LIBRARY OF THE 
 
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 AT URBANA-CHAMPAIGN 
 
 510.84 
 
 l^6T 
 
 no. 188-199 
 
 cop.3U 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/generalizeddecom194puar 
 
,v Report No. I9U 
 If 
 
 GENERALIZED DECOMPOSITION OF INCOMPLETE FINITE AUTOMATA 
 
 by 
 Arthur Ta-shiang Pu UMUBfll B> ILUWIK 
 
 December 30, 19&5 
 
 Aug i'y w>° 
 
 LIBRARY 
 

Report No. I9U 
 GENERALIZED DECOMPOSITION OF INCOMPLETE FINITE AUTOMATA 
 
 by 
 Arthur Ta-shiang Pu 
 
 December 30, 1965 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 61803 
 
 This work was submitted in partial fulfillment of the requirements for 
 the degree of Doctor of Philosophy in Mathematics, December, 1965 . 
 
Ill 
 
 ACKNOWLEDGMENT 
 
 The author wishes to express his deepest apprecia- 
 tion to his advisor Prof. David E. Muller of the Department 
 of Mathematics for his inspiring guidance, patience, and en- 
 couragement during the preparation of this thesis. 
 
 He is grateful to the Office of Naval Research for 
 their support under the contract No. NONR-1&34 (27), during 
 the early period of this investigation, and to the Department 
 of Computer Science for their continuing interest and support. 
 
 Finally, the author wishes to thank his wife, Lena 
 Chang Pu, for her unconditional support, encouragement, and 
 understanding. 
 
IV 
 
 TABLE OF CONTENTS 
 
 Page 
 
 INTRODUCTION 1 
 
 Chapter I. PAIR DECOMPOSITION 5 
 
 §0. Preliminaries 5 
 
 §1. Generalized Pair Decomposition 6 
 
 §2. ^-covers and Generalized Pair Decomposition 
 
 Triplet 11 
 
 Chapter II. PROPERTIES OF GENERALIZED PAIR DECOMPO- 
 SITION . ' 27 
 
 §1. Algebraic Properties of Congruence 
 
 Relations of Degree k 28 
 
 §2. Congruence Relations of Degree k as 
 
 Applied to Decomposition Theory 35 
 
 Chapter III. SET SYSTEMS AND *-C0VERS 46 
 
 §1. ^-covers with Substitution Property ... 46 
 
 §2. ^-covers vs. Set Systems 50 
 
 Chapter IV. MULTIPLE DECOMPOSITION 56 
 
 SUMMARY 64 
 
 BIBLIOGRAPHY 68 
 
 VITA 70 
 
GENERALIZED DECOMPOSITION OF 
 
 INCOMPLETE FINITE AUTOMATA 
 
 Arthur Ta-shiang Pu, Ph.D. 
 
 Department of Mathematics 
 University of Illinois, 1966 
 
 The problem of decomposing a finite automaton has 
 been investigated by many authors. However, their results 
 were based on the question of decomposing an automaton into 
 series and parallel connections of automata. The present 
 work is an extension to the problem of generalized decomposi- 
 tion where two-way interconnections between automata are per- 
 mitted. We are mainly concerned with decompositions for deter- 
 ministic incomplete finite automata. Our decomposition does 
 not presuppose the logical design of the circuit of an automaton. 
 
 With the new technology, the problem of economical 
 realization no longer lies in the actual complexity of the 
 logical design in each building block. Aside from a given 
 upper limit, the complexity is not reflected in the cost. Sub- 
 ject to the restraint of the given limit on each block, our main 
 object is to minimize the number of interconnections between 
 blocks of a generalized decomposition of an automaton. 
 
 We first consider decomposing a given automaton 
 M = <S,T,f> into a pair of automata M 1 = <S 1 ,T 1 ,f 1 >, 
 
 M 2 = ^ S 2 ,T 2 ,f 2^' We sive the definition of a generalized pair 
 decomposition (GPD) [M,:!^] of M. We define a --cover 
 0C = (A-., ...,kd) of S as an /-tuple such that for each 
 
 / 
 i€ 1 1, ...,/>, A i c S and U A j _ = S. We show that each GPD 
 
 of M gives rise to two ^-covers Ct, p of S. We then establish 
 
a necessary and sufficient condition for a pair of ^-covers 
 (#,/8) of S to be obtainable in this manner from a GPD of M. 
 
 In discussing the properties of a GPD (M-^Mg) of 
 M, we show that all properties are represented by (K«:Kj as 
 a partition pair decomposition of an automaton M r =<S T ,T,f t>, 
 where M is a homomorphic image of M T . We introduce the notion 
 of input configurations between component automata of a GPD 
 and define the notion of connecting lines in terms of these 
 input configurations. We deduce that to find the number of 
 input configurations between component automata is actually 
 a covering problem. In each GPD [M, :1VL] of M, we obtain a 
 lower bound k for the number of input configurations from one 
 component to the other. This lower bound k, obtained from 
 certain properties of an equivalence relation, is much easier 
 to compute than the solution of the covering problem. We then 
 show that for certain GPDs of M, the number of input configura- 
 tions from one component to the other actually achieves its 
 lower bound k. 
 
 We then prove that the known results for series and 
 parallel decomposition of an automaton are included in our re- 
 sults as special cases. We also raise the question of whether 
 it is sufficient to consider just set systems in the general 
 theory of decomposition for automata. 
 
 Finally, we extend our theory of generalized pair 
 decomposition to generalized N-tuple decomposition (GND). We 
 show that any GND of M can be obtained from successive pair- 
 wise decomposition. 
 
INTRODUCTION 
 
 The theory of decomposition for finite automata has 
 been investigated by many authors. Hartmanis f7>8,9), Stearns and 
 Hartmanis [10J have published a series of articles using parti- 
 tions with substitution property. Probing through the properties 
 of semigroups, Krohn and Rhodes (15) obtain some very interesting 
 results. Then Kohavi (14J and Hartmanis and Stearns (12 J intro- 
 duce the notion of set systems to take advantage of the state 
 splitting. However, their results and the results of others 
 (4»22,23j were based on the question of how to decompose an 
 automaton into series or parallel connection of smaller automata. 
 The information is not allowed to interflow between the component 
 automata. 
 
 The question then arises as how does one consider de- 
 compositions for automata when information is allowed to inter- 
 flow between the component automata? This is a very important 
 question. With the new technology, the problem of economical 
 realization no longer lies in the complexity of logical design 
 in each building block. Rather, they tend to depend on the num- 
 ber of connections between basic building blocks. The problem 
 arising from future practical design thus will clearly emphasize 
 the importance of having smaller numbers of connections. 
 
 With this in mind, the present work begins as an 
 attempt to study the generalized decompositions of an automaton 
 
 We consider decomposing an automaton before it is 
 actually constructed. This has the fundamental advantage that 
 one may make adjustments accordingly in the actual designo 
 
allowing interconnections. We will be mainly concerned with 
 decompositions for a deterministic incomplete finite automaton. 
 Our problem is to find, for a given incomplete automaton M, a 
 generalized N-tuple decomposition C M j_J l^i^N such that ^ or eacn 
 i, M. has a complexity equal to or less than a given bound, and 
 furthermore, the total number of interconnecting lines is a 
 minimum over all such decompositions. Our results will include 
 the known results for series and parallel decompositions as 
 special cases. 
 
 In Chapter I, we first give the definition of gen- 
 eralized pair decomposition (GPD) (M-, rNL) of a given automaton 
 M = <S,T,f>, and the definition of generalized pair decomposi- 
 tion triplet (GPDT) < S^ S 2 >f>of M. We show that each 
 GPD(M-, :M 2 ] of M naturally induces an automaton G(M-,,M 2 ) re- 
 alizing M. We then introduce the notion of,*-cover of a set 
 S. A *-cover of a set S is an /-tuple OH = (A 1 , • • «,A^), / - ({CC) 
 
 such that (Vi)(l*i*/) ( A. £ Sj and UA. = S. We show that 
 
 1 i=l x 
 
 each GPDT <Sp S 2f f) of M gives rise to a pair of ^-covers 
 ${»&)• We then proceed to obtain a necessary and sufficient 
 condition for a pair of ^-covers (tf?,jfi) of S to insure the 
 existence of a function ^f such that <E^,EU,^> is a GPDT of S. 
 E /i> E are tne index sets of 6t,"& respectively. This work 
 uses the added deleted distributive property. 
 
 In Chapter II, we introduce the definition of an 
 incomplete monadic algebra fl = <S,T,0^ and the definition of 
 a congruence relation of degree k on a. . We show that any in- 
 complete, deterministic automaton can be considered as an in- 
 complete monadic algebra, and conversely. We then find that 
 the structure of a GPD (N^.-jy of M - <S,T,f>, where 
 
M-l = <S 1 ,T 1 ,f 1 >, M 2 = <S 2 ,T 2 ,f 2 >, must be studied through the 
 structure of (M 1 :M 2 ) as GPD of an automaton M ff = <S«,T,f»> 
 where M is a homomorphic image of M», and S-,, S ? induces con- 
 gruence relations of degree k-,, k 2 respectively. We introduce 
 the notion of input configurations between M-, and M ? and show 
 that k-,, k 2 are lower bounds on the number of input configura- 
 tions from M 2 to M-, and from M-, to M 2 , respectively. We show 
 that to find the number of input configurations is a covering 
 problem. 
 
 Suppose ^ is a *-cover of S and there exists an 
 automaton M« =<S 9 ,T,f»> such that M» realizes M under "f . 
 Suppose there also exists a non-trivial congruence relation 
 9-q of degree k, on M* such that 9-,-, induces Gt on S under ^ . 
 We show that for any 22 on M» such that 0,,/^ 9 22 = 0, 9,,, 
 9 22 gives rise to a GPD (M, :M 2 ) of M. Finally., we show that 
 there exists a non-trivial equivalence relation 9 22 on S 9 such 
 that if Cm i :M 2 ) is the induced GPD of M under 9-^ and 9 22 , then 
 the number of input configurations from M 2 to M-, is exactly k-^. 
 
 In section 1 of Chapter III, we show that the known 
 results for series and parallel decompositions of an automaton 
 are included in our results as special cases. A set system 
 ^£ = (A-,, ..o,A ) of S is a special *-cover of S in which 
 (Vi,j)f(A. £ A.) £z> (i = j)J. In section 2, we exhibit examples 
 to show that in a series or parallel decomposition it is not 
 always possible to replace the pair of ^-covers by its induced 
 set systems such that the resultant is still a series or parallel 
 decomposition. We then raise the question whether it is suffi- 
 cient just to consider set systems for decompositions of an 
 automaton. 
 
Finally, Chapter IV is the extension of the theory 
 of GPD to the multiple case. We make modifications on the 
 definitions and show that the results of pair decomposition 
 do carry over to the multiple decomposition. We then show 
 that any generalized multiple decomposition of an automaton 
 can be obtained by successive usage of GPD. Thus, any dis- 
 cussion of multiple decomposition actually can be made through 
 the discussion of pair decomposition. 
 
Chapter I 
 PAIR DECOMPOSITION 
 
 We first introduce the following notations « Let E 
 be a set, then ^(E) = (EjE^f9 E). Let (A^l^n} be a col- 
 li 
 lection of arbitrary finite sets of symbols. Let TJ k< denote 
 
 i=l 1 
 
 the cartesian product of A.. For each i, 1-i-n, define 
 
 o n 
 7 ^_(a 1 , o o »*a n ) = a ± € A i . For each J £ J[ k ±s 
 
 ^±(J ) =(^ i (a 1 ,.- J a n )|(a 1 ,...,a n )y }. If T is" a finite 
 set s then denote by T* the free semigroup of all finite 
 strings formed under concatenation The empty string e is the 
 identity element in T*. 
 
 §0. Preliminaries 
 Definition 0»1 . An automaton M (non-deterministic, incom- 
 pletes finite) is a triplet <S,T,f>, where S is a finite 
 set of "states", T is a finite set of "input letters", 
 and f is a "transition function" mapping SXT into trlS). 
 Let D(f ) = { (s,t)|f (s,t)^0). Let s€ S and 
 s ^ f^i(D(f)) U f(D(f))}. Then (i) (£ t€ T)(f (s,t) / 0), (ii) 
 (^(s',t)€SxT)(f(s f ,t) = s). Therefore, we may choose the set 
 S to consist of just the states in f*^(D(f}) U f(D(f))}, and 
 the set T to consist of just the input letters in ^(D(f))o 
 In fact, D(f ) is the normal notion for the domain of the func- 
 tion f o 
 
 The transition function f of an automaton can be 
 
extended to the domain SxT* by the following recursive rules. 
 
 If we denote this extended function again by f, then f(s, e) = s 
 
 and (Vt e T*)(Vt,€ T)ff(s,tt 1 ) = ^ f(s',t n )}. 
 
 1 l - 1 s'ef(s,t) x J 
 
 Definition 0.2 . A deterministic automaton M = <S,T,f > is an 
 automaton such that V(s,t)eD(f), f(s,t) is a singleton. 
 
 Definition 0.3 » A complete automaton M = <S,T,f) is an auto- 
 maton such that D(f) = S x T. 
 
 Definition 0.4 « An automaton M 2 = <S 2 ,T 2 ,f 2 > is a subautomaton 
 of M-l = <S 1 ,T 1 ,f 1 > if and only if T 2 Q T ±t S 2 Q Sp and 
 (V(s,t)€ s 2 x T 2 )(f 2 (s,t)£ f^s^t)). 
 
 Throughout the context of this dissertation by an 
 
 automaton we shall always mean a deterministic incomplete 
 
 finite automaton, unless otherwise stated. 
 
 Definition 0. 5 ° An automaton 1VL = ^S 2 ,T 2 ,f 2 > "*" s a nomotnor Phi c 
 image of an automaton M-, = < S-, , T-, ,f, > if and only if 
 there exist an onto function /'iS-, — > S 2 , and an onto 
 function h:T-, — > T 2 such that 
 (V(s 1 ,t 1 )6(S 1 xT 1 ))C/ 7 (f 1 (s 1 ,t 1 )) = f 2 (^(s 1 ),h(t 1 ))J. 
 
 Definition 0. 6 . An automaton M, realizes another automaton M 2 
 if and only if M 2 is a homomorphic image of a subauto- 
 maton of M-j. 
 
 In the case when f and h are both one to one onto 
 
 mappings, M, and NL are isomorphic. 
 
 SI. Generalized Pair Decomposition 
 Definition 1.1 . Let M = < S, T, f> be a deterministic, incomplete 
 automaton. Let M- L = <S- L ,T 1 ,f 1 >, M 2 = <S 2> T 2 ,f 2 > be two 
 incomplete automata. Then (IYLtmJ is a generalized pair 
 
7 
 decomposition (GPD) of M if and only if 
 
 (i) There exists a function f defined on a subset 
 J 's S ± x S 2 onto S such that ^ ± ( [J ) = Sp, 
 
 (ii) There exist two one to one functions hp h ?5 where 
 
 h l :T l — "* T * S 2' h 2 :T 2 ~~ ~* T * S l° 
 (iii) D(f 1 ) 5 {(s 1 ,t 1 )|(3t€T)fh 1 (t 1 ) = (t,s 2 )J 
 
 (C^s 1 ,s 2 ),tj€ D(f)J} 
 
 D(f 2 ) 3 {(s 2 ,t 2 )| (3t6T)fh 2 (t 2 ) = (t,s)] 
 
 (C/ > (s 1 ,s 2 ),tj€ D(f)]} 
 
 (iv) (V(s 1$ s 2 )eJ)m){(f(s v s 2 ) $ t))€ D(f)J = 
 
 ~> 
 
 f(f 1 (s 1 ,h~ 1 (t,s 2 )),f 2 (s 2 ,,h 2 1 {t 5 s))J 6 
 
 T 
 
 ^" 1 (f(^(s 1 ,s 2 ),t)Jj 
 
 (v) Suppose (spSpl^Dip and ^Sp s 2 ) = f(s, t) for some 
 s € S 5 t € T. Then 
 (3(4,s^f 1 (s))( (s 1 ,s 2 ) = (f 1 (s{,h- 1 (t,sj)), 
 
 f 2 (sJ,h 2 1 (t,s]_)J). 
 
 Let (M 1 :M 2 ) be a GPD of M. Define cQc S-^ s 2 
 such that (s-,, s 2 )€c© if and only if either 
 
 (i) JP(s lf B 2 )€ f(D(f)), or 
 (ii) (3t€T) (Cs 15 hJ 1 (t,s 2 )j€ Di^)] f[s 2? h 2 1 (t,s 1 )j€ D(f 2 )J . 
 
