and ii, we show that we can

entirely eliminate the case ii. Indeed let p be a prime larger than K

and replace every coordinate Vi(f) by the pair made up of vi(f) itself

and of the value vi (f) o,f v. (f) reduced module p. Also in the com-,p ].

putation of every monomial of degree> 2, replace vi(f) by vi (f).- ,p

Thus, the (possibly unbounded) Vi(f) appears only linearily in the

polynomials and, because of the condition ii, the final result remains

unchanged. As in the previous section, it is easily verified that the

values v. (fx)'s are in fact independent of the possibly unbounded].,p

coordinates of v(f) and thus we are led to believe that the following

definition (1') is equivalent to the initial definition (1).

I.6 Formal definition.

Definition 1'. An automaton a ~:A: is the structure given by

1) a finite alphabet X =1. xl;

2) a finite set E =i (]'} ;3) for each x S X a mapping E ~ E;

4) for each x ~ X and (]' SEan integral N x N matrix ~(x,(]');

5) the following distinguished objects: (]'l~ E; v(e) ~ VN; E'C E;

V'C VN

where V'is the union of a finite number of linear subspaces of VN..

For any input word f = xi xi1 2

cessively:

xi ' the automaton computes sue­n

10

~lx. ; VeX! ) = vee) ~(x. , ~l)1 1 1 1 1

(~lx. )x. ; v(xi Xi ) = v(e) ~(xi ; ~l) ~(xi ; ~lx. )1 1 1 2 1 2 1 2 1 1

or, for short, v(f) = v(e)~f. Then,f is accepted by a if ~lf S ~' andV(f) ¢ V' •

Definition 1": The subclass/{ 0 Cfr is characterised by the following

restrictions:

1) all the matrices ~(x,~) have not negative entries;

2) v(e) has not negative coordinates;

3) the decision rule is positive~ , .. ~; i~(""'l"'~: , ,1 ' .. 1 ("; •

. ~ .

"

·,t': ..

If ~ reduces to a single state, ~f can be simply written as the product

~(Xil)~(Xi2)... ~(Xim) and we shall refer to ~ as to an homomorphism F~~.

Such an a sA with ~ = ~'= {~l\ will be said in reduced form.

We now verify the following statement.

I.6.1 Every automaton a S~iS equivalent to one in reduced form.

Proof : 'Let M be the number of states ~. of L- J

To each X S X we associate the (M x N) x (M x N) matrix ~x defined

as follows:

(~X)ij,i'j'= 0 for 1 < j, j' < N if ~ix I ~i" (1 ~ i,i' ~ M).

(~x)., i' .,= (~(X'~i)' j' for l~'j, j' < N if ~.x ='~i' (1 ~ 1; i' ~ M).1J, J J. - 1 -

11

It is easily verified that for any f = x. x •.. X, if we write1 1 i

21 m

~Xi we also havetn

Cilf) ., i I ., = 0 if 0"1'f ~ O"i IJ.J, J

(~f)i' iljl= ~(xi jO",)~(xi jO".xi )~(x jO"xi x. ) •.. Il(Xi ;O"lxi x..•. X. )J, 1 1 2 1 1 i 3 1 J.2 In 1 J.2 lm_l

if (J'i f = (J'i'

Thus, taking an initial M x N-vector vee) with (v(e))lj= (v(e))j and

(v(e)),.= 0 for i ~ 1, v(f) = vee) ~f satisfies the following relations:lJ

2) the vector v(f) which for each 1 ~ j ~ M has coordinates

~ (v(f)) .. = (v(f)). (with I' = i:O"iS ~I ) does not belong to VI, theiSI' J.J Jdistinguished subspace.

By adding eventually a dummy coordinate with value 1 for all f we

can always suppose that the sUbspace V' is defined by homogeneous linear

equations ,i.e., that v ¢ VI only if not all the vectors product vwk

are

zero where ~k~ is a fixed finite set of vectors. It is trivial that

these conditions reduce to homogeneous linear conditions on ~f since

vee) is a fixed vector.

Example r.6.l: Let ~ = {(J'} be a finite automaton with N states. To

each x S X we associate the N x N matrix ~x with entries ~xii'= 1 if

Then for each f = Xi Xi ... x. , the matrix ~f =1 2 J.m

~xi ~x, ••• ~xi1 J.2 m

also has entries ~fi,i' which

cording to rrif = rri , or not.

~ ~f is 1 or 0 accordingi¢I' Ii

Remark:

12

satisfy the condition ~fii'= 1 or 0 ac­

Thus it I' = {i:rris ~I} the linear function

to whether f is accepted or not.

As a trite consequence of I.6.1 we may observe that for the class

A there is no difference between "left" and "right" automata in the sense

of Ll.

