Table Of ContentSYSTEMS THEORY RESEARCH
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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 feed
backs - 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 ~ 1P 2 • The set rol ~ 1P 2 is called l-c o m p 1 e t e if by means of "synchronous s uperposi
tion" 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 inter
esting properties.
In particular, it turned out that in the general case not every class in 'Pz can be ex
panded 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. How
ever, 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 Z
complete system, and that the power of the set of closed classes in 1P 2 is equal to the con
tinuurn. 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 1h 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, r1) and (cp, r2) tobe equal and denote
that fact by (j, T 1) = (cp, r2) 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 in
troduce 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(x1, ••• , xi-t• xi, xi+t• •.. , xn), T) is ob
tained from the pair (1) by means of synchronous-superposition oper
ations (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),
(2)
i<.ii<.n for 1-<.j<.r and i <i < ... <i,.
1 2
Thenthe pair (f(G1, G2, ••• , Gn), r), whereGkdenoteseitheravariableora
functiongk,k=l, ••. ,n, is obtained from (2) by means of synchronous
superposition 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 synchronous
superposition 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
P2• Second, we wish to emphasize the fact that operation 3 together with the substitution öf
some pair of the form (cp, r 2), r2 > 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 1P 2• 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.
m
The set is called closed ü lm=[lm].
m
Definition. Theset iscalledl-complete if [lmJ containsallpairshaving
delays equal to l.
Definition. The set 1m is called l-precomplete if
m-
1) 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, ••• , x11) 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;
o
L:l all -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;
A all roonotonic functions;
1
D all self-dual functions;
3
L alllinear functions;
1
Y all even functions;
A all functions which are negations of roonotonic functions;
1
D allself-dual a-functions;
1
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~);
4'
F 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);
A all roonotonic a-functions and ß -functions;
2
A all roonotonic a-functions and y -functions;
3
A all roonotonic a-functions;
4
D allself-dual roonotonic functions;
2
0 all functions equal to x, and all functions derived froro it by redesignating the
1
variables without identification;
02 all functions equal to the function 1;
03 all functions equal to the function O;
6 L. A. BIRYUKOVA AND V. B. KUDRYAVTSEV
x,
04 all functions equal to the functions x or and all functions derived from them
by redesignating the variables without identüication;
05 all functions equal to the functions 1 or x, and all functions derived from them by
redesignating the variables without identüication;
Os all functions equal to the functions 0 or x, and all functions derived from them
by redesignating the variables without identification;
07 all functions equal to the functions 0 or 1;
Oa all functions equal to the functions 0, 1, or x, and all functions derived from them
by redesignating the variables without identification;
x,
09 all functions equal to the functions 1, 0, x, or and all functions derived from
them by redesignating the variables without identüication;
st alllogical sums;
Sa all logical sums and all functions equal to 1;
s5 alllogical sums and all functions equal to O;
Sa alllogical sums and all functions equal to 0 or 1;
p1 all logical products;
Pa alllogical products and all functions equal to O;
p5 alllogical products and all functions equal to 1;
Pa alllogical products and all functions equal to 0 or 1;
L2 all linear a- and ß -functions;
La all linear a- and y-functions;
L4 alllinear a-functions;
L5 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, co
incides 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 de
note 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 mc 1P 2. 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; 1P 2, 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 con
venient 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.