 Define a function F on a subset D(F) of o&xT to o© such that 
 ((s p s 2 ),t]6 D(F) if and only if ({ s 1 ,h^ 1 (t, s 2 ) ) 6 D(f 1 )J 
 ((s 2 ,h 2 1 (t J s 1 ))€D(f 2 )j((f 1 (s 1 ,h~ 1 (t s s 2 ),f 2 (s 2 ,h 2 1 (t,s 1 )]€ 9j . 
 
 For each ((s 1$ s 2 ),t] 6 D(F), define F((s- L ,s 2 ) s tJ 
 
 = (f 1 (s 1 ,h^ 1 (t 5 s 2 )), f 2 (s 2 ,h 2 1 (t,s 1 ))]. Certainly 
 
 ^(D(F))UF(D(F))S<fi. Conversely, let (s-^ S 2 )e <S . If 
 
6 
 
 y(s 1 ,s 2 )€ f (D(f )), then there exists (s',t)€D(f) such that 
 f(s',t) = )°(s 1 ,s 2 ). Then there exists (spsj)^ f~ (s*) such 
 that ((s^ f s 2 ),t)€ D(F) and F( (s^, s 2 ) , t ) = (s 1 ,s 2 ). Hence 
 (s 1 ,s 2 )€ F(D(F)). If for the pair (s 1 ,s 2 ) there exists t€ T 
 such that [(s 1 h^ 1 (t,s 1 )) € D(F 1 )K ( s 2 ,h~ 1 ( t, s 2 ) ) € D(f 2 )J , 
 then clearly (s 1 , s 2 )€ ^(D(F) ). Therefore, -7^(D(F) ) UF{D(F) ) 
 = o£) . It also can be easily shown that "^(D(F)) = T. Thus, 
 we arrive at the following proposition. 
 
 Proposition 1.1 . The triplet <o£),T,F) is an incomplete autom- 
 aton. 
 
 As a result, each GPD (M-, :M 2 ) of M naturally induces 
 a new incomplete automaton <o£>,T,F> = G(M-,,M 2 ) called the 
 generalized composition of (M,:M 2 ) . 
 
 Since <^(D(f)) U f(D(f)) = S, for each (spSg^-WJ 
 and then either jPis^ s 2 ) € f(D(f) ) or ( fj teT) Cifis,, s 2 ),t)f D(f )J , 
 i.e., (3 t€T) f(s 1 ,h^ 1 (t,s 2 ))eD(f 1 )][(s 2 ,h 2 1 (t,s 1 ))€ D(f 2 )) . 
 In either case, (sp s 2 )€oQ . Hence ^J c o© . 
 
 Proposition 1.2 . Let (M-^M^ be a GPD of M. Then the induced 
 generalized composition of (M^M^; G(M-,,M 2 ) =<?£>, T,F> 
 realizes M. 
 Proof. We shall show that M is a homomorphic image of a sub- 
 automaton of G(M 1 ,M 2 ). Since (M 1 :M 2 ) is a GPD of M, there 
 exists a function f from J c s x s 2 onto . S such that 
 "^( yi ) S^, ^(Ji ) ^2* Let us cons i der the subautomaton 
 <J >T>F\j) of G(M 1 ,M 2 ) =<c©,T,F>. There exist an onto function 
 f : J — > 3, and an identity function h:T — *• T such that 
 (V((s 1 ,s 2 ;t)6ixT)j/ ) (F(s 1 ,s 2 ,t))= / P(f 1 (s 1 ,h- 1 (t,s 2 )), 
 
 f 2 (s 2 ,h~ (t,s 1 ))jy. 
 
9 
 Since f 1 (s 1 ,h^ 1 (t s s 2 ))€^ 1 Cy" 1 (f(jO( SljS2 ),t)jJ and 
 
 f 2 (s 2 ,h2 1 (t,s 1 ))6^ 2 (^" 1 Cf() (s ls s 2 ),t)J3 , thus 
 
 y?(f 1 (s ls h~ 1 (t,s 2 ),f 2 (s 2 ,h 2 1 (t,s 1 ))J * f(/ ? (s 1 ,s 2 ) s t) There- 
 
 forQ{^{(s r 8 2 lt)^JxT)(f(?((s r s^t)) - f(/ ? (s 1 ,s 2 ),t)] Hence 
 
 M is a homomorphic image of the submachine <J,T,FL> of 
 
 G(M,,M«), and 6(M V %) realizes M. 
 
 X Q.E D. 
 
 We note that if (M-^M^ is a GPD where 
 
 M l = ^ s i» T i» jf i^ and M 2 * ^ S 2 ,T 2 ,f 2^' then there is an asso- 
 ciated triplet <S-,,S 2 ,^>. Conversely, we have the following 
 
 theorem. 
 
 Theorem 1.1 . Let M = <S,T,f>. Let <S 1 ,S 2# ^>be a triplet 
 where ^ is a function from J £ s x s 2 onto S such that 
 ^Aj) = S,, ^ 2 (>/) ■ S 2 » Then there exists a non-empty col- 
 lection of GPDs of M whose associated triplet is the given 
 (Si * S 2 , rv. 
 
 Proof . Define two automata, M x -<S 1 ,T 1 ,f 1 >, M 2 =<S 2 ,T 2 ,f 2 > 
 as follows: 
 
 (i) T 1 = TXS 2 T 2 = TXS 1 
 (ii) (¥s 1 ^S 1 )(Vs 2 tS 2 )(¥t€T)|(f 1 (s r (t, s 2 ) ),f 2 (s 2 , (t, s ± ) )j€ 
 
 ^- 1 (f(^(s 1 ,s 2 ) 5 t)j}. 
 Since f ()*(s, , s 2 ),t) makes sense and is non-empty if and only 
 if (s-^s^e./ s certainly 
 
 D(f 1 ) = {(s 1 ,(t,s 2 ))|(s 1 ,s 2 )^} 
 D(f 2 ) = {(s 2 , (t,s 1 ))|(s 1 ,s 2 )^/}. 
 
 Therefore* (M-,:M 2 ) is a GPD of M and its associated triplet 
 is <S 1 ,S 2 ,^ , >. Furthermore, the set (f 1 ^ (J^(s r s 2 ), t)jj may 
 contain more than one elements For each choice of f 1 (s 1 , (t, s 2 ) ) 
 and f 2 (s 2 (t, S]L )) in f 1 ^ (f(s v s 2 )t)J , there exists a GPD of M. 
 
10 
 
 Thus there may be more than one GPD defined this way. However, 
 clearly each such GPD has its associated triplet being 
 
 < S 1' S 2'P>- QoE . D . 
 
 We call such a triplet <S 1 ,S 2> ^>, a generalized 
 pair decomposition triplet (GPDT) of M. 
 Definition 1.2 . Let M = <S,T,f>. Let M ] _ -<S 1 ,T 1 ,f 1 > and 
 
 M 2 = <S 2 ,T 2 ,f 2 > such that (M 1 :M 2 J is a GPD of M. Define 
 a pair of non-deterministic machines M-^ =<'S 1 ,T,F 1 ) and 
 M 2 =<S 2S T S F 2 > as follows: 
 (i) (¥s 1 €S 1 )(¥t€T)(F 1 (s 1 ,t)=|s^S 1 |(3s 2 €S 2 ) 
 
 s^^ 1 Cf(^(s 1 ,s 2 ),t)j}}J 
 (ii) (¥s 2 S 2 )(¥t)[F 2 (s 2 ,t) = (s 2 6S 2 |(3s 1 €S 1 ) 
 
 3^\(fHn ( Sl ,s 2 ),t)J]}] . 
 The pair (M, ,IVL) so defined is called the pair of compo- 
 nents of this GPD (M 1 :M 2 ). 
 Proposition 1.3 . Let <S-,, S 2 ,f > be a GPDT of M. Let (M^NL,) 
 and (jVLiM,) be two GPD of M with the same GPDT (S-^S^^X 
 Then the components of (M-,:M 2 ] are the same as those of 
 (M 3 :M 4 ). 
 Proof . For each i, l-i~4, let M. = <S.,T.,F.> be the compo- 
 nents of (M-j^rMgJ and (M~ :M, ) . Since these two GPD have the 
 same associated triplet, S-, = S os S - S. and T = T-, = T = T 
 
 L } <L U, 1 £ 3 
 
 = T, . It remains to show that F-, = F-, and F 2 = F, . This is 
 quite obvious, since the transition function depends solely 
 
 on the triplet <S,,S ? ,)?> . 
 
 Q. E. D. 
 
 As the result of the above proposition, there exists 
 a unique pair of non-deterministic automata associated with 
 each GPDT of M. 
 
11 
 
 Definition 1.3 . The unique pair of non-deterministic automata 
 
 associated with each GPDT is called the pair of components 
 
 of the GPDT. 
 Proposition 1.4 . Let (M-^M^ be a GPD of Mo If (M,,M 2 ) is 
 
 the pair of components of CM-,:M 2 J, then for each i, 
 
 l-i-2, M^ is a subautomaton of M. . 
 
 Proof . Consider deterministic automata being a special case 
 
 of non-deterministic automata in the usual way. The proof is 
 
 obvious from the definition. 
 
 Q. E. D. 
 
 §2. ^-covers and Generalized Pair Decomposition Triplet 
 
 For any given set A, a cover £ of A is a collection 
 of non-empty subsets A, of A such that their union is Ao We 
 introduce the definition of a #-cover of a given set. 
 Definition 2.1 . Let S be a finite set of arbitrary symbols. 
 Then OC is called a fr-cover of S if there exists a posi- 
 tive integer / = ({Ct ) such that (i) OC is an /-tuple 
 
 (A-,,...,A*) where each A. is a non-empty subset of S, 
 
 / 
 
 (ii) Ua. = S. 
 i=l x 
 
 Definition 2.2 . Let fc = (A-,,..., Ay) be a *-cover of a finite 
 
 set of symbols S. Let f be a many to one function from 
 
 S onto S where S is another seto Let OL - {f[k^) 9 . . . >P{kd) ) 
 
 where (Vi) (l«i*/)f/>(A i ) ={f(<r) | (TSAJJ . Then OC is a 
 
 *-cover of S and is said to be induced by the *-cover OO 
 
 of S under f> . 
 
 Two *-covers OC ,(& of S are said to be equivalent if 01 
 
 can be obtained from 1p by permuting the coordinates of (D . 
 
12 
 
 Our next theorem immediately reveals the importance 
 
 of ^-covers. 
 
 Theorem 2d , For each GPDT <S 1 ,S 2 ,'P> of M = <S,T,f>, and each 
 
 ordering of states in S^ = ( a i» a 2» ' * • ,a /j' and ^2 = v- a ]_> • • • » a m J 
 
 we have that there exists a pair of *-covers CC^ = (A-^, . . . t kA 
 
 and CL 2 = (Ap...,A^) of S. Furthermore, for each i, 1-i-/, 
 
 A^ ={ s€S|(3k)f)^(aJ,a^) = sj} , and for each j, l*j*m, 
 
 k 2 . ={ s6S|Gk)(^(a^,a]) - sj } . 
 
 Proof . Since <(S-, , S 2 ^f) is a GPDT, we can represent the function 
 
 "f by a / x m table as follows: The rows are labelled by a-, 
 
 1-i-/, the columns are labelled by a., l-j'-m. In the entry 
 
 2 1 
 
 (i^j), we put the state s s if (a. s a . )€ D^) and J^(a.,a.) = s. 
 
 For each i, 1-i-/, let A. be the set containing the elements 
 
 in the i— row. For each j, 1-j-m, let A . be the set containing 
 
 th 1 a & 
 
 the elements in the j— column. Certainly A., A. have the de- 
 
 11 2 2 
 
 sired property. Let 6L-^ = (Ap...,A*) s 0t 2 = (Af, . «... ,A^). To 
 
 show &L*, Gt 2 are ^-covers of S, we need only show that 
 
 / -i m p 
 
 O A . = S and UA. = S. But this is readily so, since )P is 
 1-1 J-l J ' 
 
 an onto function. 
 
 Q. EcD. 
 
 We note that any reordering of S-, and S 2 will give 
 rise to * -covers 0t^» OC 2 which are equivalent to CC-xt ^ 2 
 respectively. Hence we can consider S-,, S- as two ordered sets. 
 
 We shall now try to establish a theorem (Theorem 2.4) 
 in the other direction, namely a necessary and sufficient con- 
 dition for a pair of ^-covers {0C 9 lB) of s t0 insure the exist- 
 ence of a function ^ such that iS^S^f) is a GPDT of M, where 
 S l' Z 2 glve rise t0 ^»"fi respectively. To do this, we first 
 obtain a weaker statement (Theorem 2.2). Using that as a step- 
 
13 
 
 ping stone* we arrive at our desired results 
 
 Definition 2.4 ° Let OL = (Apo..,A^), 1& = (B,,...,B ) be 
 
 ^-covers of the finite set So Then [dt, 1Q) is said to be 
 mutually distributive (MD) if and only if 
 (i) For each coordinate A. of CC » where A. ={s-,,...,s ], 
 there exists at least one subset |i-, s o .*i jof |l*2*.o 0> mj 
 
 such that s . € B. for each j* 1-j-p. 
 
 3 J 
 
 (ii) A symmetric condition is satisfied for each coordi- 
 nate B i of S. 
 
 To serve as an example* let S = { s-,, s 2 * ° ° ° * s ^} <> 
 Let GL= (A 1 =fs 1 , s 2 ,s^} 5 A 2 = (s^*s^J, ky £s 2 , s, * s^J ) , 
 tS = (B 1 =(s 1J s^} J) B 2 ={s 2 ,s^, s, ], B^=(s 2 »s^sJ). Then the 
 pair (&,"26) is MD. 
 
 Definition 2. 5 , Let ^ be a #-cover of S. For each s € S* the 
 multiplicity of s in ^ , denoted by m^(s), is the number 
 of coordinates in $£ containing s° 
 Definition 2o6 . Let OC = (Apo.ojA^) be a *-cover of S. Let 
 s€ S* then for any i, 1-i-/ ^ seA^ ^ 3 ^ is defined 
 as the /-tuple (A, 9 o . .A^_^,A i -(s),A^ + -^, . . oA^). 
 
 In the following definitions* $S = (Ap « <, „,A*), 
 
 = (Bp 9* B ) • 
 
 Definition 2,7 * Let (^,fi) be a MD pair of *-covers of S 
 
 such that (?s6S)(m^(s) = m^(s) = m(s)J° For each s€S 
 } m(s)> 1* {Ob,t>) is said to have the 1 st s-deleted 
 distributive property if 
 (i) For each i, l^i*/, 3 s£A t * 3j* 1-j-m 9 sfB, and 
 
 (ii) For each j, l^j^m3seB jS 3 i, 1-i-/ * si k ± and 
 «*l.,l)'^(a,3) ) isMD ' 
 
14 
 
 Definition 2. 8. Let \OC»fa ) be a MD pair of --covers of S 
 
 such that (VseSjfm^s) = m^fs) = m(s)J. For each s€S, 
 m(s) > l s ( fa^fi} is said to have the n -s-deleted dis - 
 tributiv e property if 
 
 (i) For each i, 1-i-/ 5 s e A ;i , 3 j, 1-j-m 9 s e B . and 
 ($1 '\slPi .s) has the (n-1) -s-deleted distributive 
 property. 
 
 (ii) For each j, 1-j-m 9 s€ B., 3 i* 1-i-/ 5 s € A i and 
 ( fef -\s^i -\) has the (n-1) -s-deleted distributive 
 
 v Sg 1 ) { Sj> J ) 
 
 property. 
 Definition 2.9 ° Let (^fc,g6) be a MD pair of --covers of S 
 such that (¥s)fmjs) = m^(s) = m(s)J. Then (^^3) is 
 said to have the deleted distributive property (DDP) if 
 for each s€S 9 m(s) > 1, {0C S 1&) has the (m( s)-l) th -s- 
 deleted distributive property. 
 
 The pair of ^-covers in the example following Defi- 
 nition 2. k has the DDP. The following example will show that 
 not all MD pairs of *- covers of S in which (¥s) (m c (s)=m^(s)=m(s)J 
 will have the DDP. Let S =(s ]S ...,Sc). Let 
 
 © = lB i"( s i* s 2» s /. }» B 2 = ^ s 2 9 s 3* ^l* B 3 = ( s 3» s 5p 8 It is easil y 
 checked that they are MD and (Vs)(m^{s) = rn^(s)]. Yet ( fiLffo ) 
 
 does not have DDP. 
 