I.7 We now find it convenient to introduce a few more words and

notations j by an homomorphism ~ :F~ zw we mean a mapping from F to ~,

the ring of the integral N x N matrices that issuch that ~ff'= ~f~f'

for any f, f' S F. This condition is clearly satisfied when we take an

arbitrary ~x S ~ for each x S X and define ~xi xi ••• xi as12m

By a projection ~:~~ Z we mean any fixed linear function

~ = ~ p .. ,mi " of the entries mii , of an element of Z-.. Clearly it1 < i,i'< N ~~ ~ iN

P is the matrix from ZW whose transpose has entries Pii' we have

~ = ~ (mp)i i= Tr(mp) .(the "Trace·." of mp).l<i<N '

We very explicitly state the following well known result:

L 7.1 Given ~:F~ ~ and ~' :F~ ~ we can find ~":F~ ~,,(N"= NN' )

such that for any ~:~~ Z, ~' :ZW'~ Z there exists a ~":~,,~ Z

having the property that, identically, ~"~'f is a given bilinear function

of ~~f and ~'~'f.

13

Proof. Let ~'f be the kroneckerian product of ~f and ~'f,that is, the

matrix from ~f' with entries: ~"fii' ,jjl= (~fij)(~'fi'jl)' Then, by

induction, one verifies that ~"f ~"f' = ~lIff' identically.

Since any fixed bilinear function of n~f and n'~'f 1s a fixed

bilinear function of the entries of ~f and ~'f, it is a fixed linear

function of the ent:r;:1es of ~"f and the statement is proved. Of course

if only a linear function of n~f and n'~'f is wanted, it is enough to

take N" = N+N' and for ~lIfJ the direct sum of ~f and ~'f.

r. 7.2. Let ID = ~ Fa1 (F: the set of the words accepted by a), thenV\. 1 JasJc a

Fa,Fal S CR.. implies Fa U Fas ([land Fan Fas CR:,

Proof: Let us consider firstly Fa-= if: n~f,lo } where ~ is an homomorphism

F--;>~. By taking ~2 :F---> Z~J the kroneckerian square of ~, we can find

2n2 :ZN2--;> Z such that, for all )n2~2f = (n~f). Consequently

Fa=l f: n~f ~ 0 } ={f: n2~2f ~ 0 ~; if Fa' : { f::If 1 ~'f ~ 0 } we t~ke also the

kroneckerian square ~2 of ~'. Then, if ~ is the direct sum of ~2 and ~2'

·,'.I~.~ we can choose ;r:Z~iN'2--;> Z such that identically

n ~ f = (n~f)2+ (n'~'f)2j thus Fa"={ f:'ir ilf ~ O} = FaU Fa" Since (because

of definition 11) every general a has the form F~U F

u .~ where

Fak=f:nk~f ~ o}, the result

have the side advantage that we

is proved for the operation·U. We also

know how to reduce any F to the simpler formain which a single n appears. This instantly de1ivers the result for

Fall Fa l since, if ~II is the kroneckerian product of the corresponding

~ and ~'J we just take the projection i':~,~ Z which gives the identi­

ty nil ~II f = (n~f)(n' ~If) •

14

We still go a step further in the reduction process and we prove:

1.7.3 To any pair (n,~)(~:F~ ~; n:~~ z) there corresponds one

~:F~ ~2+2 such that identically n~f = ~fl,2+N2'

Proof. For every f g F let us construct ~f as the N'x N' matrix

(N t = 2+~) with the following entries:

(i) ~fN' .= ~f. 1 for 1 < j ::: N' •,J J,

(ii) "iIf1,j+kN+l for each (j,k)(l ::: j ::: N; 1 ::: k ::: N) is equal to the

(k,j) entry of p(~ f) where p is the N x N matrix cor-

responding to n.

(ii)' fl.1 j+kN+l, N' for each (j,k)(l ::: j ::: N; 1 ::: k ::: N) is equal to the

(k,j) entry of (~ f)p.

(iii) The restriction of ~ f to the set of indices i,j(l < i,j < ~+ 2) is

the direct sum of N matrices identical to ~f.

The verification that P: is an homomorphism is straight-forward and the

result is proved because of (iv).

II Examples and counter-examples.

We want to display first an example which shows that some automata

a g;Atcan accepts sets which cannot be accepted by any finite state devices,

at least when the alphabet has two letters or more. We write F' Sd\(£Ro)for

denoting that there exist some a g~ (some finite state automaton) such

that F' :: F .a

Example 11.1. Let!: = ~ cri~ i=1,2,3,4,5' X =i x'Y1 ' and the mappings

15

(JIY = (J3Y = (J4Y = (J5Y = (J5; (J2Y = (J3'

Let also ~«J,x) be the following 2 x 2 matrices:

initial state (Jl with the initial vector v(e) = (1,0),l+nl 1+n2diagram that (Jlf = (J4 if and only if f = x Y xit is clear from the

~«Jl;X) = ~«J2;x) = (~

1 0~«J2Y) = (0 1); ~«J,x)

i); ~«J3;x)=~«J4;x)=(~ -i);o 0= (0 0) in any other case.

and that, then, v(f) = (1,nl -n2 ); in any other case the second coordinates

of v(f)is O. Thus, this algorithm can be made to accept all the input

l+n l+nwords except those which have the form f = x Y x It is a well known

fact(5)that this cannot be done by any finite automaton. Thus O(F ~.

