Table Of ContentEATCS
Monographs on Theoretical Computer Science
Volume 7
Editors: W. Brauer G. Rozenberg A. Salomaa
Advisory Board: G. Ausiello S. Even M. Nivat
Chr. Papadimitriou A.L. Rosenberg D. Scott
EATCS Monographs on Theoretical Computer Science
Vol. 1: K. Mehlhorn: Data Structures and Algorithms 1: Sorting
and Searching. XIV, 336 pages, 87 figs. 1984.
Vol. 2: K. Mehlhorn: Data Structures and Algorithms 2: Graph
Algorithms and NP-Completeness. XII, 260 pages, 54 figs. 1984.
Vol. 3: K. Mehlhorn: Data Structures and Algorithms 3: Multi
dimensional Searching and Computational Geometry. XII,
284 pages, 134 figs. 1984.
Vol. 4: W. Reisig: Petri Nets. An Introduction. X, 161 pages,
111 figs. 1985.
Vol. 5: W. Kuich, A. Salomaa: Semirings, Automata, Languages.
IX, 374 pages, 23 figs. 1986.
Vol. 6: H. Ehrig, B. Mahr: Fundamentals of Algebraic Specifica
tion 1. Equations and Initial Semantics. XI, 321 pages. 1985.
Vol. 7: F. Gecseg: Products of Automata. VIII, 107 pages, 18 figs.
1986.
Ferenc Gecseg
Products of Automata
With 18 Figures
Springer-Verlag
Berlin Heidelberg New York Tokyo
Editors
Prof. Dr. Wilfried Brauer
Institut fur Informatik, Technische Universitiit Munchen
Arcisstr. 21, 8000 Miinchen 2, Germany
Prof. Dr. Grzegorz Rozenberg
Institute of Applied Mathematics and Computer Science
University of Leiden, Wasscnaarseweg 80, P.O. Box 9512
2300 RA Leiden, The Netherlands
Prof. Dr. Arto Salomaa
Department of Mathematics, University of Turku
20500 Turku SO, Finland
Author
Prof. Dr. Ferenc Gecseg
J6zsef Attila Tudomanyegyetem Bolyai Intczctc
6720 Szeged/Hungary, Aradi v. tere I
ISBN-13: 978-3-642-64884-7 e-JSBN-J3 :978-3-642-616 JJ -2
DOl: 10.1007/978-3-642-61611-2
Library of Congress Cataloging in Publication Data.
Ge.:seg. f.
Products of automata.
(EA TCS monographs on theoretical computer science: v. 7)
Bibliography: p.
I. Sequential machine theory. I. Title. II. Series.
QA267.S.s4G38 1986 006.3 85_27900
ISBN-I 1: 978-3-642-64884-7
This work is subject to copyright. All rights are reserved. whether the whole or part of the
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banks. Under § 54 of the German Copyright Law where copies aT" made for other than private
use a ree is payable to ··VerwerlUngsgcsellschart Wort··. Munich.
10 Springer_Verlag Berlin Heidelberg 1986
Solleo," rtprinl ofltlc hardc(I>"er 151 edilioo 19S6
Bookbinding: Liideritz & Bauer. Berlin
2145/3020-543210
Preface
Both theoretical and practical considerations motivate the repre
sentation of objects as certain compositions of simpler ones. In the
theory of automata this observation has led to the concepts of pro
ducts and complete systems of automata.
In the general form of the products of automata all the component
automata are fed back to one another. With this very broad notion
of products, the realization of automata with large numbers of states
by means of compositions of basic components is a highly involved
process; this increases the possibility of errors. In order to decrease
the complexity of feedbacks, a hierarchy of products called lXi-pro
ducts was introduced some 10 years ago, where i runs over the set
of all non-negative integers. In an IXcproduct the index set of the
component automata is linearly ordered. The input of each automaton
in the product may depend on the states of all automata preceding it,
i.e., all component automata steer all those automata which follow
them in the product. Furthermore, at most the next i-I automata
(including itself) may be fed back to the input of a given component
automaton. Thus for iXcproducts the lengths of feedbacks are at most i.
The aim of this monograph is to give a systematic account of
iXi-Products. It consists of five chapters, a reference section, and an
index. The first chapter contains the necessary concepts and results
from universal algebra, automata, and sequential machines. Chap
ters 2 and 3 deal with homomorphic and isomorphic representations,
respectively. Chapter 4 is devoted to homomorphic and isomorphic
simulations with respect to a generalization of iXcproducts. The final
chapter is concerned with infinite products and representations of
automaton mappings in finite lengths.
I am grateful to the editors of the EATCS Monographs on Theo
retical Computer Science for including this book in the series. I wish
to thank my colleagues P. Domosi, Z. Esik, Gy. Horvath, B. Imreh,
and J. Vi nigh who have read the book, and who have made several
helpful suggestions and corrections. Z. Esik and Gy. Horvath also
helped in checking the final version of the manuscript. I am indebted
to P. Folberth for the excellent typing of the manuscript. The difficult
task of designing and drawing the figures was performed by 1. Adam
kovich and S. Koczkas; I am grateful to them as well. Finally, I have
to express my deep gratitude to my wife for her loving support.
