Table Of ContentINTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
C 0 U R SES AN D L E C T U R ES No. 86
SILVIU GUIASU
UNIVERSITY OF BUCHAREST
MATHEMATICAL STRUCTURE OF FINITE
RANDOM CYBERNETIC SYSTEMS
LEGTURES HELD AT THE DEPARTMENT
FOR AUTOMATIONANDINFORMATION
JULY 1971
UDINE 1971
SPRINGER-VERLAG WIEN GMBH
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© 1972 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien-New York in 1972
ISBN 978-3-211-81174-0 ISBN 978-3-7091-2802-2 (eBook)
DOI 10.1007/978-3-7091-2802-2
PREFACE
The material aontained in these leature
notes was aovered by the author during the aourse held
at the INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
UDINE; in June-July 1971.
I am grateful to all the Authorities of
this magnifiaent Centre for giving me the opportunity
of delivering the aourse and espeaially to Professor
L. SOBRERO.
I am indebted to Professor GIUSEPPE
LONGO for his kind observations, and useful adviae.
His aharm and our disaussions will remain unforgetta
ble.
I was also impressed by the high level
of teahniaal assistanae supplied by the International
Centre for Meahaniaal Saienaes.
Udine, July 1971
Introduction
Informational models given for learning systems
( [9], [12] ) , for random automata ( [10], [12], [15] ) and fcr
systems with strategies ( [11], [12]) can be managed natural] y.
In spite of seeming dJversi ty, there is a conunon mathematical
structure for all these cybernetic systemE. . This mathematical
structure proper to a large group of cybernetic systems wi th ran
dorn behaviour will be disc.t;.sse.d fully in the following pages.
Formally, this mathematical structure does not
exceed the category theory and cn the other hand the applicatior.
of the instruments of the category theory to the study of some
deterministic cybernetic systems (essentially for deterministic
automata only) is no longer a novel ty ( see [1], [2] [6] ) . Never
theless, even if the finite cybernetic systems with randombehav
iour are categories too, it is easy to see that almest all re
sul ts of category theory are not directly applicable to the study
of these systems. The reason for this strange situation lies in
the fact that the morphisms are habi tually funetions in almest
all usual applications of the category theory (and in the appli
cations to the deterministic cybernetic systems too) whereas in
the category of finite cybernetic systems with random behaviour,
these morphisms are essentially random correspondences given by
transition matrices, i. e. by stochastic matrices. On the other
6 Introduction
hand, these randorn cybernetic systerns ind.uce rnany specific prob
lerns which do not appear in the usual category theory. The prob
lern of the replacernent of the randorn rnorphisrn l:y an E, deterrnin
istic rnorphisrn is a case in point, being the core of the codific
ations in cybernetic systerns. It concerns randorn c.orrespondence
given by a stochastic rnatrix, alrnost deterrninistic correspondence
i.e. a correspondence given by a usual function with errorsrnaller
than f . Another specific problern of the study of the processes
describes the tirne-evolution of the finite randorn cyberneticsys
terns. Such a process is cornposed of a farnily of rnorphisrns charac
terizing the whole evolution of these cybernetic systerns. Theem
ployrnentof the diagram techniques and of the instrurnents of infor
rnation theory will be fruitful in the whole ca.tegorial approach
of finite random cybernetic systems.