The same example also shown that, because of our very restrictive

definition of an accepted word, it may be that F SeA and F - F ¢ rh jCt - Ct VL.

, {n+l n+17Indeed, let F = F-Fa= x Y x J n = 0,1,2, ••.. We have just seen that

FaS~ and we want to prove that F ' ¢ d{ . Let us assume for the sake of

contradiction that F'={ f:(~'f)l,N,F o}where ~I is an homomorphism

F~ ~,. Under this hypothesis Fa= F-F I ={ f:(~f)l,N,F o}. Now, since

~a is a Nix N' matrix it satisfies an equation of degree at most N' and

consequently there exist for each fixed pair (i,j) an homogenous linear

relationship independent of m S ~, between the (i,j) entries of any

2 N'sequences of N' matrices m~x, m~x , •.. ,m~x . But by hypothesis, for

every n we have

( ,n+l) ( , n+l 2) ( , n+l n) 0 d ( ,n+l) .1 0~ X yx 1,N'= ~ x yx 1,N'="'= ~ x yx 1,N= an ~ x 1,N,r;

16

thus, N' has to be larger than n+l and, since n is unbounded, N' cannot

have any finite value.

Example 11.2 Let X have a single letter x, ~ be any homomorphism

F ~ ~, and consider the usual integral power series

co¢(t) = 1 + ~ tn(~ xn)l N in the ordinary variate t.

n=l '

According to our last reduction, the set F of the words acceptedaby an automaton a sJt with associated homomorphism~, is just the set

of those integers n for which (~ xn)l,Nf O. But,as a function of t,

¢(t) is the Taylor series of an ordinary integral rational function whose

denominator is a divisor of det(l - t ~ x). Thus, according' to Sk6lem's

theorem (6), there exist natural numbers n,n' ,dl,d

2, •.• d'k(With

0< d. < n') which are such that when m > n the coefficient of t m in- ~-

¢(t) is zero if and only if m is congruent mod n' to one of the d.'s.~

Consequently, the complement F - F of F (and, because of Kleene'sa atheorem, F itself) is accepted by some finite automaton. Thus, when

a

X has a single letter our class does not differ from that of the

finite automata and there are qUite simple sets (as that of the words of

n2the form f = x where n runs over all integers) which cannot be

accepted or not accepted by some a S AExample 11.3 We want to show that the familYCR =iF - Fa1a s,A is not

closed under set multiplication. More explicitely we verify that there exist

homolllQI:Ph16malJ..':'F~zwmd.. ~':~""'~tsuc'm~!that the set F1L. of' all f"'whioh 'can

be factorised in at least one manner as fll = ffl with ~fl,N= 0 and

~lfll,N'. 0 cannot be written as F" = tf:~lIfl,NII= O}for any finite Nil.

( 1 -1)~y = 0 1;

thus, for the class~there exists a lack of symmetry even at the set

theoretic level between the accepted and not accepted types of sets.

\ 1 0 1 1Let X = t x'YJ; ~e = ~'e = (0 1); ~x = ~'x = (0 1);

~Iy = (~-i); Fctlf:~fl,2= 0 \; F~,= {f:~lf12= o}.Thus F (F ,) is the set of the words f such that If I - If I = 0a a x y

(If I - 21fl = 0) where If I denotes the number of times the letter zx y z

appears in f. By definition F - F and F - F I both belong to m anda a "\

we want to verify that this is not true of F - F" where F" =F F " LetaaW(f) = \f' :ff' SF"] for any f S F. We have

1) For any f" S Fa' W(f)C W(t"f), indeed, ffl S F" means that ffl = f l2

with flS Fa and f 2S Fa" Thus f"ff' = (f"fl )f2 with f"2f l S Fa' since,

trivially, F F C F .aa a

2) If f,f"S Fa- te") then W(f) ~ W(f"f); indeed,let f"fS Fa' that is

If"fl = If"fl = k, say. The word f"fxk satisfies If"fxkl = 2k;/f"fxk

, =k.x y x Y

Thus this- element belongs. ;to;· F' ,and since F', .C F" because e S F f'I F ,'. a a a ' • akwe have x S W(f"f). k -Now it is impossible that f x = f l f 2 with flS Fa

kwould imply f = f l f

3with If

3X Ix= If

3'x+k = 21f31y'

that is k =

Let us

given f the

If31y since f, flS Fa entails f 3S Fa and If3 'x= If3 1y

.

suppose now that F"= tf:~"fl,N"= 0} with ~:F~Zrr" For any

set W(f) of the vectors consisting of the N~th row of the

matrices ~"f' (f' S W(f»is a linear space of dimensions at most N". Since

we have seen how to build at least on infinite strictly increasing

18

W-sequence W(f) C W(fllf) C ... C W(f"mf ) •.. we also have 8.A'1\infinite- --- ~ - ---... ....", --- -- ~---_._,. ---

strictly increasing ~~po~n~ti.sequence and N'I cannot be finite.

