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Dokl. Akad. NaukSSSR

Tom238 (1978), No. 6

Soviet Math. Dokl.

Vol. 19 (1978), No. 1

A N E X A M P L E O F A N O N E R G O D I C O N E D I M E N S I O N A L H O M O G E N E O U S

RAND O M MEDIUM WITH PO SITIVE TR AN SITIO N PRO BAB ILITIES

UDC 519.217.1

G. L. KURDJUMOV

A one-dimensional homogeneous random medium

S^

(where

y

is the state-transition

function of the elements) is an infinite system of automata {s

(

-},where / ranges over all the

integers, operating in discrete time t = 0,1, 2, . . . . The state of autom aton s

t

at time f,

which we denote by s', is an element of a finite set X. It is determined with a certain prob

ability by the states at time t - 1 of the automata s,-_

x

, s

p

s

j+x

; nam ely, if the values of

s" are given for all i, u

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~

and indicates;the various stable states of this medium such that their outp ut trajectories

never intersect; after this one proves that if all the transition probabilities of the'medium

so constructed are made positive by adding a sufficiently small independent random noise,

then th e trajectories issuing from the stable initial states , as before, do not m eet. This en

able^ one to prove that to each stable state,corresponds itsown

v

invariant measure; under

certain additional conditions one proves linear independe nce of those states . In our case

both the deterministic medium itself and its stable initial states are very complica ted. We

will describe them simultaneously and depict all the constructions together.

Description of the construction.

The cons truction in the large is an organized structure

of infinite length with an infinite number ofhierarchies

t

of cell-like divisions . Cells of the first

level are just the elements of the

medium*

All cells of the second level are constru cted out

of those of the first level and consist of tw o parts: a zone of difference and a reference,

direc tory. Along the edges a second-level cell is borde red by auto ma ta in the br ac ke t -

state, among which a single head of a Turing machine freely moves.and^operates. The head

can never cross over the bracket of a neighbor but it can "look in there, having sent its

do ubl e across the boun dary , i.e., a similar head capable only of looking in somewhere

within the bounds of a cell and communicating about "what it sees to its master. The .

general assignment of a cell is to make a comparison with an automaton similar,to those of

which it consists itself. For th is a word is wr itten in the z one of difference of the cell en

codingthe/state of one automaton, and the operation of the head consistsin (and not only

in this, as we see in wha t follows) investigating with the aid of the doubles the contends

of the zone s of difference of its nearest neighbors, com paring the set. of three w ords ob

tained with the information in the directory, and altering accordingly the contents of its

zone of difference. The process is repea ted infinitely ma ny times.

}

In order

foi

the entire system to be resistent to noise, all information is copied no

fewer, than,three times in constructing{he cells of the second level from those of the-first.

The head and brackets are also duplicated so that no single malfunction canprevent the,nor-

mal operation of all cells; it willbe.corrected w ithout a trace at the very next instant. The

magnitude of thenoisein the medium is made small enough for the,probability of two.fail

ures in an entire cell over the whole period of its operation to, be at most halftiie probabil

ity of failure of one element per unit of time . Thus the probab ility of incorrect -functioning

of a cell of the second level is at most half the probability of.incorrect operation of a cell

of first level.

In a completely analogous way cells of the third level are constructed from those of

the second, and from those cells of the f ourth, etc. The probability of incorrec t functioning

of a cell at each (i + l)th' level is at least two times less than the corresponding probability

for a cellat th e ith level; on this account stability of the entire construction is achieved.

It should be mentioned that, according to the scheme described above, only code

failures can-be corrected, i.e., only those that do

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influence of a group of failures occurring in it itself.

In order that this kind of pathologica l aggression might not be prolonged indefinitely,

special measures are introduced, aimed at curbing the speed of any (including positive) aggres

sion and at localizing any patho logies. In particu lar, if a certain cell discovers tha t its neigh

bor on the right is,not con structed th e same as it is, the n it sends to its. destru ctor a special

signal, which attenuates at a fixed distance,equal to tw o cell lengths, erasing all inform ation

there and creating two new,cells comprising only brack ets along the bo rders a nd heads in the

appropriate states. The left o ne of the newly generated cells begins to observe the right on e,

andtlje right one to mark the time; this continues throug hout the nex t five periods. This

delay is necessary so tha t in case the aggressor is a single patholog ical cell, the left neighb or

is able to recognize the pathology in it and to destroy it before the resulting new cells ac

quire the power to oper ate actively .