Szeged, May 1986 Ferenc Gecseg
Contents
1. Basic Concepts and Preliminaries
1.1 Sets and Relations. . . . . 1
1.2 Algebraic Structures. . . . 3
1.3 Automata and Sequential Machines 9
1.4 Products and Complete Systems of Sequential Machines
and Automata . . . . . . . . . . . . . . . . .. 14
2. Homomorphic Representations 23
2.1 A Homomorphically Complete System for the lXo-Product 23
2.2 A Minimal Homomorphically Complete System with
Respect to the lXo-Product. . . . . . . . . . . . . . 27
2.3 Homomorphic Representations of Automata by lXo- and
lXI-Products of Smaller Automata . . . . . . . . . . 29
2.4 Homomorphically Complete Systems for Q(cProducts with
i > 1 . . . . " ............... 31
2.5 Comparison of the Homomorphic Representation Powers
of lXi-Products. . . . . . . . . . . . 47
2.6 Homomorphically Q(cSimple Automata. 55
2.7 A Decidability Result . . . . . . . . 60
3. Isomorphic Representations. . . 61
3.l Embedding into IXcProducts of Automata with Fewer
States Than a Given Integer. . . . . . . . . . . . . 61
3.2 Isomorphically Complete Systems for the lXo-Product. . 66
3.3 Isomorphically Complete Systems for lXi-Products with
i ~ 1 . . . . . . . . . . . . . . . . . . . . . . 67
3.4 Comparison of the Isomorphic Representation Powers
of lXi-Products. . . . . . . . . . . . . . . . . . . 69
3.5 Isomorphically Complete Classes for Nilpotent Automata 71
VIII Contents
4. Generalized Products and Simulations 75
4.1 BasicConcepts . . . . . . . . 75
4.2 Simulations by Generalized (Xo-Products 77
4.3 Simulations by Generalized (Xl-Products. 79
4.4 Simulations by Generalized Products and Generalized
(Xi-Products with i > 1 . . . . . . . . . . . . . . . 81
4.5 Homomorphic Representations by a Restricted Form of
Generalized Products . . . . . . . . . . . . . . . 86
5. Representation of Automaton Mappings in Finite Length. Infinite
Products. . . . . . . . . . . . . . . . . . . . . . . 93
5.1 Metric Completeness. . . . . . 93
5.2 Equational Classes of Automata. 95
5.3 Metric Equivalence of Products . 98
Bibliographical Remarks . . . . . . . . . . . . . . . . . 101
References . . . . . . . . . . . . . . . . . . . . . . . 103
Subject Index . . . . . . . . . . . . . . ...... 105
1. Basic Concepts and Preliminaries
In this chapter we survey some basic concepts and results from the theories of auto
mata and universal algebras. It is presumed that the reader has a certain routine in
the structural theory of automata and universal algebras. Thus, the results of this
chapter are stated without proofs. The only exceptions are certain special group
theory results and two statements concerning products of automata, the constructive
proofs of which will be needed in our later discussions.
1.1 Sets and Relations
We shall use the following concepts and notations from elementary set theory.
Sets will generally be denoted by upper-case Latin letters with or without indices
and their elements by the corresponding lower case Latin letters.
If there is no danger of confusion, then we simply write a for the one-element
set {a}.
o
denotes the empty set.
IAI is the cardinality of the set A.
a E A means that a is an element of A. The opposite case is expressed by a ¢ A.
A ~ B means that A is a subset of B. Proper inclusion will be denoted by A c B.
A - B denotes the complement of B with respect to A, i.e. A - B consists of all
elements of A which are not in B. We sometimes write A \ B for A-B.
Let (Ai liE 1) be a family of subsets of a certain set indexed by the elements of [.
Then (l (Ai liE 1) stands for their intersection, and u (Ai liE 1) is their union. If [
is finite, say [= {I, ... , k}, then we frequently write Al (l ... (l Ak and Al U ... U Ak
for (l (Ai liE 1) and U (Ai liE 1), respectively.
We sometimes define a set A as the collection of all elements a satisfying certain
properties PI' ... , Pk• For such A, we use the notation
A = {a I a satisfies PI' ... , a satisfies P k} .
For two sets A and B, let A xB = {(a, b) I a EA, b E B}. A subset (] ~ A x B is a
(binary) relation from A to B. If (a, b) E (] for a pair (a, b) E A x B, then we also write
a == b«(]). The opposite case will be expressed by a 'i= b«(]), too.
2 1. Basic Concepts and Preliminaries
If (! ~ A x B is a relation, then its converse is the relation (! -1 from B to A, defined
by
(!-1 = {(b, a) I (a, b) E (!} .
Take two relations (! ~ A x Band ! ~ B x C. The product of (! by ! is the relation
(!! from A to C for which (a, c) E (!! if and only if there is abE B with (a, b) E (! and
(b, c) E!.