Remark: In the next section it will be proved that for any ~:F~ ~, the

formal power series ~ f(~f)l N is in a sense a rational (non-commutative)fSF '

function of the input letters x. This, of course, is true in particular

not in general a rational function when

~ f is an algebraicf¢F 'a

However for

of the sum ~ Pwhere F I is the set of words accepted by a finite auto­fSF

,i

maton. Of course also ~ f isfSFa

F is the set of words accepted by an arbitrary a S~.a (~)

a very special subclass it is easily verified that

but not necessarily rational function of the XIS and that apart from

this special subclass the same sum can be a transcendant function. This

again points to the asymmetry mentioned above between "accepted" and

"not accepted" sets of words.

III KLEENES I S THEOREM

III.1 Although this part could be written without explicitly using the

notion of the ring Aof the formal integral power series in the non-

commutative variates x S X, it seems more natural to do so and we recall

without proofs a few definitions and results on A. These are very

special and shallow cases of theorems used by many authors, in the study

of other problems. Two especially valuable references are (7) and (8).

Definition 1: A is the ring of all formal infinite sums a = ~ f(a,f)fSF

with integral coefficients (a,f). The addition and mUltiplication

are defined respectively by:

a + a l = ~ f«a,f) + (a'f»; aa l

fSF

19

= ~ f ( ~ (a, f 1 )(a I ,f" ) )fSF f'f"= f

where, as always in this section, ~ means a summation over allf'f"= f

factorizations f = f'f" of f. It may be easier to visualize any element of

A as a generating function in which every_word fhas a (positive or

negative) integral coefficient (a,f). Thus, in particular to each sub-

set FIeF there correspondsthe formal sum (its characteristic function)

~ f = ~ f ~,(f) with X-,(f) = 1 or 0 according to fgF ' or not.fgF' fSF 'F 'F

The mUltiplication is simply the ordinary multiplication of series

using infinite distributivity, that is aa l can also be formally expressed

as

~ f(a,f)a ' = ~ a f"(a ' ,f) _fgF fgF

~ ff' (a,f)(a I ,f ' ).f,f'SF

It may not be unnecessary to stress that this product is B2! the

Hadamard product ~f(a,f)(a',f) to which we were led by the construction of

the kroneckerian product of matrices in 1.7.

We shall always denote the empty word by e and byA*.the subset, of' a.ll

a g A in which (a,e), the coefficient of e, is zero. The elements of

A* are usually called quasi regular and we denote by a* the mapping

A~A* defined by a - (a,e)e. If and only if a is quasi regular

(i.e. a = a*), it has a quasi inverse aO= ~ an which satisfiesn>l

000aa + a = a a + a =a. In a perfectly equivalent manner an element

- -1 -1 -1a g A has an inverse a (a a = a a = e) if and only if it belongs to

the group GCA of the elements a l which are the sum of e and of the quasi

-1 0regular element a l *; then a' =e + (-a'*) . We shall find it more

20

convenient to deal with t~e quasi notions because if a has non­

onegative coefficients the same is true of (a*) but generally not of

(e + a*) -1.

All the above operations are legitimate because A* is a continuous

topological algebra where the distance between a and a' (a Fa') is

the supremum of the inverse of the length of those f for which

(a,f) F (a' ,f).

III.2 Definition 2: R~A is the subset of all formal power series which

have the form r: f(l-lf)l N for some homomorphism I-l:F~~ (N < co).fSF '

Proposition III.2: R is the smallest subring of A such that its inter-

section with G is a group and whicb:.contains everyi.x ~ x.

The proof is simple but we find it clearer to break it into four

independent statements:

III.2.1

Proof:

R is a submodule.

if a = r: f(l-lf)l N'fSF '

a'= r: f(I-l'f)l N' (for short, if afSF '

is produced by I-l and a' by I-l') we take the direct sum 1-l":F~~+N' of

I-l and I-l' and we apply I.7 for reducing to the desired form.

III.2.2

Proof:

R is a subring.

We have to prove that if a is produced by I-l and a' by I-l' we

can construct some ~' which produces aa'. It will be simpler to prove

the result under the additional assumption that a, a' S A* and to observe

that the general case follows from II.2.1 because

aa'= a*a'* + (a,e)a'* + a*(a' ,e) + (a,e)(a' ,e)e. We can also assume that

I-le and I-l'e are the identity matrices of ~ and ~,respectivelY. After

these preliminaries we proceed to the actual construction.

21

For each x ~ X we define j.LII X g ~+N' as the matrix (6X (~~~u)

where by (j.Lx)u we mean the NxN' matrix in which all columns are zero

except for the first one which is equal to the n-th one of j.Lx. Then,

after taking j.Lll e as the identity matrix of ~+N" we extend j.L1I to

an homomorphism j.L1I :F-->~+N' in the usual manner.

Because of our assumptions the following relations are surely true

if f = e or x:

""f I'f when 1 < i <_ N,' IllIf L: IIfi II Ifll when.... l,i= .... l,i .... 1,N+l flfll=f.... 1,N.... l,i

1 < i < N' . Let us verify now that if they hold for f they ,also hold

for fx. Indeed we have for 1 < i < N:

j.L1I fx = L: j.L1I f j.L1I x, . = L: j.L If j.Lx. = j.Lfxl

.l,i l:::j::;,N+N' l,j J,J. l:::j::;,N l,j J,i ,J.

and for i = N + i I < N+N' :

j.LllfXl,N+il= L: j.Lfl j(j.LXU). i l + L: j.Lllfl N+,j.L'Xj '1'

l~j~' J, l~j~' ,J ,J.