 Definition 2.10 . Let ( #£, jfi ) be a MD pair of *-covers of S. 
 Let A^ : {s 1( ,». v s J be a coordinate of OL . Let s. 6 A. . 
 Then A i is said to require s^ in a c o ordina t e B , o f ft£> 
 to dist ribute if whenever / i ... , . . . , i } ^ (l, 2, . . . ,m} satis- 
 fies the condition (i) in Definition 2.4, then i. - k Q . 
 
 Definitio n 2.11. Let ^= (A^ . ..,Aj be a *-cover of S. By 
 
15 
 E^, we shall mean the set of indices of 0C , i.e., 
 
 E^jr = ( 1, 2, . » . , / J. 
 
 In the following, let fit = (A-,, . ..,A^), 
 O = (B-^, • • • , B ) o 
 
 Lemma 2.1 . Let (^£,#3) be a MD pair of '--covers of S 
 } (Vs€ S)Cm^(s) = m fi (s) = m(s)]. Let E^ E fi be defined as 
 above. Then there exists a function \P defined on a subset 
 Ej c e^* Eg onto S such that 
 
 (i) (VDil^/llA^s^jgE^f^i^-) = sj}) 
 (ii) (¥j)(l^j%)(B j =(s|(3i6 E^)fP(i,j) = sjj) 
 (iii) For each s € S, define D g ()P) = ((i,j)€ E^* E^fli, j)=sj; 
 ohen the cardinality of D ^]P) is m(s). 
 if and only if thexe exists a i eccangular table witn columns 
 labeled by j, 1-j-m, and rows labeled by i ; 1-i-/, such that 
 (i) Each s€S appears in the table exactly m(s) times 
 and in exactly m(s) columns and m(s) rows, 
 (ii) Each column j, 1-j-m, gives an exact list of the 
 elements in the j coordinates B. of tj . Each 
 
 «J 
 
 i row, 1-i-/ gives an exact list of the element s 
 
 +■ v> 
 in the i coordinates A. of OL • 
 
 Proof . The "if" part is trivial. Simply define ^(i,j) = s, 
 
 if s occupies the entry (i,j). Conversely, suppose there 
 
 exists a *f satisfying (i), (ii), (iii). Assign to each (i,j) 
 
 of EjC E^x Ejg the element j^(i,j). Since )^is a function, 
 
 the assignment is unambiguous for each entry. Further, it is 
 
 clear that the table obtained satisfies the desired properties. 
 
 Q.E.D. 
 
 Lemma 2.2 . Let (#,j6) be a MD pair of ^-covers of S. Let 
 
 s, s»6 S , s / s' and m^(s) = m^(s) =1, m^(s') = m^s') = 1- 
 
16 
 Then there does not exist (i,j) where i€ [l,2, . . .,/}, 
 
 je [i 9 ...,m\ such that fs,s ? } 9 k^^\ b.. 
 
 Proof . Suppose there exists (i,jh i^fl, ...,//, jg{l,...,m} 
 
 such that (s,s'} c A. n B.. Since m^( s) = m fi ( s ? ) - ra^(s) 
 
 = ra-(s ? ) = 1, B . is the only coordinate of 16 containing s and 
 
 s f . Further, since A. 2 fs,s ? } 5 A. can not be distributive 
 
 over ^8 • This contradicts the MD property of ( ££, jG)« 
 
 Q. E. D. 
 
 Lemma 2.3 . Let ( <&, 6 ) be a MD pair of *-covers of S. Let 
 
 A. be a coordinate of 0C» Let s€ A, and A. requires s in a 
 coordinate B, of tO to distribute. Suppose s v e A. n B, and 
 
 s f ^ s. Let A, = { s-. , . ..,s | where s-, = s 5 s~ = s*. For each 
 
 subset {i,,...,i } of (l, 2, ...pin} such that (Vj ) (1-j-p) (s •€ B. ), 
 
 we have that i-, = k and s = s-, ^ B. . 
 
 Proof . Suppose there exists a subset {i-,,.. ,i \ of {l, . ..,mj 
 
 such that (Vj) (1-j-p) (s .€ B. ) and that seB. . Then certainly 
 
 3 lj i 2 
 
 the subset (i!,...,i' J of {l, 2, ...,m}, where ±1 = i 2 ? i? = ^i# 
 and i. = i. for 3~j-p> will have the same property* Further- 
 more* i! / k, since il = i, = k and i« = in / i| = ip° This 
 contradicts the assumption that A. requires s in B, to distribute. 
 
 Q.E.D. 
 Lemma 2.U * Let (#,fi) be a MD pair of ^-covers of S. Let A i 
 be a coordinate of ^3s€A. and A. requires s in a coordinate 
 
 B, of l£) to distribute, Suppose s*6 k^n B . , s 1 ^ s, then A i 
 
 can not also require s' in B. to distribute. Furthermore, if 
 
 j 
 
 f ^, 6) has DDP, then B. can not require s* in k. to distribute. 
 Proof. Let A i ={s,»...,s } and s = s,, s' = s,, h / k. Since 
 A i requires s in B i to distribute, £J{i,,.o.,i } - l 1 * 2 > ' * * > m j 
 such that (Vj) (1-j-p) (s .£ B, ). Suppose A. also requires s» 
 
17 
 in Bj to distribute. Then i h = i k = j which is a contradiction. 
 
 Suppose (^,6) has DDP and B. requires s 1 in A. to 
 distribute. Then B. does not use s in A. to distribute. Let 
 Bj ■{8 1 ,...,a q } and let s = s h , s» = s k , k / h. Let {jp---, jq} 
 be the subset of (l, ...,/} such that (¥*<)( l^^q )( s e A. ). 
 Then j fc = i and j h / i. Since (^, j6) has DDP, for the set 
 Aj , there exists j Q € {l, ...,m} 3 ( fo*, jg« ) = (^ Sj • ),6( s . )) 
 also has DDP. Yet A. requires s in B. to distribute and j, f i, 
 thus Jq ^ j. Therefore, B. does not require s in any coordinate 
 A , r / i, of fit to distribute. 
 
 Now, consider the new pair (^£ ? s j6> ? )° The conditions 
 of this lemma still apply. The multiplicity of s in GO or ^ ? , 
 denoted by m'(s) is (m(s) - 1). As long as m»(s)>2, b. does 
 
 ■J 
 
 not require s in any A , r f i, of OV to distribute for the 
 
 same reason. We may similarly arrive at a new pair [tfC"»@")* 
 
 Suppose m w (s)> 2 S we proceed to obtain [crL" % > $"* ) and s0 
 
 forth until m**(s) = 2. Then, since (A^B.) still contains s, 
 
 there is only one other A* of OC 'b s6A». By a similar 
 
 argument, B. does not require s in any A , r f i of @L to 
 3 ~ 
 
 distribute. However, B. can not use s in A. to distribute and 
 
 J 
 
 i 
 
 there is only one other A* containing s, B. must require s in 
 
 A 9 to distribute. This contradicts the fact that B. does not 
 
 require any A , r / i, to distributee Therefore* B. does not 
 r j 
 
 require s ? in A. to distribute. 
 
 1 Q. Eo D. 
 
 Lemma 2.5 » Let { M, jS) be a pair of *-covers of S with DDP. 
 
 Let i€ {1,2,...,/} such that s € A^_ and k ± requires s in a 
 
 coordinate B, of -j6 to distribute. Then for any j / i, 
 
 j€fl, ...,/}, A. can not require s in B k to distribute. Fur- 
 
 thermore, for any r / k, r€{* 1, . . .,m},B can not require s in 
 
IS 
 
 A. to distribute. 
 
 Proof . Suppose 3 j / i, j6 (l, . . . ,/j , A. also requires s in 
 
 B, to distributee Then for s € B, , there does not exist any- 
 coordinate A^ of 6L such that s€A , and ( 01, %, $8, ^J ) 
 again has DDP. This is a contradiction. 
 
 Suppose there exists r / k such that B requires s 
 in A. to distribute. Then ( &* . \, S/ . \) can not have DDP. 
 However, since A. requires s in B, to distribute, for s in B. , 
 B, is the only coordinate of o from which s can be deleted 
 together with s in A. to give a new pair of ^-covers with DDP. 
 Thus, if B requires s in A. to distribute, then [CL 9 ~&) can 
 not have DDP. Since for s in B. , there does not exist any 
 coordinate A of 0l3s€A, and { OL ^\»*G( k \) has the 
 (m(s)-l) -s-deleted distributive property. 
 
 Q E. D. 
 Remark 2.1 . Let s € S. Let (^£>q) be a MD pair of ^-covers of 
 S ^ m fA 5 ^ = ni^(s) = 1. Let A, B be the only coordinate in 
 OL, $ respectively containing s. Then A requires s in B to 
 distribute and B requires s in A to distribute. 
 Lemma 2.6 . Let [0C,fB) be a pair of ^-covers of S with DDP. 
 Let A^ be a coordinate of OC . Suppose seA. such that (i) A. 
 
 does not require s in any coordinate B . of (fi to distribute, 
 
 (ii) For any B. of tfB & sf B ., B. does not require s in A. to 
 j j j - 1 - 
 
 distribute. Then there exists j n , l-j n -m 3 seB. (1) B. 
 
 u ^0 J 
 does not require s in any coordinate A of ^ to distribute, 
 
 2) Suppose s»€ S, s» / s and s'eA.AB. , then A. does not 
 
 require s T in B. to distribute and B. does not require s T 
 J ^0 
 
19 
 in A. to distributee 
 
 Proof . Let (Gt 9 fQ) be given and that s6A. of &C satisfying 
 the conditions (i), (ii). Suppose for every coordinate B. of 
 )S 3 s 6 B ., B. requires s in some coordinate A of 0C to dis- 
 tribute. Then, by (ii), r / i. There are only m(s) - 1 co- 
 ordinates A in which s£ A and r / i. Yet there are m(s) 
 coordinates of GL containing s° By Lemma 2.5, there should be 
 m(s) distinct coordinates A such that s€A and r ^ i. This 
 leads to a contradiction. Therefore the set J =/ j | jsfl, • • «,m}, 
 s £ B ., Bo does not require s in any coordinate A of ftt> to dis- 
 tribute} / 0. We note that for any j € J s j satisfies the con- 
 clusion (1). 
 
 To show the existence of j Q satisfying (1), (2), 
 we need only show the existence of j Q e J such that j Q satis- 
 fies (2)o Suppose there does not exist any j Q € J = { j-j.* • • • » J p } 
 satisfying (2). Then there exists fs. , .««,s. } where s. f s 
 
 for any k, 1-k-p, with the property that either B. requires 
 
 J k 
 
 So in A. to distribute or A. requires s. in B. to distrib- 
 ^k x J k J k 
 
 ute. Suppose s = s 1 in A., then j° / 1 for each k, thus^ by 
 
 Lemma 2.3, Lemma 2.4^ for any j k € J, and any subset {i 1 ,...i } 
 
 of {l,2,...,/j satisfying Definition 2.4, i 1 / j k * We shall 
 
 say that A. must require s = s. to be not in B. to distribute. 
 1 i l J k 
 
 But, by definition of J, for all k £ J* = fk |(l*k*m) (s6B k ) (k^J)} , 
 
 B, requires s in some coordinate A^ to distribute, where 
 K r k 
 
 r k / i. Let (^» } g?) be the *-covers of S obtained from de- 
 leting s in B, and A for each k£J«. Then (<£»,$') must 
 
 r k 
 still have DDP, and we have ( s € k ± ) ( ( s€ B^ ) £=£ (j 6 J)) where 
 
 k ± , Bo are the i th and j th entry of 0C , 6 * respectively. 
 
20 
 
 Furthermore, for each j £ J, A. still requires s to be not in 
 
 any B. to distribute. But by the construction of )6 T , there 
 
 exists no coordinate B, of -Q 1 , k 4 J which contains s. Thus, 
 
 A. is not distributive over -fe T , a contradiction to the fact 
 
 that {&*,-&') has DDP. Therefore, there must exist a j Q 6 J 
 
 } B. does not require s ? in A. to distribute and A. does not 
 J x 1 
 
 require s* in B. to distribute for any s* / s. This proves 
 the lemma.. 
 
 Definition 2.12 , Let <K= {k^ • . , ,kjf) , tS = (B- L ,...,B m ). Let 
 ( #, $) be a MD pair of ^-covers of S. Let A k be the k th co- 
 
 frf th 
 
 ordinate of UL $ s € A. . Let B, be the h coordinate of 
 (6 3 s € B^. Let A. = { s-, , s 2 » • • • » s. =s, s. +1 , . . . , s j. Let 
 (n-,,...,n } C fl, ...pj be such that n. / i for each j, 1-j-r. 
 Let o^ be a one to one function from {n J 1-j-r/ into {l, 2, . . . , m). 
 Then A, is said to require s*. = s in B, to distribute with 
 
 fixed B i/ )' " ' >^a(ln ) ^ anc * on ^y ^ whenever {k,,...,k } is 
 a subset of (l, 2,...,mj 3 (Vj ) (1-j-p) ( s . € B, ), and for each q, 
 1-q-r, k =^(riq), then k. = h. 
 
 Remark 2.2 . The statement "A, (or B, ) requires s. in B h (or A^) 
 to distribute" and the statement "A, (or B, ) requires s. in 
 B h (or A,) to distribute after fixing B^/ )> ,,, * B <( n ) (° r 
 
 ^U(n )' ' * " *^©<(n P" are t ^ le same wnen ni(s. ) = 1. 
 
 Remark 2.3 . The results of Lemma 2.2. to Lemma 2.6 are still 
 
 true when one replaces the statement "A, (or B,) requires s^ 
 
 in B h (or A k ) to distribute" by the statement "A k (or B h ) 
 
 requires s. in B, (or A, ) to distribute after fixing 
 
 B ^(n 1 )--'» B ^(n r )C or ( V(n 1 )^'" A o<(n r ) ) - Su PP°se m ( s i) = X > 
 then the result follows naturally. Thus we may assume m(sj)>l« 
 
 Then first, whenever m(s ) > 1, 1-j-r , simply delete s from 
 
 n j J 
 
21 
 
 A k and {s n , s.J from B^^ * and s ± in their corresponding 
 
 <*(n ) sucn that after the deletion, the new ^-covers of S 
 have DDPo Since A, can be distributed without using the s. in 
 B o<f(n ) ,0,#, ^o<(n )' certain ly ^ w n \) f can always be chosen 
 not to denote A,. Let {00,15*) De the new Pa ir of ^-covers 
 after all such deletion, then s. € A, r\ B,. Now we can apply 
 Lemma 2.2 to Lemma 2.6 to ( CO , 1Q f ) to obtain the desired 
 results. 
 
 Theorem 2.2 . Let Cft = {k^ . . . ,kf) , &= (B 1 ,...,B m ) be a MD 
 pair of ^-covers of S. Let E^ = { 1,2, . . . ,/}, E^ = ( 1,2, . . . ,m] 
 be as defined in Definition 2.11. Then {dC,1&) has DDP if and 
 only if there exists a function W defined on a subset Ej of 
 E.X E* onto S such that 
 
 ( i ) For each coordinate A^^ of OC , k ± ={s€s|3j€E j6 9 
 f(±,3) = a}. 
 (ii) For each coordinate B . of 6 * B . = {s6S|3i€E^^ 
 f(i,j) - s}. 
 (iii) For each s € S, the number of elements in the set 
 
 {(i,j) lf(i,j) = s) is m(s). 
 
 Proof . For the only if part, by Lemma 2.1, we need only show 
 
 that there exists a (/x m) rectangular table, not necessarily 
 
 all filled, with each column labeled by i, giving the exact 
 
 list of the elements in the coordinate A i of CL , and each row 
 
 labeled by j giving the exact list of the elements in the 
 
 coordinate B. of n. 
 J 
 
 We shall describe a process in assigning states into 
 the entries of the table. We will show that after each step 
 the assigned state in the entry (i,j) is in k ± H B. and such 
 that no one state repeatedly appears in the same row or same 
 
22 
 
 column. We will then show that this process is exhaustive; 
 i.e., every s€S is assigned to exactly m(s) squares. And 
 certainly we must show that each step described in the process 
 is possible to complete. 
 
 (I) (a) For each column j, assign s€B. to the entry (i,j) 
 
 if B. requires s in A. to distribute. 
 