If (! ~ A x A, then we say that (! is a relation on A. The following two special
relations on a set A will frequently be used:
'A
= {(a, a) I a E A} (diagonal relation on A), and
W A = {(a, b) I a, b E A} (total relation on A) .
For any relation (! on a set A and non-negative integer n, we define the power (!n by
The relation (! ~ A x A is said to be
(i) reflexive if I A ~ (!,
(ii) symmetric if (! -1 ~ (!,
(iii) antisymmetric if (! n (!-1 ~ 'A' and
(iv) transitive if (!2 ~ (!.
A relation on A which is reflexive, symmetric and transitive is called an equivalence
relation on A. If (! is an equivalence relation on A, then for every a E A we set
al(! = {blbEA,(a,b)E(!}.
This notation is extended to an arbitrary subset B of A by
BI(! = {bl(! IbEB}.
The cardinality of AI (! is called the index of (!.
Let (! and! be equivalence relations on a set A with! ~ (!. Then, for every a E A,
we set a[(!I!] = {bl! I b E al(!}.
A partition Qf a set A is a set n of pairwise disjoint nonvoid subsets Ai (i E l) such
that u (Ai liE l) = A. Each Ai (i E l) is called a block of n. For every a E A, n(a)
will denote the block of n containing a. It is well known that, if (! is an equivalence
relation on A, then AI (! is a partition of A, and every partition of A can be given
in this way.
A reflexive, antisymmetric and transitive relation (! on a set A is a partial ordering
on A. It is easy to see that the converse of a partial ordering is also a partial ordering.
For two partitions n1 and n2 of a set A, let us write n1 ~ n2 if and only if each block
of n1 is contained by a block of n2• Then ~ is a partial ordering on the set of all par
titions of A.
1.1 Sets and Relations 3
A partial ordering g on a set A is a linear ordering if a == b(g) or b == a(g) for
arbitrary two elements a and b of A. We can easily prove
Theorem 1.1. Let g be a partial ordering on a finite set A. Then there exists a linear
ordering r on A such that g s r. 0
A mapping or function from a set A to a s,et B is a relation cp s A x B such that,
for every a E A, there is exactly one bE Bsatlsfying a == b(cp). If cp is a mapping from
A to B, then we usually write cp : A ~ B. The fact that a == b(cp) is also expressed by
cp(a) = b or cp: a ~ b. If cp(a) = b, then b is the image of a, and a is an inverse image
(or counter image) of b under cpo We extend the notation cp(a) to an arbitrary subset
A' s A by cp(A') = {b I b = cp(a) for some a E A'}. If B' s B, then cp-1(B') denotes
the subset of A consisting of all a E A for which cp(a) E B'. A mapping cp : A ~ B
is onto if cp(A) = B. It is one-to-one if different elements from A have different images.
The composition or product of two mappings cp : A ~ Band ljJ : B ~ C is the pro
duct cpljJ of cp and ljJ as relations. Clearly, cpljJ is a mapping from A to C.
The restriction ofa mapping cp : A ~ B to a subset A I S A is the mapping cp I A I : A I
~ B defined by cp I A' = cp 11 (A' x B). If a mapping ljJ is obtained by a restriction
of cp, then cp is an extension of ljJ.
Let (Ai liE /) be a family of sets. The Cartesian product A of Ai (i E l) is the set
of all mappings cp : I ~ U (Ai liE /) such ntha t, for every i E I, cp(i) is in Ai' For this
Cartesian product we use the notation A = (Ai liE /). If lis finite, say I = {I, ... ,k}
(k ~ 0), then we also write A = A1 X .•• X Ak• In this latter case it is convenient to
think of the elements of A as k-tuples a = (a1, ... , ak) with ai E Ai (i = 1, ... , k).
If k = 1, then we generally write a for (a) (a E A1). Moreover, if Ai = B for each
i( = 1, ... , k), then A is also called then kt h Cartesian power of B; in notation, A = d.
Take a Cartesian product A = (Ai liE /). For each i E I, we define the ith
projection pri: A ~ Ai by pri(cp) = cp(i) (cp E A). If cp and ljJ are two elements from A,
and J is a subset of I, then the notation pri cp) = pri ljJ) means that prlcp) = pri(ljJ)
for every i E J.
In the sequel the elements of a Cartesian product will generally be denoted by
lower case Latin letters with or without indices. N will stand for the set of all non
negative integers. Further, for every integer k ~ 0, [k] denotes the set {I, ... , k}.
If k and n are positive integers and i E N, then n + i (mod k) is the least positive
residue of n + i modulo k. Finally, lcm [m1' ... , mk] will denote the least common
multiple of the integers ml' ... , mk•
1.2 Algebraic Structures
We start with some general concepts and results of universal algebras needed in the
later chapters. °
Let A be a nonvoid set and m ~ an integer. A mapping (J : Am ~ A is called an
m-ary operation on A. The pair d = (A, :rd ), where :rd is a set of operations on A,
is termed a (universal) algebra.