1 and zero otherwise.The first sum is just j.LfXl,N when i =

induction hypothesis the second sum is

By the

~ ~ f' 'fll I '" f' 1I 1 fll~ ~ j.L 1 -~ 1·1J. Xj i'= ~ j.L 1 N.... xl i"f'ftl=f l~j~ ,N ,J , f'f"=f' ,

When i ~ 1 this can also be written as L: j.Lf'l Nj.L' g"l i'

sincegl gll=fx' ,

j.L' e1,i'= 0; on the contrary when i = 1 we have

j.LllfXl,N+l= j.LfXl,N+ L: j.Lgi j.LI g" L: j.Lgi IJ.' gilglgll=fx 1,N 1,1 g'g"=fg 1,N 1,1

gil ~ e

above relations are true for all cases. Since they imply that

and the

22

t f(l-l"f)l N N'= t f t (I-lf')l N(I-l'f")l N'= aa l the result is proved.fSF ,+ fSF f' f" =f ' ,

III.2.3 R contains the quasi inverse of each of its quasi regular

elements.

Proof: As above we assume that I-le is the identity matrix and we

define iie as I-le. For each x S X, we take I-lx equal to the sum of I-lX and of

a matrix (I-lx)u S ~ which has all columns zero except for the first one

which is equal to the n-th column of I-lX. For f = e or x we have

I-lf1, i = I-lf1, i + f' f~=fiif ' 1,N I-lflll,1. As in the lastproof above:

-f 0:" Ilf ii 0:" 0:" iif' "f" 1 ., III-l xl ·= ~ ~ xl .~x., i+ ~ ~ ~ 1 N ~ ~x. i", ~ 1<i<N ' ~ ~, f' f" =f 1<i ' <N' , ~ ~ ,

Thus»

if i = 1) iifXl

,; = IJ.fx + I-lfx + t iif' I-lfll x + t iif' _J.lfll x.. •... 1, i 1,N f' fll =f 1,N 1,1 ft fll =f 1,M 1.,N

Thus the~initial relation is valid in all cases. Let us now compute

In

particular for i = N we have aN= a + aNa, that is e = (e + aN) (e-a) and,

since a was assumed to be quasi regular, aN= (e_a)-l_ e = aO

•

III.2.4 Reciprocally, any element a = t f(l-lf)l N' of R, can be obtainedfSF '

from the generators x S X by a finite number of ring operations and

formation of the quasi inverse (of quasi regular elements).

Proof: As usual we can assume that a g A* without loss of gener-

ality. Let us introduce as a tool the ring ~ of the formal sums

23

~ fmf with coefficients mfS ~ and consider the element s ;fSF

~ Xllx ofxSX

AN' If we extend in a natural fashion to ~ the notions introduced

~ sn does exist since eachn>l

o rinfinite sum s ; U ;

for A, s can be considered as a quasi regular element of ~ and the

rentry Ui,j of u is the

element ~ f~fi j from A.fSF '

Thus, the problem reduces to that of showing that any entry u, j~,

can be obtained from the generators by the above listed operations.

This is trivial if N = 1, because, then, AN= Aand we assume that

the result is already proved for N - 1. Let us consider t S ~-l

obtained from s by replacing by zero all the entries sl,i and si,l

o(l~i~); by the induction hypothesis t = v exists and all entries

vi ... (2 < i, j < N) satisfy the desired conditions. We define u S "L_,J - - -~

by the following expression for its entries:

If i, j ~ 1

uli= sl'+ ~ sl' vJ'~; uis; siS+ ~ v' j sJ's~ ~j~ J ... j:;:29f ~

uij ; vij+ uiS(e + ull ) Ulj .

By the induction hypothesis, all the u .. 's can be obtained from~J

generators by the specified operations and we verify that u ; sO

by showing that us = u-s. We have

24

(US)ll= ullsll+ ~ ul·s· l = ullsll+ ~ Uljs. l + U ~ U S2:9::N J J ~j~ J 11 2:;:j::N lj j 1

=ullsll+ (e + Ull)( ~ sl,s'l+ ~ sl"v"j S., )2:;:j~ J J 2:;:j,j'::N J J JS

= ullsll+ (e + ull)(e - sll- (e + ull)-l) = ull- sll'

If 2 <: i < N

(US)li= ullSli+ (e+ull) ~ Ul'S ji. 2:;:j~ J

= ullSli+ (e+u1l)( ~ SljS"+ ~ Sl"vj , .S .. );2<j<N J~ 2<j j'<N J J J~- - -' -

Because of the induction hypothesis, this is equal to

= u11Le+(e+ull)(sll+ ~ 61 ,S'l+ ~ sl"v" .s'l)_7- sil. 2:9:::N J J 2:;:j, j , :;:N J J J J