 »J 
 
 (b) For each row i, assign s 6 A. to the entry (i,j) if 
 A. requires s in B. to distribute. 
 
 This process is possible to complete. As a result 
 of Lemma 2.4, any coordinate A. of OC can not require two 
 different s, s f € S in the same B. to distribute. Again, by 
 Lemma 2.4, if A. requires s in B. to distribute, then B. can 
 not require any other s' / s in A. to distribute. Thus (b) 
 can be done without conflicting with (a), and vice versa. 
 
 Moreover, if A. requires s in B. to distribute and 
 [0C,D) is MD, by Lemma 2.5, we have that A. does not require 
 s in any other B. , k / j, of {Q to distribute, and for each 
 B, of £J, k f j> B v can not require s in A. to distribute. 
 Thus, after the assignments are done in this step, no state 
 s repeats in any row or any column. 
 
 (II) (a) For any row i, assign s € A . to the entry (i,j) if 
 
 A. requires s in B. to distribute after fixing 
 
 B *(n ) ,,,,,B *<(n ) where {(i*^(n.) ) 1 1-j-r} are the 
 entries occupied by states assigned in the previous 
 step, 
 (b) For any column j, assign s€B. to the entry (i,j) 
 
 if B. requires s in A. to distribute after fixing 
 
 j « 
 
 k <*{m )''"' k o({m )' where {(°^( m i)#j) 1-i-k) are entries 
 occupied by the previous step. 
 
23 
 
 By the rsult of Remark 2.3, this process can be com- 
 pleted as for step I. 
 
 (Ill) Repeat step (II) until no new assignment is reached by- 
 such repetition. 
 (IV) Suppose for some row i, 3 s 6 A. which has not been 
 assigned by previous steps. Then, by Lemma 2.6 and 
 
 Remark 2.3, there exists a column j 3 s£B. which also 
 
 j 
 
 has not been assigned. Assign s to this entry (i,j). 
 Then perform steps II, III again, until no new assign- 
 ment is reached in III. 
 (V) Repeat step (IV) until every s in every coordinate A. of 
 has been assigned. 
 It is not hard to see that the result of these steps 
 will give a rectangular table having no state s appearing re- 
 peatedly in any row or any column. Certainly, each s appears 
 once in some row. But, it also appears once in some column. 
 This completes the proof in one direction. 
 
 To prove the if part, we need only show that if such 
 a table as described in the beginning of this proof exists, 
 then (&>,$ ) has DDP. Let E^, Eg be defined as in Definition 
 2.11. For each s, if s appears more than once in the table, 
 we can delete the state s in the i row and j column if s 
 is in (i,j). Then the resulting new pair of ^-covers is still 
 MD. Since each state s appears exactly m(s) times, we can per- 
 form this deletion for all s € S up to (m(s)-l) times. Hence 
 for each s£S , m(s)>l, ( 6t, ft ) has the (m(s)-l) th -s-deleted 
 distributive property. Therefore ( ^£,S ) has DDP. This com- 
 pletes the proof. 
 
 Q. E.D. 
 
24 
 
 Using Theorem 2.2 as a stepping stone, we shall now 
 establish a necessary and sufficient condition on a pair of 
 ^-covers (^£,#3) of S to insure the existence of a function ^ 
 satisfying just (i), (ii) of Theorem 2.2 such that <E^,E^,^> 
 is a GPDT of M. 
 
 First, we introduce the following definition. 
 Definition 2.13 . Let <£, 6 be --covers of S. Then {fiC y ^>) 
 
 has the added deleted distributive property (ADDP) if and 
 
 only if 
 
 (i) There exist a finite set of symbols S and two 
 
 --covers fit, j& of S. 
 (ii) There exists a function P from S onto S such that 
 frC is a --cover of S induced by the --cover fit of 
 S under P, and $ is a *-cover of S induced by the 
 *-cover 6 of S under/ 5 , 
 (iii) (#L,S ) has DDP. 
 
 We shall show an example of ^-covers with ADDP. Let 
 S - ( s i' s ?' ' ' '* ^M ' Let UL - ((s-i, S2>s^j,[s-,,s^|, [ s, , s r ) J , 
 VD = (fs-j,s,j, {s-ijSpfS, }, (s,|Sr] ). Then there exists a finite 
 set of symbols S = (s-, *, s 2 S • • • > S/-* }. Let 
 5t= ({s 1 «,s 2 »,s 3 *j, fs 1 ',s 7 »}, (s 4 ',s 5 *,s 6 ']), 
 
 t5 = ( ( s i tjS 4 ? }' K^V'^'J' (V'V'Vf ■ Let ^ : s — * s 
 
 such that((Vi)(l^i^5)(/ ? (s i M - s i )J(y^(s 6 ») = sj(/>(s 7 ») = s 3 ] . 
 Then ^C,$ are --covers of S with ADDP. 
 
 Theorem 2.3 . Let dC= ( A-,, . . . , A//) , ?S =(B-., . . . ,B ) be --covers 
 of 3. Let E^, E^ be defined as in Definition 2.11. Then 
 (££,$) has the ADDP if and only if there exists a function y> 
 on a subset E/ of E^x Eg onto S such that 
 (i) For each coordinate A. of fiC , 
 
25 
 
 A i = ( s|3j€E^(i,j) = s } 
 
 (ii) For each coordinate B. of to , 
 
 Bj -{ s | 3i€E^a)C(i,j) = S K 
 
 Proof * Since (^,{S) has the ADDP, there exist (a) a finite 
 set of symbols S = { s. J, (b) two ^-covers dl,Q of S, (c) 
 a function P f rom S onto s such that <#-,$ are ^-covers of S 
 induced by ^,j& respectively under P. Furthermore, ( $S,^o ) 
 has DDP. Let E=, E:g Be defined as in Definition 2.11. By 
 the result of Theorem 2.2, (<fc,^) has DDP implies that there 
 exists a function £> from a subset Ej of E- * E^ onto S such 
 that 
 
 (i) For each coordinate A. = | s | 3 j e Er B J^i, j )=s~j. 
 (ii) For each coordinate B. = ( s | 3i£ E^ * J^i, j ) = sj. 
 (iii) For each s € S, #{(i, j ) | f{±, j ) = s) = m(s), where 
 m(¥) is the multiplicity of s in fiCor-fc . 
 Since ^fc j6 are ^-covers induced by ^£, jQ respectively under 
 f>, (Vi)(W/)(A i =/°(A i )),, and (¥j ) (l*j*m) (Bj =/ ? (B J .)). 
 Therefore Er = E^, Er= E* and E T = E,* Define a function 
 f=Pf from E^ onto S such that jP(i,j) = s if /°(^(i,j)) = s. 
 Thus, for each i, A^^ ={ s |( 3i € EJ {f[±, j ) = s) * , and for 
 each j, B. ={ s |(3i€E )()°(i,j) = s)}. This proves the 
 only if part. 
 
 To show the if part, we note that the function JP gives 
 rise to a /* m rectangular table in which there may be columns 
 or rows containing repeated appearances of certain states. In 
 this table, the rows are labeled by i, 1-i-jf, and the columns 
 are labeled by j, 1-j-m. We shall construct a new table where 
 rows are again labeled by i, 1-1-/ ; columns are labeled by 
 j, 1-j-m. For each occupied entry (i,j), we associate a new 
 
26 
 
 symbol* s(i,j). Let S ={ s(i,j)|(i,j) is occupiedj. For each 
 (i,j), let s(i,j) occupy the entry (i,j) in the new table. 
 Then we obtain a /x m table not necessarily all filled. For each 
 i, let A. = f s(i, j ) I s(i, j ) occupies (i,j)j. Similarly, 
 B. = { s(i„ j )| s(i, j) occupies (i,j)j. Let CLr (A 1 ,...,A^) and 
 58 = (B-|,.«o,B )« Define a function P from S onto S as follows. 
 For each s(i,j) £ S, P(s(i,j)) = s if s occupies (i,j) in the 
 original table. Certainly £C»Jo are ^-covers of S. And for 
 each i, 1*1*/, ^(A ± ) = {/°(s(i,j))|l(i,j)6 A ± } 
 
 = {s|s is in the i rowj 
 
 = A.. 
 
 l 
 
 Similarly, for each j, 1-j-m, ^(B.) = B.. Therefore, 01,^ 
 are ^-covers of S induced by the ^-covers OC,^ of S respec- 
 tively under P . It is obvious that (fit,jo) has DDP. There- 
 fore, (^-,Jd) has ADDPo This completes the proof. 
 
 QcE. D. 
 We have now reached the following main theorem. 
 Theorem 2.4 ° All GPDT of M =<S,T,f>are of the type 
 <E^,E£, f>, where tit, fa are ^-covers of S,and ( 4C,]& ) 
 has ADDP. 
 
27 
 
 Chapter II 
 PROPERTIES OF GENERALIZED PAIR DECOMPOSITION 
 
 At the beginning of this chapter, we shall intro- 
 duce the notion of a monadic algebra and its relation to an 
 automaton. Our definition for an algebra (3 ) will be a 
 rather general one. This allows us to state many theorems 
 with appropriate degree of generality. 
 
 By an algebra -T2, we shall mean a triplet 
 (S,T, 0= {fo<jX where T is a set of symbols called operation 
 indices, S is a set called the carrier of elements, and is 
 a set of single-valued functions, =/fJ-<eT9f : S n — > S 
 and n = n(«<)}. If nM = 1 for all ^6T, then 12 is called a 
 monadic algebra. 
 
 Let M =<S,T,f / ) be a deterministic complete automaton. 
 Then M is a monadic algebra in the following way: Let T be 
 the set of operation indices and S be the carrier set. For 
 each o^eT define an operator f^iS — » S such that 
 (Vsjff^s) = f(s,eO). Let ={ f^ | *€ T }, then<S,T,0> is 
 clearly a monadic algebra. Conversely, any abstract monadic 
 algebra <fs,T,0 = f f. \) can be considered as a deterministic 
 complete automaton. Let S be the set of states, T the set 
 called the input alphabet. The transition function f is de- 
 fined such that (¥(s,oO € S * T)[f (s,o<)=f < ( s )) . As a consequence, 
 any result of monadic algebra will hold for an automaton. 
 
 We shall now introduce the notion of an incomplete 
 monadic algebra. In an incomplete monadic algebra (S,T,0), 
 each operator f^ is defined on a subset of S. Certainly a 
 
23 
 
 monadic algebra is also an incomplete monadic algebra. If 
 we follow the analogous formulation as discussed before, we 
 can show that an incomplete monadic algebra is an incomplete 
 deterministic automaton and vice versa. Therefore, in the 
 following,, we make no distinction between an automaton and 
 monadic algebra. 
 
 §1. Algebraic Properties of Congruence Relations of Degree k, 
 Let M = <S,T,f)> be an incomplete automaton. Let 
 (M 1 :M 2 ] be a given GPD of M, where M. ± = < s 1 * T 1 * f 1 > and 
 M 2 = < (S 2 ,T 2 ,f 2 ) are incomplete automata. By the result of 
 Chapter I, M is thus a homomorphic image of a sub-automaton 
 ^/»T,FL> of the generalized composition of M-,, IYL. (See 
 definition on page g. ) We will call <y,T,F «> the induced 
 automaton of the GPD (M-^M^. 
 
 In fact, Cm 1 :M 2 ) is also a GPD of </,T,fL> , and 
 the automaton M as related to the GPD C]VL:M 2 ) is completely 
 represented by <>/, T,F /> e Thus, in order to find properties 
 of a GPD (M 1 :M 2 J of M, we must study (1^:1^) as a GPD of the 
 induced automaton <>/,T,fL> or any isomorphic image <S f ,T,f f )> 
 of <^/,T,fL> . If M' = <S',T,f»> is isomorphic to (J,T,fL> 
 where (T is the isomorphism, then (M-,:M 2 J is also a GPD of 
 <S»,T,f»>. Furthermore, if <S 1 ,S 2 ,/ / > is the associated GPDT 
 of (N^iMg) for M», then ft is the isomorphism <r from <^,T,fL> 
 onto <S',T,f v >. Thus, we need only study the sort of GPD 
 CM 1 :M 2 ) of an automaton M where in its associated GPDT 
 C'-'l»^2 , Y ) ^ the homomorphism )P is an isomorphism. 
 
 Let (M 1 :M 2 ) be a GPD of M =<S,T,f> . Let the asso- 
 ciated GPDT be <3 r S 2 ,^> where f is one to one from J onto S. 
 
29 
 Define a relation 9-^ on S such that (¥s,s f £ S) ((s9,,s t ) < — ^ 
 
 [ ^ 1 ()^ 1 (s)) = ^ 1 ( ^ 1 (s')))j. Similarly, we define a relation 
 922. It can be easily shown that 9-^ and 9 22 are equivalence 
 relations. Further, in the lattice of all equivalence relations 
 on S, the g./.b. of 9-^ and 9 22 = 9 n n 9 22 = 0. Since if s / s' 
 
 and s(9 i;L ^ 9 22 )s», then ^ i f^" 1 (s)J = ^J^s' )] for 1^2, 
 
 -1 -1 
 therefore y> {s) =<f> (sM. Yet f is one to one onto, there- 
 fore it leads to a contradiction. Hence, automata M-. , M ? give 
 rise to two equivalence relations 9-,-,, 9 22 on S (thus two par- 
 titions on S) such that 9-,-,/^ 9 ?? = 0. 
 
 We shall now study properties of certain equivalence 
 relations on a monadic algebra which are pertinent to the 
 theory of decomposition of automata. 
 
 Let JT2 = {S,T,0) be an incomplete monadic algebra. 
 Let T = {©<.. , . . . f o(A. Let 9 be an equivalence relation on S. 
 For every s € S, f(s,T) denotes the /-tuple (s,,...,s^) where 
 (¥i)fs. =f, (s)l. On the set of all /-tuples of states of S, 
 9 induces an equivalence relation 9 such that 
 
 [ ( s lf . . . , s*)Q{ s ± i , . . . , s f )£=M Vi ) Cs i 9s i 'j). 
 
 Definition 1.1 . Given an incomplete monadic algebra 
 
 0- = <^S,T, (f^}>, 9 is a congruence relation of degree k 
 
 if and only if 
 
 (i) 9 is an equivalence relation on S. 
 
 (ii) Let N(s,9) be the number of distinct equivalence 
 
 classes under 9 having a non-empty intersection with 
 
 {f(s»,T)|s»0s}. Then k = k(9) = max N(s,9). 
 L ; seS 
 
 We note that a congruence relation of degree 1 is 
 
 the ordinary congruence relation (i.e., an equivalence relation 
 
30 
 
 satisfying the substitution property (3))« 
 Proposition 1.1 . Let © 1$ © 2 be congruence relations on 
 of degree k , k 2 respectively. Then (©^ © 2 ) is a congruence 
 relation of degree ^ min(k 1 k 2 ,#(R) ) where #(R) is the number 
 of distinct equivalence classes under 0^ in the range space 
 R =«[ f (s,T)| s € SJ where © 3 = (© x ^ © 2 ). 
 
 Proof. It is obvious that (©j^ ^ 1S an ec l uivalence relation 
 on S. It also can be easily shown that the number of equivalence 
 classes of (0^ 9 2 ) in the set {f ( s' ? 5 T j | sM© 1 H © 2 )s} for any 
 s 6 S is ^ k-,k 2 » However*, the number of equivalence classes of 
 (9 r\ © 2 ) is also restricted by the number of equivalence 
 classes in the range space. Therefore, (0-|_^ © 2 ) is a con- 
 gruence relation of degree $ minl.#(R) 9 k 1 k 2 ) . 
 
 Q.E. D. 
 
 The following example will show that if ©p 2 are 
 congruence relations of degree k, s k 2 respectively, then the 
 l.u.b. of 1 and © 2 = (0 :i U © 2 ) may be of degree higher than 
 k 1 k 2 . 
 
 Let M = <S,T,f>. Let S ={ 1, 2 S . . . , lg}, T = {<*} . 
 The transition table is : 
 
 f 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 d 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 o( 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 10 
 
 
 
 1.1 
 
 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 Let 1 = ({l23},{456l/7d9},{l0t{llUl2j / ..{ld», then Q ± is of 
 degree 3- Let 2 = f (12}X34U5},{67};[89},{10}, . . {l8}) , then ? 
 is of degree 2. Then (G^U 0) = ( (123456789},{10},{11 , . . . ,18}) 
 is of degree 9 > k-,k ? = ^" 
 
 Definition 1.2 . Let 12 = ^S^T.ff^}) be an incomplete monadic 
 ;ebra. Let be an equivalence relation on S. Then 
 
31 
 f" (9) is a relation defined as follows: 
 
 (¥s,s'€ S)[(sf 1 (9)s0^=^(¥ O <)(;f >< (s)9f,(s»)j) . 
 