The contents of the paper can l:e summarized as
follows. The first chapter contains the definition of thefinite
randorn categories, abl:reviated by FR-categories, including the
rnain classes of rnorphisrns and their techniques which will be uti
lized in the paper. The secor:d chapter includes exarnples of FR
categories, narnely the noisy cornrnunication systerns without
rnemory, the finite random automata, the two-person garnes andthe
lea.rning systems. Frequent reference will be made to these exam
ples throughout the paper in order to apply the general theory
for arbitrary FR-categories, whose processes will be studied in
Chapter three. Such a process in a given FR-category contains
Finite Random Categories 7
those morphisms which characterize completely the whole time-ev
olution of the randorn cybernetic systern described cy the respec
tive FR-category. In the sarne chapter the general theory is ap
plied both to the learning process frorn an arbitrary learning
system, an~. to the process describing the evolution frorn the
random autornata as well as to the process occurring in an arbit
rary two-person game. The fourth chapter, which together with
chapter three is one of the largest sections of the paper, and
considers the problern of the reduction of one arbitrary randorn
morphisrn to an t-deterministic rnorphisrn containing applications
of the general theory to the codification in Markovian communica
tion systems and to the codification in random autornata. Applic
ations of the rnost rational algorithm of recognition to the cod
ind and decoding problern are also given in this chapter togeth
er wi th some Observations about the algebraic and probabilistic
theory of codes.
Chapter 1
DEFINITION OF FINITE RANDOM CATEGORIES
To give a category ~ it rneans to give:
a) A set Ob(t) whose elements are called the ob-
e ;
jects of
b) For every pair of objects X,Y E Ob(e) a set
Home(X,Y) {or simply Hom(X,Y) ) whose elements are called morphisms
(or arrows) from X to Y i.e. with the source X and ending Y.
In denoting the source and the ending of an arbitrary morphism
u by S(u.) and respectively E(u,), given that uEHom~(X, Y) one
gets S(u}=X and E(u)= Y. The sets Home(X, Y) are mutuallydisjoint,
i.e. every morphism has a single source and a single ending;
c) For every three objects X,Y,ZEOb(e) an appli-
cation
Hom-e(X,Y)xHome(Y,Z) __. Home(X,Z)
called the composi tion of the morphisms which associates wi th ev
ery pair of morphisms u.EHom-e,(X, Y), \)"EHomeCY,Z) one morphism
from Home(X,Z) being denoted by V"0u. or V"U..
e
Given that category is associative the cornpo-
sition of the morphisms is associative,
no matter what the morphisrns
Identical Morphism. Dual Category 9
u.EHome(X,Y) , U"EHome(Y,Z) , 'lltEHom~(Z,U).
e
Let us suppose Category to be a category with
identical morphisms if for every object XE Ob(t:) there exists a
morphism 1xEHome(X,X) called the iclentical morphism of the ob
ject X , or the identity of X , so that ix0u. = u. and v01x ='l>'
for every morphism u. with the ending X and every morphism '\t
X .
with the source
We shall denote an ar'hitrary morphism u.EHomeCX,Y)
by u.:X___.Y or frequently by
X~Y.
We sball also use .M.(e) to denote tbe set of all
e'
morphisms of the category i.e.
cM,(f) = UHome(X,Y)
where the union is taken over all objects X,Y from Ob(t). Obvious
ly, in a category with identical morphisms there is one-to-one
e
correspondence between the objects of the category and the i-
dentical morphisms X ,. ., 1x •
e e
If is a category, let the dual category of
e0
be denoted by ,defined as follows such that
a) Ob(e0) = Ob(t) ;
b) Homy:O(X,Y) = Home(Y,X)
whatever be the objects X , Y ;
10 Definition of FR - Categories
e0
c) the composi tion of tr and u. in is equal
e .
to the composi tion of u. and \1 in
A category ~ I is called a subcategory of e if
a) Ob(e') c Ob(e) ;
for every pair of objects X, Y from Ob(e1
);
c ) The composition of the morphisms in ~~I is in-
e .
dicated, is induced, by the composition of the morphisms in
e e
A subcategory of is called a full subcate-
I
gory if
Hom ~AX, Y) = Ho m oe(X, Y)
whichever be the pair of objects X, Y from Ob(e1
) •
A subcategory eI of 'e is called a rich category
if
=
Ob(e1 Ob(e) .
)
e e
Obviously, a subcategory 1 of the category
which is at the same time both rich and full will coincide with
e.
We shall say that a category is an FR-category
of ~ -~ (finite random category of ~-type) if:
a) The objects are finite sets;