= uilLe+(e+ull)(e - (e+ull)-1)_7- sil= uil(e+ull) - sil= uil- sis'

Finally, if i, j ~ 1

(US)'j:::; U. si'+ ~ uij,s.,.= UilSl'+UilullSi'+v, .-S,.+ ~ u'l(e+ull)uljlS<l'~ ~s J 2<j'<N J J J J ~J 1J 2:;:j':::N ~ JJ

25

Proposition III.2 can be interpreted as meaning that R is the ring of

the rational function with integral coefficients in the (not com-

mutative) variates x. As a further justification of this terminology

we verify the following property that has some application in problems

involving probabilistic considerations.fin

Let A ~ A be the subring of the (not commutative) polynomials i.e.,

the subring of the a S Awhich have only finitely many coefficients (a,f)

different from O.

-finLet also ~ be the canonical homomorphism which sends A onto

the ring of the ordinary polynomials with integral coefficients in the

variates AX = x; A extends to Ain a natural fashion and we prove

III.3 For each r g R, Ar is a power series, converging in some domain

around zero and representing there a rational function of the variates

X = AX.

Proof: We consider the matrix E x~x = sxSX

and the ordinary polynomial det (I - s) in the commutative variates

x = AX. For small enough €/det(I - s) has its value arbitrarily close

to 1 when all the IiI are less than E. Under this condition the matrix

I + E sn= (I - s)-l= (det(I - s»-l Adj(I - s) exists and its (l,n)n>l

entry, that is Ar, is a rational function of the ordinary variates x.

Example III.2: Let us use the notation of example I.6.1 and consider

the matrix s = E~x S~. Since for any f S F and rriS E, the state

rrif is unique, the elements (so)liS Aobtained by the construction of

26

III.2.4 have only coefficients 0 or 1 and the same is true of any

sum E (so)l" Thus, by this construction we can exhibit a strictlyHall J.

algebraic form of the sum E f(~f)li= E f which is the character-iQI' f~Fa

istic function of the set F accepted by the automaton.aDefinition 3: Let 'Rreg be the smallest subset (in 'fact, the smallest

semi-ring) of Awhich satisfies the following conditions:

(i)

(ii)

(iii)

x ~ 'Rreg for any x g X and e ~ 'Rreg ;

-reg -regif a, a' ~ R then a + a' and aa' alS0 belong to R ;

if a g 'Rreg , then a*o~ 'Rreg •

Proposition III.3.1 A necessary and sufficient condition that

-reg pos posa S R is that a = E f~fl N where ~:F~ ZN and where ZN denotes

f~F '

the subset (in fact, the semi-ring) or the integral NxN matrices with

non-negative entries.

Proof: It is enough to revert to I, III.2.2 and III.2.3 and to

observe that if a, a' are produced by homomorphisms into ~os the same

ois true of a + a', aa' and a ; also trivially ~ and all the matrices

~x belong to ~os. The construction performed in III.2.4 does not use

-regsubtraction either, and consequently E f~fl,N~ R .

III.3·2 - -regR is the smallest submodule of A that contains R and any

r S 'R can be written under the form r = r' - r ll with r' ,r"~ Rreg .

Proof: Since every r ~ R can be obtained from the generat~rs x

by a finite number of additions, subtractions, multiplications and

formation of inverses it is enough to prove that if the result is

true for r l , r2~ R it is still true for r3= r

l+ r 2; r 4= r l - r 2 ;

27

r = r 1- r" where r l r l r" and r" belonD's to Rreg . We have:2 2 2 l' 2 1 2 c

r = (r'+ r')-(~'+ ~,). r = (r1+ r")-(~'+ r 1). r = (r1r1+ ~'r")-(r'~'+ ~'r')312 12' 4 1 2 12' 5 1 2 1 2 1 2 1 2

-regwhere again all the elements between brackets belong to R since

implies (e - r*)-l_ e

_ s)-l_ e.

r*= r'*- r"* and that r S Rreg111

-reg (then, r* S R since r*= e

Now, for any a, b g R with a = a*, b =b* we have

e - a+b = (e - a) (e +(e-a)-~)

= (e - a)(e+(e-a)-~)(e -(e-a)-~)(e _(e_a)-~)-l

• (e - a)(e _«e_a)-~)2)(e - (e_a)-~)-l. From this we get the

identity

(e-a+b)-l= (e-(e-a)-~)(e_(e_a)-~(e_a)-~)-l(e_a)-l

= Lre-(e-a)-~(e-a)-~)-1(e-a)-:7

- Lte-a)-~(e-(e-a)-~(e-a)-~)-.1(e-a~-:7.

Thus, taking, a = ri* and b = ri*, we can display (e - ri*+ ri*)-l as

the difference of two elements from Rreg and the result is proved.