 In general, f (9) may not be an equivalence relation. 
 However, if 12. = < S, T, {f^} > is complete, then f _1 (9) is also an 
 equivalence relation. 
 Definition 1.3 * Let Op © 2 be two relations on S, then 
 
 (O-^©^ if and only if (Vs,s»6 S) ((s0 2 s» ) =» (sOjjs* )) . 
 Definition 1.4 . Let 9p © 2 be equivalence relations on S such 
 that 9-^2 ©2» Then (0j/0 2 ) is an equivalence relation on 
 the set j£ , where Js is the family of equivalence classes 
 of S under ©«• For each s€ S, let s € 2 • Then we define: 
 (sjLfOj/O^Ig J <£=£ 
 
 (Vs 1 » ) ( Vs 2 » ) I [(s^s-^ ) (s 2 =i 2 » )J => (s 1 »9 1 s 2 0j . 
 For any equivalence relation 9 on S, let m(9) denote 
 the maximum number of elements in any class of 9 and n(9) the 
 number of equivalence classes of 9. 
 
 Proposition 1.2 . Given a complete monadic algebra 12 =(S, T, I f. j), 
 then 9 is a congruence relation of degree k if and only if 
 (i) 9 is an equivalence relation on S 
 (ii) ■'(— T - 9 ) == k. 
 
 f" ± (9)ne 
 
 Proof . We need only show that condition (ii) here is the same 
 as condition (ii) in Definition 1.1 when SI is a monadic alge- 
 bra. This is easily done. Since SI is complete, f~ (0) is 
 
 
 also an equivalence relation on S. Therefore, m( — * ) 
 
 f" 1 (0)A0 
 is the maximum number of distinct equivalence classes under 
 
 on the set {f (s», T) | s»©s}, for any s € S. 
 
 Q. E. D. 
 
32 
 
 Proposition 1.3 . Let 9,, 9 2 be congruence relations of degree 
 
 k-,, k 2 respectively such that 9-^2 Q 2 * Tnen k ]_ ^ k 2 ^ for 
 each fixed s 6 S, for every s»€ S, (s'O-j^s), Bt€ S, t9 2 s, 9 
 
 Proof . If for each s», (s»9 1 s), 3 t € S, t9 2 s, ^ 
 
 (V°()(f (s»)9X(t)], then, in particular, this is true for 
 
 state s*, s'9-iS, yet s f 2 s. Thus, all such elements s f are 
 
 not mapped by f to any addition equivalence classes of 9 2 « 
 
 Hence k-, ^ k~. 
 
 1 * Q. E. D. 
 
 We shall show by an example that the restriction 
 
 9 2 £ 0-, in the above proposition can not be lessened. Let 
 
 -^ = (SfjT^f^} be given as follows: 
 
 f 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 6 
 
 7 
 
 c< 
 
 7 
 
 7 
 
 5 
 
 6 
 
 7 
 
 5 
 
 7 
 
 7 
 
 Let 9 1 ={{0,1},{2,3,4},[5,6K(7}}, 9 2 ={{o}, il,2,3,b} 9 ft}. (6), {7}}. 
 
 Although all other conditions are satisfied except 9 ? £ 9-,, we 
 
 have that the degree of 9 2 is 3 and the degree of 9-, is 2. 
 
 Definition 1. 5 » Let 9-,-,, 9 22 be equivalence relations on S 
 such that 9-^° 9 22 = 0. Then 9 21 is a symmetric, re- 
 flexive relation on S defined as follows. For s-,, s ? € S, 
 
 (s i®2i s 2^ ^ anc * on ^y ^-^ 
 
 (V<*) (Vs 1 » ) (Vs 2 * ) [O s i , © 22 s 1 ) ( s 2 ?9 22 s 2 ) ( V 9 llV ^ 
 
 > Cf(s 1 S^)9- L1 f(s 2 S^<)j] . 
 Proposition 1.4 . 
 
 (i) 9 12 2 9 n and 9 21 2 9, 2 
 (ii) 9 12 9 f _1 (9 22 ) and 9 21 £ f" 1 (9- L1 ). 
 Proof. Part (i) is easily shown by noting that {q*A 
 
 are 
 
 equivalence relations such that f\ 9 
 
 i = l 
 
 11 
 
 0< 
 
 To prove part (ii), 
 
33 
 let s^ 12 s 2 » Then for each s-^ and for each s 2 T such that 
 (s 1 »9 11 s 1 ) and ( s 2 fe n s 2^ and ^ s l t9 22 s 2^ we have that 
 (^r f ^ s i ?)9 22 f <=< (s 2 t ^ • In Particular, 
 
 (¥cO[f,( Sl )0 22 f,(s 2 )]. 
 
 y. h. d. 
 
 With the result of the above proposition and the 
 discussion on pages 2#, 29 of this chapter, we have the fol- 
 lowing theorem. 
 
 Theorem 1.1 . Let (M-^M^ be a GPD of M = <S,T,f >. Let the 
 associated GPDT be <S-,,S 2 ,) D > such that P is one to one from 
 J c s^ S 2 onto S and 1£{Ji) = Sp ^ 2 {J) = S 2 . Then 
 (M-,:M 2 ) gives rise to a set of relations { 9. . |l-i, j-2 J on S 
 such that 
 
 (i) Q -. -. , 9 22 are equivalence relations on S and 
 
 9 11 A 9 22 = °* 
 (ii) 9 12 3 9 n 
 
 y 21 - y 22 
 (iii) 9 12 ^ f" 1 (e 2 2 ) 
 9 21 c f -l( Qll ) 
 
 For the result in the other direction, we have the 
 following theorem. 
 
 Theorem 1.2 . Let M = <S,T, f>. Let { 9 ± . |l-i, j*2 } be a set of 
 relations on S such that 
 
 (i) 9-.-,, 9 22 are equivalence relations on S and 
 
 9 li n 9 22 = 0# 
 (ii) 9 12 3 n 
 
 9 21 - 9 22 
 
 (iii) 9 12 £ f _1 (9 22 ) 
 
 21 ^f" 1 (9 11 ). 
 
34 
 Then there exist two incomplete automata M-, = <^S-, ,T-, ,f , ) , 
 
 M 2 = ^ S 2 ,T 2 ,f 2^ such that 
 
 (i) [M 1 :M 2 J is a GPD of M 
 
 (ii) y> is 1 -* 1 where <S ±l> S^j^is its associated GPDT. 
 (iii) (M 1 :M 2 ) gives rise to the set ^9. . |l-i, j-2}. 
 Proof . Let M- L =<S 1 T 1 ,f- L >, M 2 =<S 2 ,T 2 ,f 2 ) be defined as fol- 
 lows. Let S-, be the family of equivalence classes under 9-,-,. 
 Similarly, let S 2 be the family of equivalence classes under 
 9 22 . For each s € S, let (s)-,£ S-, and let (s) 2 ^ S 2 » Let 
 
 T l = S 2* T * T 2 = S 1 X T * Define f i :S i x T i * s i such that 
 
 for all ((s)-^ (s') 2 ,«')if (s) i r\ (s») 2 f 0, then 
 
 f 1 ((s) r (s') 2 ,°<) = ff((s) 1 /°» (s') 2 ),°<). This is well-defined 
 
 Since ©ni^ ©22 = ® 9 ^ (^i^ ^ s? ^2 ^ $' then tnere exists 
 exactly one unique element in the intersection. We define 
 f 2 similarly. 
 
 We now proceed to show that (M-, :M 2 ) is a GPD of M. 
 Define J £S ± * S 2 such that J = { ((s)-^ (s») 2 ) | (s)-^ (s») 2 /0). 
 Let y be a function from Jf onto S such that 
 Mslp (s») 2 )^J[y ? ((s) 1 , (sM 2 ) = (s) 1 A(sM 2 ]. Clearly, 
 (M-^M^ is a GPD of M and if <S ][ , S 2 ,)°> is its associated 
 GPDT, f is 1 — > 1. 
 
 By the construction of M-. , M 2 , it is obvious that 
 M-^, M 2 induces two equivalence relations 9-,-,, 9 2 2 res P ect i ve ly 
 and 9-^n © 22 = 0. Let Sp s 2 € S. Let (s-jO-j^s^. Let 
 
 s^jS^e S such that ^ S 1 T9 11 S 1^ s 2 f9 ll s 2^ s l te 22 s 2 ? ^ ° Then 
 since 12 C f" 1 (9 22 ), (Vo< € T) (f (s^JO^f (s 2 ,*)] . Also, 
 
 G ll- 9 12- f " 1(9 22 ) * thus (^)(f(s 1 »,o<)9 22 f(s 1 , <)] and 
 ^V^)(f (s 2 », G <)9 22 f (s 2 ,o<)J . Since 9 22 is transitive, 
 v 0ff (s 1 «,«<)0 22 f (s 2 »,o<)] . Hence, if (sjO-^Sg) then 
 
35 
 (V o 0(¥s 1 O(¥s 2 M((;(s 1 »9 11 s 1 )(s 2 '9 11 s 2 )(s 1 »9 22 s 2 )3 
 
 ==>P(s 1 f ,-00 22 f(s 2 So0j) • 
 We can construct a similar proof for 9 2 -, . Thus, [M-, :M ? ] gives 
 
 rise to the set { 0. . 1 1-i, j-2 }. 
 
 3 Q.E.D. 
 
 We define a cover ^ of a given set A as a col- 
 lection of non-empty subsets A, of A such that their union 
 is A. Any subcollection of elements in Ot, whose union is A 
 is a subcover of OC* A minimum subcover 0t*. is a subcover of 
 a given cover OC such that it has the least cardinality. The 
 classical covering problem is to find a minimum subcover 0C*. 
 of a given cover OC. 
 
 For any reflexive, symmetric relation relation & on 
 the set S, we can represent the relation A by a cover C- of S. 
 Each subset of S in this cover C# is a collection of elements 
 which mutaually bear the 6 relation. Thus, if we consider 
 9. ., i / j in particular, we can also represent this relation 
 by a cover of S. In the case when the relation A is an equiv- 
 alence relation, the cover is represented by the collection of 
 its equivalence classes. In this particular case, the cover 
 is itself the minimum subcover. We shall let n{£) denote the 
 number of elements in a minimum subcover of the cover Cg, where 
 A is a symmetric and reflexive relation on S. This number 
 n(^) is unique. 
 
 §2. Congruence Relations of Degree k as Applied 
 to Decomposition Theory 
 In §1, we have developed properties of congruence 
 relations of degree k. In this section, we consider its appli- 
 cation to the theory of decomposition. Again, we shall study 
 
36 
 
 GPD (M- | :MJof M = <S,T,f) where ^f is a one to one function 
 from J 9 S ± x S 2 onto S £ ^( yf ) = S ±$ ff 2 { J) = S 2 . In this 
 case, M is an isomorphic image of a sub-automaton ^/, T,F^> 
 of G(M-,,Mp). Automata M-,, M 2 induce two equivalence relations 
 
 6 11' 9 22 on S > 9 11 A 9 22 = °- 
 
 Let S, = { a-, , . . . , a } , S 2 = f b i» * • * » b n } * For each 
 
 a.€ S n , define f :S XT into S~ such that 
 1 1 a- <. £ 
 
 f?(b^)6S 2 XT)(f ai (b.^) =^ 2 r^ rl (f(/ :, (a i ,b j ),-<)j]J . Simi- 
 larly, (Vla^Jes^Tj^U^) =^ 1 /'^Yf(^a i ,b j ),^])J . 
 
 «j 
 
 We note that for each a., D(f ) = / (b .,o<) I c<6 T, (a. ,b .) 6 V{f) 
 
 i i ^ ± J 
 
 and {f{a ± ,b .),*()€ D(f ) J, and for each b., 
 
 D(f b ) ={ (a^cK) |o<6T,(a i ,b j )€D()p) and ( ^ ) (a i ,b j ),<*) 6 D(f ) }. 
 
 j 
 Definition 2.1 a For any a-,, a 2 e S-,., f is compatible with 
 
 f ; f **** f if and only if they are the same mapping 
 a 2 a-L a 2 
 
 over their common domain (i.e., D(f ) A D(f )). 
 
 a l a 2 
 Similarly, we define for b-,, b 2 , f , *>** f . . 
 
 We note that compatibility is not an equivalence 
 
 relation. It is an equivalence relation when n-,n 2 = #(S). 
 
 In that case, the common domain for all f is S 9 X T, and for 
 
 a <~ 
 
 all f b is S 1 X T. 
 
 Suppose we consider a set of mutually compatible 
 
 functions f as one function. As a. ranges over S-,, we ob- 
 a^ i e 1* 
 
 tain a collection of such distinct non-compatible functions. 
 
 Let p be the number of elements in this collection. Since 
 
 compatibility is not transitive, as a. ranges over S-. we may 
 
 obtain several collections of such distinct functions. We 
 
 denote the minimum p by F q as we range over all possible 
 
 b l 
 
37 
 
 collections. Similarly we use F q • In view of the nature of 
 
 b 2 
 these functions we shall call F s the number of input configur- 
 ations from M-^ to M 2 , and Fg the number of input configurations 
 from M 2 to M-, . 
 
 For each such collection of non-compatible functions 
 from M-, to M 2 and from M 2 to M-, , we must encode it into alpha- 
 betical information from M-, to M 2 and from M 2 to M-, respectively. 
 These, in turn, must be coded into binary signals. If we denote 
 by fxl the least non-negative integer £ x for any real number 
 x, then we need flog 2 F s "1 and |log 2 Fo | numbers of binary 
 signals. We shall call flog 2 Fo ^ , flog 2 Fol the number of 
 connecting lines from M-, to M 2 and M 2 to M-, respectively. 
 
 We have so far introduced the notions of input con- 
 figurations and connecting lines between machines. The next 
 natural question is how does one find these numbers? First, 
 we show the following proposition. 
 Proposition 2.1 . Let s-, , s 2 6 S 3 *%( & (s-,)) = a-^, 
 / ^_(^ 1 (s 2 )) = a 2 . Then f ^ f a if and only if s-jO-^s^ 
 Proof . The statement really says that Definition 1.5 and 
 Definition 2.1 are equivalent. This is quite straightforward. 
 Since 9-^ is defined such that (s^sj if and only if 
 ^(^ 1 (s i )) ="^ 1 ( ^ 1 (s.)), and 9 22 is defined such that 
 (s.Q^s^^^s.)) =V 2 (f(s.))], f ai ~ s *. 2 ^Uy 
 
 means s n 0-, o s o , and vice versa. 
 
 1 ^ z Q.E.D. 
 
 Similarly, we have a proposition for f fe . Let 
 
 Sl ,s 2 6S 3^ 2 (^ 1 (s 1 )) = b^ ^ 2 (f 1 (s 2 )) ■ V Then t.~> f. 
 
 if and only if Si©oi s 2* 
 
 As a result of the above proposition and the discussion 
 
3S 
 
 at the end of §1, the problem of finding the numbers of input 
 configurations becomes a covering problem (l9)« This covering 
 problem is the same problem as, for example, the state minimi- 
 ation of incomplete machines. Thus a detailed study of this 
 type of problem can be found in several articles (2,13,16,18). 
 Using the notation in §1, we denote by nlO^) and n(9 2 -,) the 
 numbers of input configurations from M-, to M 2 and M 2 to M-, 
 respectively. 
 
 Let M 1 = <S 1 ,T 1 ,f 1 >, M 2 =<S 2 ,T 2 ,f 2 > such that 
 CM 1 :M 2 ^ is a GPD of M. Let <S ls S 2 ,y>> be the associated GPDT, 
 and let y? be one to one. For each a € S-, , each b € S 2 , there 
 exists an /-tuple (a-,, •••«a/) such that (i) For each i, 
 Up©^) 6D(f b ), then a ± ={^(3,^)] (ii) For each i, 
 (a,°C.) ^ D(f, ), then a. - 0. For each a€ S-,, let the collec- 
 tion of all such /-tuples as b ranges over S 2 be the successor 
 set of a. Define two /-tuples (a-,>»«°>a^) and (an',...^/) 
 as equal if and only if for each i where a. f and a- ? / 0, 
 
 then a. = a . ? . Let L q denote the maximum number of distinct 
 ii b x 
 
 /-tuples in any successor set of a as a ranges over S-,. 
 