-Localization of the movement of a destructor (and hence of the zone of its destruction)

is achieved by degradation of the destructor itself according to the degree of its movement.

The states of the network elements associated with the destructors are indexed; the index of

a destructor is reduced at each step of its progress. The to tal num ber of possible indices is

equal to twice the length of a cell. In order not to bring about a contradiction w hen one

element of the medium is modeled by a cell (the length of a cell depends upon the structure

of tr\emedium,,and therefore also upon the number of possible states of an element of it),

a special me thod of coding the destructive states is established. If each nond estructive state

is coded.by a word to which correspond definite places in the directory (all normal cells

have the same directory) indicating the required transitions from this state, then each state

corresponding to a destructor is coded by a word which, in the first place, is explicitly dif

ferent from all codes of the nondestructive states and, in the second place, contains the

entry- of a natura l numb er called the index of the des truc tor. The destructive state with

the largest index can be referred to in the directory not as an initial state, but as one to

which one must pass. The movem ent of a destru ctor of the highest level takes place also on

account of the process of aggression.

The result of agression is the b irth of two new cells. These cells have a definite matu ring

period. If in this time their paren t is destro yed , they remain in a rudimen tary s tate ; bu t if

not, then after completing their period they copy his transition directory for themselves.

If in this pvent the parent cell (not having been destroyed) represents a destructor (at a unit

of the highest level) not-expiring and yet removed from it by a distance (equal to two cell

lengths), then the newly created cells require aggressiveness , send ou t to their right neigh

bors a destructor and also begin to represent a destructor (at the highest level) themselves;

however, they note in their zones of difference that this destructor traverses one more step

of its path.

,If th eparent,cell .represents a destructor proceeding through itsentire,path, then the

right one of its ch ild ren begins to represent a right brack et and aggression is not resum ed.

In this case the.right one of the newly generated cells looks to the right, and if it discovers

an imm ature cell the re, it rapidly becomes aggressive and starts for its-destructor. If there

proves to be afully developed cell on the right of the newly created dells, then the righ t

one of those newly created compares the structure with its own. If coincidence is.detected,

then the newly generated cells acquire full deve lopm ent: the y become aggressive , fill ou t

their zones of difference, and the correction is completed in this cycle.

i

213

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In case the newly created cells, having copied the directory of their parent, do not de

tect-similaritybetweenthemselves-and their'fully developed right neighbor, they- freeze ,

i.e.,they'do not take on aggressiveness, and their zones of difference remain uncompleted.

If after this the double of a higher level is set in motion al6ng the cells in the form of a

singular wave from left to right and is driven in to the pair of fro ze n cells, then it vanishes;

from this itsmasterthe metacell'-on the left makes an inference about the necessity of

destroying all the corresponding metacells: In order for a*single failure not to be able to cause

destruction of acell, while any failures inside

j

a single cell would not'bring about destruction

of a metacell, etc., the destructors are duplicated inthe'sameway that all of the other'useful

information in the cells is duplicated.

The lastquestionthat of "search"is the question of recognition by the'cells of un-

coded malfunctions in their neighbors. This ques tion is resolved as follows: the double sent'

into the territory of any cell sees not what is going on there at the present mo me nt but

what has taken place immediately before, just as in self-photographya singular operatibn

which the head can undertake and during whi&h information contained in allthe-working

elements of the cell is transferred 'to special trace registers,but'thednformation contained in

the head itself is'lost Self-photography is performed each time after'the" conclusion of the

regular working cycle. It iSno t difficult to see tha t iri such a systemho'overlooked devia

tion passes except those which quickly vanish of their own accordeitherbecauseof*their

special nature or because of repeated failures.

We still require that each element of the'mediumstore the additional information 0 or

1, and in case at least two elements differing by this factor from all the remaining elements

of the cell ariseinside'the ceil, the head would die out (which would lead to the'destruction-

of the whole cell).

'*

We now estimate the admissible value-of the' noise. Let

T

be the duration tof a period,

L

the length of a cell (of first level),

p

th evalue'of the noise or probability of failure in a

single element at-one-instant. The*probabilityp

LT

of two failures in'a single cell during one

period is bounded, obviously, by the expressionp

LT