III.4 We now revert to the proof of Kleene' s theorem for (R = { Fa1tfs A'

III.4.10 If F ,F IS 6\, then F F IS OZ.a a aa

Proof: Let Fa= {f:lJ.flN~ o} ; Fa ,= if:IJ.'fl,N'~ 0 }. We can

assume that for all f,lJ.fl,N and lJ.'flN, are both negative (cf I.7). Then,

28

if r = I: f(~f)lN and r' = I: flJ.f lN" we have (rr',f) = I: (lJ.f')l N(IJ.'f''), N'l'SF l'SF 1" 1''' =F ' .... ,

that is, (rr' ,f) F 0 if and only if there exist at least one factori­

zation l' = 1"1''' for which IJ.f'l,NF 0 and lJ.'fi,N,F o. Thus

Fcla,={f:(rrl,f) F OJ; since we know by III.2.2 how to construct

~':F~~+N' such that (zz' ,f) = ~'fl,N+N' the result is proved.

III.4.2 If F S OZ then F* S rn .a a V'\..

~: Let Fa={f:(z,fL= IJ.fl,NF O}With (z,f) ~ 0 for all 1', as

above. We can write ~= { e) U (F0:" {e1)* and consequently we can

assume that Fa dOBS not contain e. Then,as in 111.4.3 it is easily

checked that ~ = { 1': (rO ,f) F OJ and the result follows from 11I.2.3.

Let us recall that according to the definition ~' of 1.6, and the re-

ductions to simpler form carried out in the same section an automaton

a belongs tofto if and only if the associated homomorphism IJ.:F-'>~

is a ~pping into ~os. Thus Rreg= tflJ.flN

where IJ. corresponds to some

a S IJHt/lt.o .111.4.3 A necessary and sufficient condition that F SID (the familya \.1\0

of the sets of words accepted by some finite automaton) is that

Fa= tl' : (r ,f) F 0 1for some r S 'Rreg

.

~: The condition is necessary beacuse as we have seen in

example 11I.2, the sum r = I: f(;~~felOngS to 'Rre~1'l'SF ....... -~---. "----------"a

posFor proving the sufficiency we start with any IJ.:F~ ZN and we

consider the mapping ~ which sends 0 onto 0 and every positive element

29

onto 1, where:: Q and 1 are boolean elements, i.e., QQ = 2. 1 .=1 Q

=Q =Q + Q and 1 .1 = .1 =1 + Q = Q + 1 = .1 + 1j t3 is an homomorphismposof semi-ring and it can be naturally extended to ZN by defining

f3m when m g ~os as .the matrix whose entries are f3(mij ) g {Q,~} .I pos I posTrivially, for any m,m g ZN we have f3mm' = f3mf3m and f3ZN has at

N2 1most 2 < co distinc t elements. Thus, {f3lJ.f fW is a finite monoid-

M and Fa ={f:lJ.fNf 01 = \f:f3lJ.fl ,Nf Q1is the inverse image by the homo­

morphism t3IJ.:F~ M of a subset of M.

It is enough now to verify the slightly more general statement:

-1A necessary and sufficient condition that F'g 6\0 is that F' = ¢ ¢ F'

where ¢ is an homomorphism of F into a finite monoid G.

The necessity is trivial since the mapping ¢ defined by ¢f = ¢f '

if and only if crf = crt' for all states cr of the automaton is an homomor-

phism of F into a finite monoid and since F' is union of ¢-classes.

In order to prove that the condition is sufficient let E =tcrg} be

a set in one to one correspondence with G =¢F and for each xgX define

cr x = CT ¢ j the initial state is s¢ and E' =(cr l . Becauseg g x e l g J g€¢F ' =G

¢ is an homomorphism, cr f g E' is equivalent to ¢e¢f =¢fg¢F ' and thee -1

result is proved since by hypothesis F'= ¢ ¢ F. In consequence, we have

shown that(R"o={Fa1~ J+.o

•

Another simple characterization of 0<..0 is the following one:

III.4.4 A neeessary and sufficient condition that Fag~ is that

30

Fctif:(Z,f) I: °~ where r S R is such that (r,f) is bounded flDr all f.

Proof: This is just a more formalized version of the'heuristic

reasonings of I.5. If l(r,f)1 = l~fINI < K for all f we take Yp the

homomorphism that sends every integer upon its residue mod p where

p ~ K. Then, as above, Yp~F is finite and Fct t f :~f1,NI: O}..:= {f: Yp~flNl: 01is the inverse image of a subset of it.

III.5 An intuitive interpretation of

Let us consider a finite automaton with set of states E = ~ and

input alphabet Y = fyI .We assume that an initial state ~l and a Bink\mo.. s:W::};l;~tp.~11l'~\:)Y' ;;: ~o

. . ..... "....for",a,;)jJJt,·'j' ~,y:~.. nave:. b<;~n ·'tl1st1LguislJedr. To each state ~ I: ~ is

0- •.• o' t,. ~.r 0

associated a word f = Qo- ~e; in a certain output alphabet X ={x} ; it

is ~ required that Q~ I: Qo-' when ~I:~' .

Given an input word g = Yi y.•1 1.2

y. we compute successively theJ.m

.• , ~i =~ig·

mIf ~ = ~i 0

m

there exists a maximal index m'< m such that ~i I: ~ and we definem' 0

We call this process a (right) transduction Q;

in fact it is a "right coset IDaFPing."