 Proposition 2.2 . The number L Q is a lower bound on the number 
 
 b l 
 of distinct non-compatible functions (input configurations) 
 
 from M 2 to M-,. 
 
 Proof . This is a direct result of the definition of L Q . 
 
 b l 
 There are at least L c distinct f, , since these f, map the 
 
 s 1 bj bj 
 
 same input-state pair to distinct next state. Therefore L Q 
 
 b l 
 is a lower bound. 
 
 Q. E. D. 
 
 Recall the definition of a congruence relation 9 of 
 degree k. The number N(s,0 11 ) is thus defined as the number 
 
39 
 
 /**/ 
 
 of distinct equivalence classes of 9-,, in -ff (s',T) | s , 9 1 s } The 
 maximum of N(s,9-,-,) for any 5 is < = ^(O.,-,) and is exactly the 
 
 number L^ - Therefore, we have the following proposition. 
 
 b l 
 Proposition 2.3 » There exists a congruence relation 9-,-, on S 
 
 of degree k if and only if there exits a GPD [M-,:M 2 ) f M such 
 that 
 
 (i) M-, induces 0-, -. . 
 (ii) f is 1 — > 1 where (S.^ S 2 ,J*> is the associated GPDT. 
 (iii) k is a lower bound on the number of input configura- 
 tions from M 2 to M-, . 
 Let 9-,-, be a congruence relation of degree k, k>l, 
 on S. Suppose 9 22 is anv equivalence relation on S such that 
 9 r\ 9 22 = 0. Then n(9 2 -,) is not necessarily equal to k, 
 i.e., k is only a lower bound on the number of input configura- 
 tions from M 2 to M-,. This can be easily explained by an example. 
 Let M = <S,T,f> where S = f 1, 2, . . . , 6} and T = {^}. Let the 
 transition function be: 
 
 f 1 2 3 4 5 6 
 o< 3 4 5 6 2 4 
 
 Let Q 1± = f (1,2,3), (4, 5,6j), then 9 n is of degree 2. Let 
 
 Sj^ = fa-^a^ where a ± denotes {l,2,3}, and a 2 denotes (4,5,6/. 
 
 Suppose we choose 9 22 = { {l,4J, (2, 5J, (3, 6} j. Let S 2 = {b-^b^b^ 
 
 where b ± denotes {l,4}, b 2 denotes {2,5}, b 3 denotes {3,6}. 
 
 It can easily be shown that n(9 2 -^) = 3» 
 
 Our next result will show that if k = 1, k is the 
 number of input configurations from M 2 to M 1 « 
 
 Proposition 2.4 . There exists an ordinary congruence relation 
 9-,-, on S if and only if there exist two incomplete automata 
 
40 
 
 M 1 =^S 1 ,T 1 ,f 1 > s M 2 = <S 2 ,T 2> f 2 ) such that Cm i :M 2 J is a GPD 
 of M and 
 
 (i) M-, induces 0-. -. • 
 (ii) f is 1 — > 1 where <S 1 ,S 2 ^>is the associated GPDT. 
 (iii) The number of input configurations from IYL to M-, 
 is 1. 
 Proof . (i), (ii) can be obtained directly from Proposition 
 2.3. Tow show (iii), we need only show that (Vs-,,s 2 € S) (s-,9 2 , s 2 ), 
 i.e., n(9 2 -j) = 1« This is quite simple. Let s-,,s 2 6 S. Let 
 ^ s l 9 ®22 s l^ s 2 ?e 22 s 2^ - s l ?e il s 2 ? ^* Since 9,, is an ordinary 
 congruence relation., (s-, '©itSo' ) == ^ (¥<^) (f (s^W)©-. nf ( s 2 f ,<=*)J . 
 
 Thus (¥s is s 9 6 S)[s-,9 9 i s 2 ). 
 
 1 * -l ^J- <: Q.E.D. 
 
 Remark 2.1 o We shall show in Chapter III that the result of 
 this special case includes the known results [l2] about 
 series decomposition of an automaton. 
 
 If we combine the results of Theorem 1,2 and Propo- 
 sition 2o3 5 we reach the following. 
 
 Proposition 2. 5 » There exist two congruence relations 9-, -,, 9 22 
 of degree k,, k 2 respectively £ 9^,0 9 22 = if and only if 
 there exists a GPD (M.,:M 2 ) of M 
 
 (i) M-., M 2 induce 9-,t> 9 22 respectively, 
 (ii) ^ is 1 -> 1 where <S- L ,S 2 ^> is the associated GPDT. 
 (iii) kp k 2 are the lower bounds of the numbers of input 
 
 configurations from ]VL to M-, and M, to M 2 respectively. 
 Finally, with the aid of proposition 2.4* we have 
 Proposition 2.6 . There exist two ordinary congruence relations 
 ^ll 1 9 22 on 3 such that 9-^A 9 22 = if and only if there 
 exists a GPD (1^:1^) of M such that 
 
41 
 (i) Mp M 2 induce 9^, 9 22 respectively, 
 (ii) f> is 1 — > 1 where <S- L ,S 2> ,y>is its associated GPDT. 
 (iii) The numbers of input configurations from M 2 to M-, and 
 from M-, to M 2 are 1. 
 Remark 2.2 . We shall show in Chapter III that the result of 
 this special case includes the known results [l2] about 
 parallel decomposition of an automaton. 
 
 We have shown that if 9-,-, is a congruence relation 
 of degree k on S and if (M-^M^ is a GPD of M = <S,T,f> and 
 the states of M-, induce 9-,-, on S, then in general K is just 
 a lower bound on the number of input configurations from JVL 
 to M,. We shall now establish the result that this lower bound 
 k can always be reached if we allow M 2 to vary. We shall obtain 
 an upper bound on the number of states of such M 2 and show that 
 it is always less than the number of states in the original 
 automaton Finally, we show by an example that, in fact, this 
 upper bound is sometimes the least upper bound. 
 
 Let M =<S,T,f) and #{S> = n. Let Q^ be a given con- 
 gruence relation of degree k. Denote by f P i | l-i-nO^)} the 
 set of equivalence classes of 9^. Let Yl{? ± ) be the number of 
 distinct f~ 1 (© 11 ) classes in P j _. Let f ± denote the number of 
 elements in each equivalence class P. . We shall choose P^ 
 to be a particular equivalence class of 11 such that ^(Pj) = k 
 and if, for some j, >J(P.) = j> then /,• S /]_• 
 Definition 2.2 . For each i, l-i-n(9 u ), define 
 
 r /. - (k+1) if L> k+l 
 
 g(i) -{ X 
 
 v otherwise. 
 
 Theorem 2.1 . Given a congruence relation 11 of degree k, there 
 exists an equivalence relation 9 22 on S which, together with Q 1±t 
 
42 
 
 gives rise to a reflexive, symmetric relation 9 21 as defined in 
 Definition 1. 5s such that 
 (i) 9 i;L n 9 22 = 0. 
 
 r k if (Vi)(/ ± « k) 
 
 (ii) n(0 2? ) = j n(9 n ) 
 
 1 (k+1) + J2 S^ otherwise 
 
 i=l 
 
 (iii) m(0 11 )^ n(9 22 ) ^ n 
 
 and n(Q 22 ) = n only if n(9 11 ) = 1. 
 (iv) 21 2 9 22 
 (v) n(9 21 ) = k. 
 Proof . For each P., order the elements in P. such that the 
 first ^(Pj) states give a complete set of representatives for 
 the f~ (Qyi) equivalence classes in P.. For each i, let s. . 
 denote the t state in P.. 
 Case I . (¥i)(/^ k). 
 
 Let Q,,...jQ, be k subsets of S defined as follows: 
 For each i„ l-i^n(9-, 1 ) 
 
 (1) For any t, 1-t-/., s. . € Q. for one and only one 
 j> 1-j-k. 
 
 (2) For any t, t» 3 t / t» and 1-t, t»^/. , and if ( s . .6 Q.) 
 
 1 1, X) j 
 
 and (s i<t? € Q. f ), then j/j ? . 
 Clearly, by (1), {Q.j -,^±^ir is a collection of disjoint subsets of 
 S. By (2), no two distinct states in any 9-, -. -equivalence class 
 
 belong to the same Q.. Since at least /, = k, we have 5 for any 
 
 k 
 j, 1-j-k, Q is non-empty. Clearly, U Q. = S. Further, { Q. } 
 
 J i = l ± 
 
 forms a partition on S; thus induces an equivalence relation 9 22 « 
 
 Therefore, we have 9 n ^ 9 22 = and n(9 22 ) = ko From Q 1± , 9 22 , 
 
 define 9^ as in Definition 1.5. Hence 9 21 2 9 22 <, by Theorem 1.1. 
 
 Thus n(9 22 ) = k * n(9 21 ) - k. Thus n(9 2] _) = k. 
 
43 
 Case II . (3i)(l-i-n(9 11 ))(/.>k). 
 
 Let m be the minimum such i. Since fy(P )-k<k+l, 
 
 and by the ordering we took, 3t , l^t -n(P )-k 
 
 m m m such that 
 
 s m,t ^ f ~ ^°11^ s m,k+l' simi larly, for each i such that /. > k, 
 3t ±J l*t 1 *n(P 1 )<k such that s ijt Cf" 1 (9 11 )]s i ^ k+1 . 
 
 Let Q 1? ...,Q k+1 be disjoint subsets of S defined as 
 follows. 
 
 (1) Let m be designated as above. Let s , A , € Q, , and 
 
 m, k+1 k+1 
 
 ^" et s m 1* * e,,s m k ^ e assi S ned t0 Qi»»»»*Qi, respectively. 
 
 (2) For each i, f ± > k and i / m, let s ± k+1 e Q k+1 and let 
 
 s i,t. e v 
 
 i m 
 
 (3) For each t, t f t ± , l^t^k, 3 a unique j / t ffl , l^j^k 
 
 9 s i>t €Q.. 
 
 (4) No two states s. . , s . +.. with 1-t, t f -k belong to the 
 
 same Q. for any j, 1-j-k. 
 j 
 
 Finally, each of the remaining states constitutes a set by 
 
 n(9 u ) 
 
 itself. Then there are exactly r = 2_ g(i) more sets, 
 
 i=l 
 
 denoted by R,,...,R . Together with Q. , they form a disjoint 
 
 partition on S. This, in turn, induces an equivalence relation 
 
 n(9 11 ) 
 
 9 22 on S. Certainly G^^ © 2 2 = ° and n ^ 9 22^ = ^ k+1 ^ 4- s(i)» 
 
 Furthermore, define 9 21 from 9,^ 9 22 as in Definition 
 1»5« Thus we have 
 
 (1) Every state in Q k+1 is Q 21 -related to every state in 
 
 Q t by definition of Q k+1 and Q t . 
 m m 
 
 (2) For each j, l^j-r, 3 i, 1-i-k 3 the only state s <= R. 
 is 9 21 related to states in Q^. 
 
44 
 (3) If s€Q ±9 s ? 6Q i , and i / i\ 1-i, i'-k, then s0 21 s» 
 
 Hence we have n(9 01 ) = k. 
 
 £1 Q. E.Do 
 
 The following example shows that the number 
 
 r k if (Vi)(/.=k) 
 
 n(0 22 ) - n(Q^) 
 
 ^ (k+1) + }_ g(i) otherwise 
 i=l 
 
 may be the least upper bound. 
 
 Let M = <S,T,f> be given, where S = (l,2, ...,$}, 
 
 T = [°i\ and the transition table is as follows: 
 
 f 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 3 
 
 <* 
 
 1 
 
 6 
 
 4 
 
 5 
 
 3 
 
 7 
 
 7 
 
 5 
 
 Let n ={{l,2,3,4},{5,6,7»3}} and 9 U is of degree 2. In 
 
 order to find 9 22 ^ ®n^ ®22 = ^ s n ^®22^ must be at least 4° 
 
 However a choice of 9 22 where n (©22^ = ^ ^H result in a ©oi ^ 
 
 n(9 2 -, )> 2o In order to obtain n(9 2] ) = 2 a 9 22 may be 
 
 n(9 n ) 
 
 f{l5U26],{3},{V7}{8}} and n(9 ?? ) = 5 = (k+1) * Zl " g(i)« 
 
 > ^ i=1 
 
 The results of this section are very important. 
 We shall summarize the significance as follows. Let 
 M =<S,T,f ) be a given automaton. If (M- L :M 2 ) is a GPD of M s 
 then to find the number n(9 2 -,), n ^io^ ( in P ut configurations 
 from M 2 to M-, and from JYL to M 2 respectively) is a covering 
 problem. Let 0,, be an equivalence relation on S. If k is 
 the degree of 9-^, then k is the lower bound for n(0 21 ) for 
 any choice of the equivalence realtion 9 22 such that 0-q^ e 22 =0 * 
 If k is the degree of 9-,-. and k is larger than the desired bound 
 on the number of input configurations going into an automaton,, 
 then without finding the decomposition one may disregard 0,-. 
 
45 
 
 immediately and pick another 9-,-,* such that 9-,-j 1 is of degree 
 less than or equal to the bound. If 9-,., is a non-trivial con- 
 gruence relation (i.e., 1 <n(9-, •,) < n) of degree k, then there 
 always exists a non-trivial equivalence relation 9 22 on S such 
 that 0,, A 9 22 = °> and n(9 21 ) = k. Finally, an algorithm 
 can be easily obtained to find an equivalence relation 9 22 on 
 S3(i) 9,,^ 9 22 = (ii) n(9 21 ) = k and (iii) n(9 22 ) = minimum 
 of all n(9 22 »), where © 22 ' satisfies (i), (ii) (i.e., there 
 exists a minimal state automaton M 2 such that [M-, :M«] is a 
 GPD where M-,, M 2 induce 9-,-,, 9 22 respectively and n(9 21 ) = k). 
 
46 
 
 Chapter III 
 SET SYSTEMS AND *- COVERS 
 
 §1. ^-covers with Substitution Property- 
 Let M =<S,T,f) be a given incomplete automaton. 
 Definition 1.1 . A set system CC= (A,,...,A^) of S is a 
 
 *-cover of S such that (¥i,j')(l*i, J*/) CUj* Aj£=Mi=jj) . 
 Definition 1.2 . A *-cover &C= (A,,..., A*) of S is said to 
 have the substitution property (SP) if and only if 
 
 (¥t€T)(¥i)(l^i^/)(3J)(l-J-/) 
 
 [f(A i ,t) = {f(s,t)|s€A i ,(s,t)eD(f)} Ca). 
 
 In the following, we shall show that the known results 
 (12J for series or parallel decompositions are included in the 
 results of Proposition 2.4 and Proposition 2.6 of the previous 
 chapter. 
 
 Proposition 1.1 . Let M = <Cs,T,f)> be an incomplete automaton. 
 Let 0C- = (A,, ...,Av) be a *-cover with SP. Then there exists 
 an automaton M* = <S,T,f »,), and a homomorphism p from M r to M 
 and an ordinary congruence relation 9-,-, on M f such that 9-,-, 
 induces OL.. under p. 
 
 Proof. For each i, l*i</, let k ± = { s^ ± 1 j , . . . , Sj^ j} . 
 For each pair (i,j) such that s /. . \€A. , we associate a dis- 
 tinct new symbol S/. .y Let A. = f s\ . -.>.,... ,~s, . \J. Let 
 S be the collection of all new symbols as i ranges over 
 (1,2,...,/}. Define a mapping P f rom S onto S such that for 
 
 a11 ^(i,j) € &' P^[± i)* = s (i i) # since &i has SP > for 
 each t € T, each coordinate A. of QC->, we choose a particular 
 
 number h(i, t ) € { 1, 2, . . . ,/ } such that f (A ± ,t) £ k h , ± t y Let 
 
47 
 
 f be a function from SxT into S such that if f(s ( . n ,t) 
 
 i i* J / 
 
 = s (h(i,t),k)' then f,(i (i,j)' t) = ¥ (h(i,t),k)- Let 
 
 M« = < S,T,f>. Since t*f> ~ f • t* 9 f> la * homomorphism from 
 
 M* to M. The collection of subsets {I.} forms a partition 
 
 on S, and thus induces a unique equivalence relation 9-,-. such 
 
 that ((s^ .)Q i;L S( h k j) £=> (i=k)J. Certainly Q ±1 induces tit^ 
 
 under f> . It remains to show that 9,, is a congruence relation. 
 
 Let s/. -j) e n s (i k)* For each t, by construction, f f (s,. ■\)> t ^ 
 
 = ^h(i,t),p) and * , <»(i f k)' t) = »(h(i,t),q)' HenCe 
 ^ f ^ s (i 1 )' ^ ^11^' ^ s (i k) ,t ^ anc * ®11 is a congruence relation. 
 
 Q.E.D. 
 As a result of this proposition and Proposition 2.4 
 of Chapter II, we have that if 0C-, is a *-cover of S with SP, 
 then there exists a series decomposition for M = <(S,T,f). 
 Since a set system is a special *-cover, our result in Propo- 
 sition 2.4 includes the known results for set systems in (12} . 
 Corollary 1.1 . Let Ct be a *-cover of S with SP. Let S" be a 
 set of new symbols. Suppose OC x = (A-,", . . . ,A^" ) is a partition 
 on S" such that there exists a function & y from S" onto S and 
 such that OC n induces OL under P». Suppose there exists a 
 function T f rom S" onto S such that 0C" induces CL-\ under f 
 and <^y°= f>* where P, S, <fc, are as defined in Proposition 
 1.1. Then there exists a mapping from f" on S"x T into S" 
 such that M" = <S",T,f"> realizes M and fa," is a congruence 
 relation on M n . 
 
 Proof . Simply define f" as follows. For each s^ .j€ A i ,f , 
 choose f"(s (i j)ft)€ <r" 1 (f f (^Tls (i j)»t)). It is easily 
 shown that p 1 is exactly the homomorphism from M" to M. 
 
 Q.E.D. 
 
4S 
 
 For the case of parallel decomposition, we have the 
 following proposition. First, we introduce the following 
 definition. 
 Definition 1.3 . Let fit = (A- L ,...,A^), ^6= ( B i>"*> B m ) be 
 
 --covers of S. Then {6tn^) = if and only if for each 
 pair (i,j), l-i-/> 1-j-m* (A.AB.) contains at most one 
 element of S. 
 Proposition 1.2 . Let M =<S,T,f). Let CL^ 0Z 2 be two --covers 
 of S with SP such that {0t^\ 01^) = 0. Then there exist an 
 automaton M T = <( S, T, f f ^ and a homomorphism *f f rom M 1 to M and 
 two ordinary congruence relations 9,,, 9 22 on ^ suc ^ that 
 0-, -j A 9 22 = 0, and 9-, , , 9 22 induce OL, OC respectively under JP. 
 Proof . Let Ot^ = (A- L , ...,Ay), OC 2 = (Bp ...,B ). Let 
 E x = {l,2,...,/}, E ={l,...,m}. Let J^E^XE such 
 
 that ^0 = { (i, j) I (A i AB.) / 0}. Define a mapping f such 
 that )<U,j) = s if s€(A.nB.) / 0. Thus D(^) =>f . Since 
 0C ±f Ot 2 are ^-covers of S, ^{J) = E , ^ 2 {J) = E^ . Now 
 
 consider ^ as a table in which the rows are labeled by i, 
 1-i-/, and columns are labeled by j, 1-j-m. If V?(i,j) = s, 
 then s occupies the entry (i,j). For each occupied entry 
 (i,j), we introduce a new symbol S/. . \. Let S be the collec- 
 tions of all new symbols. For each i, 1-i-/, let 
 
 A. =4 S/4 -J the entry (i,j) is occupied j. 
 
 For each j, 1-j-m, let B. ={ ~s,. .\ the entry (i,j) is occupied}. 
 Let 01^ = (A ] _,...,A^), 0t 2 < - (B-l, ...,B m ). Thus {h ± }, [bA 
 are two partitions on S and hence induce two equivalence rela- 
 tions 9-^, 9 22 on S. Since ( fo^n 01 ) = 0, by construction, 
 
49 
 
 ©JjTS ©22 = ®* It: is olDV:i - ous that 9,-,, 9 22 induce (TC -, , <^ ? 
 respectively under p . 
 
 By Corollary 1.1, for the *-cover fit-, with SP, we 
 have an automaton M-, =<Cs,T,f-,) realizing M, and Ot, r is a 
 congruence relation on M-, . Similarly, we have an automaton 
 M 2 = ( S,T,f 2 ) realizing M, and ^ 2 f is a congruence relation 
 on M 2 « Define an automaton M ? = < S, T,f T > as follows. For 
 
 sS 
 
 each s/. . \€ S and each t € T, define f f (s/. . \>t)=s > where 
 v i» 3 i \i-t 3 ) 
 
 f _i (i (i,j)' t)eA ^(i,t)' r 2 (i (i.3)» t)6B h 2 (j,t) and 
 
 "? = AtI / . . v n B» . We shall show that f» is well- 
 h l (l ' tJ h 2 (j,t) 
 
 defined on SxT whenever f, and f« are defined, i.e., we must 
 show ^(i.t)^ B h 2 (j,t) ' *' B ^ Corollary 1,1, 
 
 }fl[f 1 (5 (lfJ ',.t))e A hi ,. )t) , and flf 2 <"Cl,j)' t))<B h 2 (j.t)- 
 
 Further, 
 
 ^(■(Ljj.t)) ■r(« (1 ,j,.t) -) 9{f 2 ( »li.j)' t,) - 
 
 Hence* if ?= f(s/. ,wt), then £ is the unique element 
 
 in A, , . . \ f\ B, / . . x which implies ? exists and is unique 
 h 1 (i,tj h 2 (j,tj r > 
 
 inA' ,. +1 HBi ,. fl . Clearly $T-, » is still a congruence 
 h 1 (i,t) h 2 (j,t) ' 1 
 
 relation on M», since for each A^ and S|. .jU^, we have 
 
 f,(B {i,j)' t)€ *i 1 d.t) wheref i (i (i,j)' t)e A ni (i,t)- simi - 
 
 larly, ^ 2 f is a congruence relation on M* . 
 
 Q. E. D. 
 
50 
 
 Again with the result of Proposition 2.6 of Chapter II, 
 this includes the known results [12) for the parallel case. 
 
 Any attempt to obtain a stronger statement than that 
 in Proposition 1.2 of this section, such as replacing the con- 
 dition "(Jt^r\Cl 2 =0" by "(#-,/&) has ADDP" will not be success- 
 ful. We show this by an example. 
 
 Example 1.1 . Let M = <S,T,f), where S ={l,2,...,7J and 
 T = {°<} and the transition table be as follows. 
 
 f 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 o( 
 
 5 
 
 4 
 
 1 
 
 2 
 
 1 
 
 4 
 
 2 
 
 Let 61^ = (A ] _,A 2 ,A 3 ) where k ± ={4,6}, A 2 = (2,4,7}, 
 
 A 3 ={l,3,5}. Let CZ 2 = (B 1 ,B 2 ,B 3 ,B 4 ) where B ± = {4,5,7}, 
 
 B 2 = {l,2/, B-3 = {3,4/, and B, = { 6j. It is easily shown that 
 
 Ct lt CC 2 are set systems with SP, ( ^i 3 ^2 > has ADDP and 
 
 OC-^ 0t 2 { 0. Let dt ± = (A^A^X) where I 1 = f 4,6}, 
 
 A 2 ={2,4 f ,7}, A 3 ={l,3,5} and let 6C 2 = (B^Bg,!-,!, ) where 
 
 \ =(4,5,7J, B 2 ={l,2}, B 3 ={3,4 T }, B 4 ={6). Clearly 
 
 0L]> OC 2 are Partitions on S ={ 1, 2,3,4,4', 5, 6,7} and 
 
 0L-^\ ^2 = °* In order t0 make ^C-, a partition with SP, we 
 
 must require f(4, c <) = 2, f(6,»0 = 4 T , f(2,<*0 = 4 T , f(4 T ,°<) = 2, 
 
 f(7,°0 ■ 2, f(l,*.) = 5, f(3,°0 = 1, f(5,<*) = 1. In this case, 
 
 fr- 2 does not have SP any more, since f(l,°<) = 4% f(2,©<) = 5, 
 
 and 4 T , 5 are not in one block of OCy. 
 
 §2. ^-covers vs. Set Systems 
 Definition 2.1 . Let frt = (A 1 , ...,A^) be a *-cover of S. Let 
 (A 1 T ,...,A *) be a set system of S. Then 0O is said to 
 
51 
 be induced by CC is and only if 
 
 (i) (¥i)(l-i*/)Oj)(lM«p)£ A ± £ A.') 
 
 (ii) (Vi)(l^p)(3j)(l^/)[A.' --Aj] 
 
 *-covers differ from set systems in that they may 
 contain coordinates A., A. where A. £ A . yet i / j. From the 
 development, we see that each coordinate in a *~cover repre- 
 sents a state in an automaton. The natural question is then 
 if Ai S A ., do they represent the same state? Can we always 
 replace the *-covers by their induced set systems and obtain 
 essentially the same decomposition? In particular, if (M-,:M 2 ] 
 is a series or parallel decomposition of an automaton M, and 
 0C$'£> are the ^-covers that [M, :Mj gives rise to, then if 
 0t»^ are replace by their induced set systems, is the resul- 
 tant still a series or parallel decomposition of M? 
 
 We shall consider first the series decompositions. 
 Let M = <S,T,f>. Given (^fc,$) a pair of ^-covers of S with 
 ADDP such that OC has SP. Then by Proposition 1.1 of Section 
 1, there exists a GPDT <E^ t ,E J0 , ( f > of M and a GPD C^rM^ such 
 that it is a series decomposition of M, that is, there exists 
 a GPD Cm 1 :M 2 J of M with M» = GfM^M^ = <S»,T,f»> such that 
 GfMnfML) is a series decomposition and OC induces a partition 
 with SP on S». 
 
 Theorem 2.1 . Let (<£,£>) be a pair of ^-covers of S such that 
 {0C,& ) has ADDP. Let 01 have SP. Let M» = <S»,T,f> be de- 
 fined as above. Then the induced set system OC 1 has SP, and 
 there exists a *-cover jg» of S' such that 
 
 (i) The number of coordinates of B' is no more than that 
 of $. 
 (ii) <E M E At ,/*> is a GPDT of M with an associated GPD 
 
52 
 
 (M'.M^J whose G(M 1 ',M 2 ») = M" realizes M, and [M 1 « iK^} 
 is a series decomposition of M. 
 Proof * Since ^ is a *-cover with SP, by Proposition 1.1, 
 there exists an automaton M» =<S»,T,f f >> a congruence relation 
 OC on M f , an equivalence relation j8 on S ? , and a homomorphism 
 Z 7 such that M» realizes M, and Ct,fi induce OL,fo under f 
 respectively. Hence OC,i?> give a partition pair series decom- 
 position CM-,:M 2 ] of M». And G(M 1 :M 2 ) realizes M under *f . 
 
 Let OC 1 be the set system induced by OC. Clearly, 
 tit* is a set system with SP. Define & s from $6 such that 
 •$» = (B 1 f ,...,B f ) where for each j, 
 
 B.» = B.-{s€S |(3i)(A. in&)(A. not in OV ) {f{k. ,B . ) = s) 
 
 and(^k)(A k in X')'[f{k k ,B*) = s}. 
 
 We note that B.' may be empty. In such cases, instead of re- 
 numbering the coordinates for ^8 *, we will not have a j co- 
 ordinate f or jg 1 . For each coordinate A. f of CO and coordinate 
 
 B.» of t6 f (i.e., for each j, B ' / 0), define 
 j j 
 
 y , (A i »,B.») = ^ , (A i ,B.), if (A i ,B.)€ D(^). Let S" be the set 
 of symbols obtained from the table of V". Then there exists 
 an injection map A from S" into S» such that f{3{%))= f* (%) 
 for each f 6 S". Define f '» (f, t ) € / 5" 1 (f » (fi(f ) , t ) ) where 
 (£t)€S"XT and {&{% ),t ) € D(f • ). Then M" = < S",T, f"> clearly 
 has a partition pair decomposition (M-, t :M 2 t j arising from 
 OV > 43 '> two partitions induced by 01, p under tf» . Further- 
 more, y?' is a homomorphism from M" to M. 
 
 Q. E. D. 
 
 Therefore, if {#C,]3) is a pair of ^-covers with ADDP 
 such that &L has SP, by replacing the *-cover OC by its induced 
 
53 
 set system OV > there always exists a *-cover j$» such that 
 0L r > iS T give rise to a series decomposition and -0 T has no 
 more coordinates than fp . However, the total number of con- 
 necting lines of the new series decomposition obtained from 
 OV , (/& f may be more than that of the original decomposition. 
 We further note that not both 0£>^5 can be replaced by their 
 induced set systems jO , 76 T respectively, since ^. T , ^ r 
 may not have ADDP any more. 
 
 Let M = <S,T,f>. Let 6C,-0 be two '--covers with SP 
 such that [4Z,j6) give rise to a GPD (M^lMgJ of M where GfMpMg) 
 is a parallel realization of M. We shall show by an example 
 that even when OC is already a set system, we still can not re- 
 place ft by its induced set system jg f and correspondingly Ot by 
 a new set system ft* such that the new resulting decomposition 
 is still a parallel decomposition. 
 
 Example 2.1 . Let M = <S,T,f> where S = { 1, 2, . . . , 6}, T ={<*} 
 and the transition function f is given as follows: 
 
 f 
 
 1 
 
 2 
 
 3 
 
 4 
 
 — 
 
 5 
 
 6 
 
 <* 
 
 5 
 
 4 
 
 1 
 
 2 
 
 1 
 
 2 
 
 Let K= (A 1 ,A 2 ,A 3 ) where k ± ={4,5,6}, A 2 ={2,l}, k^ ={3,4}. 
 Let ^6 = (B 1 ,B 2 ,B 3 ) where B 1 ={1,2,4}, B 2 ={6,2,4}, B^ ={ 1,3,5} 
 Let E ic = {a 1 ,a 2 ,a 3 }, E^ ={b 1 ,b 2 ,b 3 ] denote OC,fi respectively. 
 Let y> be a mapping from E^x E^ onto S as follows.. 
 
 f 
 
 b l 
 
 b 2 
 
 b 3 
 
 a l 
 
 4 
 
 6 
 
 5 
 
 a 2 
 
 2 
 
 2 
 
 1 
 
 a 3 
 
 4 
 
 4 
 
 3 
 
54 
 
 Consider the specific GPD (M-^M^ whose GPDT is 
 <E^,E^,^> such that G(M 1 ,M 2 ) = <S,T,F>, where 
 S - { 1,2,2', 3, 4,4', 4", 5,6} and 
 
 F 
 
 1 
 
 2 
 
 2' 
 
 3 
 
 4 
 
 4' 
 
 4" 
 
 5 
 
 - 
 6 
 
 o4 
 
 5 
 
 4 
 
 4 
 
 1 
 
 2 
 
 2 
 
 2 
 
 1 
 
 2 
 
 Further, let d£= (ApAg,!,), where k ± = {4,5,6}, A 2 = {l,2,2»j, 
 A 3 = {3,4 t ,4"}» Let jg = (B^B^Ij) where B^ ={ 2,4,4"}, 
 B 2 = {6, 2»,4 ? }, B\ ="{l,3,5}« Then ^T,^6 are partitions with 
 SP on S such that d£n<j& = 0. Hence this GPD (M-^JVy is a 
 parallel decomposition of M. 
 
 We note that jp is not a set system, since B-, £ B 2 » 
 The induced set system then is ffi = (Bo'jB-*), where 
 B 2 ^ =f6,2',4'}, B 3 * ={l,3,5>. Then ^' = (A 1 ',A 2 «,A 3 » ) 
 where A-^ = {5,6}, A,/ = {l,2»}, A 3 » = {3,4'}» Then 00, fg t 
 are partitions on S* = { 1, 2*, 3,4*, 5, 6}. The transition func- 
 tion F* is 
 
 F» 
 
 1 
 
 2' 
 
 3 
 
 4 ? 
 
 5 
 
 6 
 
 ©< 
 
 5 
 
 4» 
 
 1 
 
 2» 
 
 1 
 
 2« 
 
 and W is defined by 
 
 y" 
 
 b 2 
 
 b 3 
 
 a i* 
 
 6 
 
 5 
 
 V 
 
 2» 
 
 1 
 
 ... 
 
 4' 
 
 3 
 
 It can easily shown that f, , f, are two distinct 
 
 b 2 b 3 
 
 non-compatible functions. Hence the resultant is not a paral- 
 
55 
 lei decomposition of M any more. 
 