Given a transduction, Q, we can count for each word fSF how many

distinct words g are such that Qg = f and ~lg I: ~o' This number which,

of course, can be zero is denoted by (t,f) and we have:

III.5.1 The infinite sum

( -post = E f t;f) belongs to Rf~F

Proof:

31

Let N be the number of states in E distinct from (f .,\'.o

and consider for each y S Y the NXN matrix vy with entries vYi'= Q(f.J J

if (flY =(fj and =0 otherwise; also let m =E vy. The matrix m belongsySY

to the ring ~ of the NxN matrices whose entries belong to Aand as

4 0 nabove in III.2. we can define its quasi inverse m = Em.n>l

Because

We revert to the proof

Q(f.~ e for all (Ji~ (f no word fSF of length less than n appears in1 0

n' 0m when n'> n and consequently each entry (m )i,j is a sum

aij= E f(aij,f) with finite (but unbounded) coefficients (aij,f). Again,

al . is equal to the sum of all words Qg where g satisfies the equation,J

Thus, t = E f(t,f) = .E Qg = E aljfSF gSG j'(f. g/:cr"

1 0

of III.2.4 and we verify by performing the same constructions that

any entry aij belongs to Rreg •

III.5.2 To any FaS ~, there corresponds at least one pair (Q,Q') of

transductions such that fSFa

1fand only if (t,f)~(t',f':)'

~: By definition there exists some r S R which is such that

F = f: (r,f)~Oa-posAccording to III.3.2, there exist r', r 1l S R

which are such that r = r'· r 1l and, consequently, that Fcttf:(r' ,f)~(rn,f)}

Thus, it is enough to show that to any r S RPos there corresponds at

least one transduction Q such that r =t. As usual, we verify that

this is trivially true if r = 0 or r = x S X and we perform a construction

corresponding to each of. the,threer·operations.

i) Addition

Let r = E f(t,f)jfgF

32

r' = E f(t' ,f) where t and t' correspondfgF

respectively to transductions 9:G~ F and 9' :G'~ F. We can assume

that the two input alphabetsY~or Q) and Y' (for 9') are disjoint and

also that E (\ E' =15.

Now let 'I' = y V Y' and E" = E u E' v1a~ ~where a~ is a new in!tial state j

we define a~ y to be O"lY if Y g Y and to be a1y if y g y'. We also define

the other transitions are kept and Q(]' and 9 1 a' are defined as in the

original transductions. The verification that this process gives r + r'

is trivial.

i1) Multiplication.

We use the same hypothesis as above and we take 'I' = y u Y' and

E" = E VEl. For each y g Y we define:

ay = O"y when 0" g E,

=a' when (j g E' •0

For each y' g y' we define"

ay' = a' yl for each agE - { a0 1)1

a y'= a' for agE'o 0 o 1

a'y'= aryl for each a'g E'.

Trivially again this gives rr' when the initial state is alg E.

iii) Quasi inverse

We take two copies E1 and E" of E and two copies Y' andY' of- Y.

33

All the transitions of the form {!:~ y' )~!:' or (!:ll ,!:yll)~ !:" U'~nthe

same as in (!:,Y) but we define

(j' y" =(j" y" and (j" y' = (j' y' for any (j' g !:' , (j" g !:", y' g y' and y" g. '1' ,1 1

The initial state is (jig !:l' With this process (t",f) is equal to the

sum!: f(t",f)m>l

o= r ,

~X =1

Example III, 5

Let x = txl ,x2 ~ and consider the mapping F--.;> Z4 defined by

1"1'00 1000100 0 0 000

(0 01 0); ~x2= (0 0 1 1) ,O· 0 0 0 0 0 1 0

induction shows that (~f)ll is just the number of ways of factorizing

f as a product of xl ,x2' and xlxl and, similarly,for (~f)33 with respect

to xl ,x2 and x2x2

'

any l~i, j~3; Q(jl= ~; Q(j2= X2 j Q(j3= xlxl ' Then (t,f) is precisely

equal to (~f)ll' and a similar construction gives (t' ,f) = (~f)33'

Thus F = f:(t,f) ~ (t' ,f) and, for instance,ex

REFERENCES

(1) Automata Studies, Princeton, 1956.

(2) M. Rabin and D. Scott, IBM Res. Journal 2, 1959, p.114.

(3) cf s. C. K1eene's paper in ref. (1).

(4) W. J. Harrington, American Math. Monthly, vol. 58, 1951, p. 693-696.

(5) C. c. E1got, Decisions Problems of Finite Automata Design,Univ. Michigan, 2722, 2794, 2755, 6 T, (June 1959).

(6) T. Sko1em, Comptes Rendus du 8-eme Congres des MathematiciensScandinaves. stockholm 1934, p. 163-168.

(7) M. Lazard, Anna1es Sci. Ecole Norma1e Sup (3) 72, 1955, p 299-400.

(8) K. T. Chen, R. H. Fox and R. C. Lyndon; Ann. Math. 68, 1958,P 81-95. -

I

M. P. schutzenberger, SeminaireDUbre11-Pisot Institut H. Poincare(Paris Dec. 1959).