 In [12], Hartmanis and Stearns have studied series 
 and parallel decompositions of an automaton by considering 
 just set systems. We shall make some comment on the suffi- 
 ciency of using just set systems. By the question of suffi- 
 ciency we shall mean: are there series or parallel decompo- 
 sitions obtained by using ^-covers which are not obtainable 
 from set systems? From our development, it seems to cast a 
 strong doubt for the sufficiency. As illustrated before, a 
 series decomposition using *-cover with SP can also be ob- 
 tained by its induced set system with SP. However, by so doing, 
 the total number of input configurations may be increased. 
 Since the number of input configurations is of great importance 
 in the general theory, we can not be satisfied with decompo- 
 sitions using just set systems. Our Example 2.1 also demon- 
 strates this point. Consider it as a series decomposition; 
 by using the induced set system, the number of input configur- 
 ations from IVL to M, is increased from 1 to 2. However, this 
 example does not answer the question of sufficiency completely. 
 The desired example would be one for which there exists a 
 decomposition using ^-covers having the minimum number of input 
 configurations among all possible decompositions using set 
 systems. At the present time, such an example has not been 
 found. The trouble is mainly that the number of decompositions 
 using set systems is very large and one has to resort to an 
 exhaustive search. We thus leave this as an open question. 
 
56 
 
 Chapter IV 
 MULTIPLE DECOMPOSITION 
 
 In this chapter,, we extend our definition of pair 
 decomposition to multiple decomposition,. We shall show that 
 the results for pair decomposition can be equally extended 
 to the case of multiple decomposition. We shall further show 
 that any multiple decomposition can be obtained by successive 
 pairwise decompositions. Thus any discussion of multiple 
 decomposition actually can be made through the discussion of 
 pair decomposition. 
 
 Definition 1 . Let M = ^S,T,f > be a given deterministic, in- 
 complete, finite automaton. Let {m. = < S. ,T. ,f .> |l-i-N } be a 
 set of incomplete automata. Then [M, :M 2 :• . . :M«J, denoted by 
 
 i^ l-i-N* is a 6 enera l ized N-tuple decomposition (GND) of 
 M if and only if 
 
 (i) There exists a function ^ defined on a subset 
 
 N 
 
 J Q lis. onto S such that for all i, 1-i-N, 
 i=l 
 
 \{J) = s r 
 
 (ii) There exists a set of one to one functions 
 
 {h ll^i^N) such that 
 
 N 
 
 (Vi)(l^i^N) fh. :T. — ►TxTTs.l. 
 
 Jfi J 
 3 = 1 
 
 (iii) (¥i)(l«i*N) D(f i )={(s i ,t i )|h i (t i ) 
 
 It, Sp • ° • , s. _-. , s. ^^t • • • j s fjJ 
 3[(/ ? (s 1 ,...,s N ),t)6 D(f)j} 
 
57 
 (iv) (V(s 1 ,...,s N )€ > i)(Vt€T) [(^s lf ...,s N ),t)€D(f)] => 
 
 r N 
 
 (¥i)(l^i^N)[[TTf i (s i ,h i - 1 (t,s 1 ,..., Si _ 1 , 
 
 a i+1 , • • • , s N ) )J € (Y"\f {?( s r ..., s N ,t ) )]] 
 
 (v) Suppose (s 1 ,...,s N )€y and )^(s 1 , . . . , s«)=f (s, t ) 
 
 for some s€ S, t€ T. Then 
 
 [3(s 1 f ,«««»s N l )€ f (s)J [(s 1 ,...,s N ) 
 
 N 
 = ( "[J f i ( a ± « , h." 1 ( t, S;L » , . . . , s i _ ] _» , s i+1 » , . . . , s N » ) ) )J 
 
 Definition 2 . Let GND tuple (GNDT) of an automaton M = <S,T,f> 
 
 be an (N+l) tuple (S-,, Sp* • • «,S«,^> where each S. is a set of 
 
 N 
 
 symbols, and (^ is a mapping from Ji £ I S. onto S such that 
 i i=1 i 
 
 (¥i) O^i) = S ± ]. 
 
 The following results are similar to those in Chap- 
 ter I. Since the proofs of these results are completely 
 analogous to the corresponding ones in Chapter I, we shall 
 omit them. 
 
 Theorem 1 . Let hi[ ± = (s.,T ±f f ± ) | 1-i-NJ- be a set of incom- 
 plete automata.. If [M^ t^j^w is a GND of M, there exists a 
 GNDT <S 1 ,S 2 ,...,S N ,^> of M. Given a GNDT <S 1S . . . , S N , f> of M, 
 there exists a non-empty collection of GNDs of M whose asso- 
 ciated GNDT is the given (S-. , . . . , S«, jP^ . 
 Theorem 2 . For each GNDT <S 1 , S 2 , . . . , S N ,^> of M = <S,T,f> , 
 
 and each ordering of the states in each S^ = ( a^, . ..,ay ]$ 
 
 ) k 
 
 there exists a set of ^-covers { 0L^» • • >,0C^\ of S. Let A i 
 
 denote the i th coordinate of the *-covers # k » For each 
 
 fixed k« for each i. 
 
 .g J-^J. CCH/H J. f 
 
58 
 
 A^{s€S|(3(a n 1 ,...,aj-* ,...,aj ))W(sJ ,...,s* , . . . , s^ )-a)f. 
 
 1 ^ 1 1 x k-l X N x l 1 k X N ' 
 
 Definition 3 ° Let |^ i |l-i-N} be a set of *-covers of S. 
 
 For each i, let fc. = (A, , ...,A. ). Then J^Jl^i^N} 
 
 i i -J ip *• i 
 
 i 
 is said to be MD if and only if for each i, for each 
 
 j entry, A.. ={s 1 ,...,s, \ of 0C ., there exists a 
 
 set of k. ^.disoinct (N-l)-tuples such that 
 (i) For each k, l^k. ., let the k th (N-l) -tuple be 
 
 (A l,j(i,k)'*""' A (i-l),j(i-l,k)' A (i+l),j((i+l),k)'"-' 
 
 A N °(N kP where for each q g 1-q-N, q/i> we have 
 
 i-j(q,k)-p q . 
 
 (ii) a.€ A„ o,- , » for each q and each k. 
 Definition 4 * Let {^J 1-i-N ) be a set of MD "--covers of S 
 
 such that (Vs) (Vi, j)(m^ (s)=m^(s) = m(s)J. For each 
 
 r i i i J 
 s € S, m(s)>l, {^• i |l-i-N| is said to have the first-s- 
 
 deleted distributive property if and only ± r there exists 
 
 a set \k, . ,...,A M . \, where for each i, l-j.-p. and 
 
 A. . is the j" coordinate of ul. y such that 
 1 > J A i 
 
 (i) s e A. . for each i. 
 1,J i 
 
 (ii1 (^.((l.^)^)' — ' ^.((H.^J.b)) 1 is -gain MD. 
 
 Definition 5 . Let j^Jl-i-N} be a set of MD ^-covers of S 
 
 such that (Vs) (Vi, j) f m (s) = m ^(s) = m(s)]. For each 
 
 i *j 
 s, m(s)>k, {^Jl^N} is said to have the k th -s-deleted 
 
 distributive property if and only if there exists 
 
 'A, . , ...,A.. . ) such that 
 
 Recall definition 2.4 of Chapter I for notation. 
 
59 
 (i ) s€ A. . for each i. 
 
 (ii) C«l,((l.j 1 ) f8 )"— ^((H.j^.a) 5 hasthe (k " 1)th " 
 s-deleted distributive property. 
 
 Definition 6 . Let {^ i l 1-i-NJ be a set of MD *~covers of S such 
 
 that (¥s)(¥i)[m^ (s) = m(s)]. Then {^.|l^N| is said 
 
 i 
 
 to have the deleted distributive property (DDP) if for each 
 s, m(s)>l, (^\|l^N} has the (m( s)-l) th -s-deleted dis- 
 tributive property. 
 Definition 7 * Let {fc ± \ 1-i-N} be a set of MD ^-covers of S. 
 Then f^J 1-i-NJ- is said to have the added deleted dis- 
 tributive property (ADDP) if and only if 
 (i) There exists a set S* of symbols, 
 (ii) There exists a set -[d^ 1 1 1-i-N } of ^-covers of S». 
 (iii) There exists a mapping p from S f onto S such that 
 
 for each i, 1 - i - N, 6L. is induced by ^£.' under p. 
 
 With all these definitions, we can show, by induction, 
 
 N 
 
 that there exists a mapping <p from a subset Jj of E ^ onto 
 
 ' i=l i 
 
 S if and only if [fit] l*i*NJ has ADDP. The proof is similar 
 
 to that in Chapter I» Hence, we arrive at a result similar 
 
 to Theorem 2.4 of Chapter I. 
 
 Theorem 3 * All GNDT of M = <S,T,f> are of the type 
 
 (E , ...,E , p> where E^. denotes the *-cover fc ± of S and 
 
 { fo ± \ l*i*N} has the ADDP. 
 
60 
 
 With an argument similar to that in Section 1 of 
 
 Chapter II, we shall simply study GND [M.) nz-^ N of an automaton 
 
 M = <S,T,f>, where in its GNDT (Sp . . . , S^, f> , f is an iso- 
 
 N 
 
 morphism from J £ T s . onto S, and (?i) (^ (J )=S. ). For such 
 
 i=l x x x 
 
 a GND CM.] -, *. .*« of M, it naturally induces a set of N equiva- 
 lence relations \9. J 1-i-NJ" on S as follows: For each i, 1-i-N, 
 define 0. . on S such that 
 
 (Vs 1 ,s 2 €S)((s 1 O ii s 2 )^=^('^ i ()o- 1 (s 1 )) = ^l(f 1 (s 2 )))) 
 
 N 
 
 It can be easily shown that ' « 9. . = 0. 
 
 i«l X1 
 
 Definition & » Let (9. . | 1-i-Nf be a collection of equivalence 
 
 N 
 
 relations on S such that A 9. . = 0. For each i, each j, j/i, 
 
 i=l X1 
 
 define a symmetric, reflexive relation 9. . on S as follows: 
 
 For each s-,, Sp€ S, s-,9. .s 2 if and only if 
 
 (Vo(eT) (Vs^ ) (Vs 2 M((s 1 '9 ii s 1 ) (s 2 '9 iiS2 ) (?k) (l*k*I) ( k A) (s 1 , 9 kk s 2 ») 
 
 =» [f(s 1 »,^)O jj f(s 2 ',^)j). 
 Theorem 4 * Let /m. - < S. ,T.,f.) | i=l, 2, . . . ,n} be a set of in- 
 complete automata. Let CMj t^^m be a GND of M ■ <S, T,f > such 
 
 N 
 
 that in the GNDT, V> is a one to one mapping from ic TT S. 
 
 * i=l 1 
 
 We recall and generalize that given a GND CM.) i«4«v 
 
 of M = <S,T,f>, we have an automaton M 1 ■ (s»,T,f f ) such that 
 
 (1) M» realizes M, and (2) Cm.) 1-4 *- is a GND of M' , (3) the 
 
 9 N i -l i « 
 
 homomorphism from Jl c TT S. onto S f is an isomorphism, (4) 
 
 i = l x 
 
 the properties of [mJ t*j*w as GND of M are completely repre- 
 sented by that of [Mj J**** as GND of M» . 
 
61 
 onto S and for each i, l^i^N, "fr ± { J ) = S ± . Then CM.],^.^ 
 
 gives rise to a set of relations { 9 . . (l-i-NJ on S such that 
 
 (i) For each i, 9- i is an equivalence relation on S 
 
 N 
 
 and A 9.. = 0. 
 i=l X1 
 
 (ii) (¥j)(vi) Ce i;j 2 9.J 
 
 N 
 
 dii) (?,) [ r\ Q s f-i( Q )]. 
 
 i = l 
 
 Proof . We need only show (ii) and (iii). Part (ii) can be 
 
 easily shown. To prove (iii), let s-,,s 2 € S such that 
 
 N N 
 
 s 1 [/09 i .] s 2 . Then (Vs 1 » ) (Vs 2 » )■ [C s i' < .0 9 ii } s l^ 
 
 N N , 
 
 Cs 2 »(n e ii )s 2 )fs 1 ' ( ^ Q ±± )s 2^ z= ^ [f(s 1 t ,^)e jj f(s 2 s^)JJ. 
 
 N N 
 
 Yet Ho.. =0. Therefore, if sjH^Js, then s-, Cf _1 (9 . . )] s~. 
 
 Q. EoD. 
 Theorem 5 * Let M = <S,T,f>. Let {9..} l^i,i^K be a set of 
 relations on S such that 
 
 (i) For each i, 9. . is an equivalence relation on S. 
 N 
 
 (ii) r\Q = o. 
 
 i=l i:L 
 
 (iii) (Vj)(Vi)(9 ±j 2 Q ±± ] 
 
 N 
 (iv) (¥j) (Ho S f" 1 (9 ,)J. 
 
 i = 1 -"-J JJ 
 
 1*1 
 
 Then there exists a set of incomplete automata (M i = (S^T^f^l 
 i=l, 2, . . . , N J such that 
 
62 
 (i) C M ± ] !^i^ N is a GND of M 
 
 (ii) In the GNDT < S^, . . . , S^, f>, f is one to one from 
 N ' 
 
 J Q TTs. onto S. 
 
 i=l x 
 
 (iii) [Mjj ^ ± ^ gives rise to the set {q ± . |l-i, j-N}. 
 
 Proof . The proof is very similar to that of Theorem 1.2 of 
 
 Chapter II. Q. E. D. 
 
 To show that any multiple decomposition can be obtained 
 
 by successive pairwise decompositions, we consider the following 
 
 approach to the formulation of GND of an automaton. Let 
 
 M = (S,T s f^. Let S-p...,S M be given sets such that there 
 
 N 
 exists a function from Ji Q TT S . onto S such that for each 
 
 i> ^:(>J ) = So. We may consider the transition function f 
 
 N N 
 
 as a mapping from T xTTs. into TlS., where f is a single 
 
 i=l 1 i=l 1 
 
 N 
 
 valued function on D(f) =J Q T x"|TS., and f maps to 
 
 otherwise. Let { i-, , . . . , i, j £ { 1, 2, . . . ,N/. Define 
 
 i-,i 9 ...i, = foP such that P. , is the projection 
 N 1 k k 1 k 
 
 ma 
 
 p from Ts. on TT S . . Clearly, we may consider f. 
 
 i-l j=l J Ik 
 
 N k 
 
 as being defined on T x TT S. to TTs. such that 
 
 J-k+1 X j j=l x j 
 
 Jip . . . , 1., . . . , i„ J = {l,2,...,Nj. Let M. be the automaton 
 
 N 
 with the set S. of states, T X TT s . of inputs and f . the 
 
63 
 transition function. Then f = (f,, ...,f ). Thus, (M-,:...:M ) 
 
 is a GND of M and G(Mp . ..,M N ) is the automaton with the set 
 N 
 
 IIS, of states, set T of inputs and f the transition function. 
 i=l 1 
 
 In general^ if (i-,, . . . ,i,} 9 {l,...,N}, then G(M. ,...,M. ) is 
 
 k X l x k 
 
 -rr N 
 
 the automaton with the set Ms. of states, T * TT S, of in- 
 
 J-l X j h=l h 
 
 h^{i 1 , ...,i k } 
 
 puts and f . . as the transition function. Since cartesian 
 1 l , ° ,1 k 
 
 products are associative, we have 
 
 G(M-,,...,M, T ) = G(G(M. ,...,M. ),M. ,...,M. ) 
 
 where { ip . . . , i^, j^, . . . , J N _ k } = { 1, • • • , N }. Hence any multiple 
 decomposition can be obtained from successive pairwise decom- 
 position. This result enables us to obtain results for N-tuple 
 decomposition by successive use of the results for pair decom- 
 position. 
 
64 
 
 SUMMARY 
 
 This work is aimed at developing a theory of de- 
 composition for deterministic, incomplete, finite automata 
 allowing interconnections between component automata. In 
 general, a decomposition for an automaton, even for a com- 
 plete automaton, gives rise to a set of incomplete automata. 
 Thus, in order to arrive at a theory allowing further decom- 
 positions, it is necessary to deal with incomplete automata 
 instead of just complete automata. 
 
 Let j& be a measure function for the complexity 
 of an automaton such that if M f , M are two automata and M f 
 realizes M, then £>{W) ^ ^(M). Let B be a given bound for 
 the complexity of each component automaton of a decomposition 
 of M = <S,T,f>. Then the final object of this work is to ob- 
 tain an algorithm which generates all decompositions GND 
 (M 1 :...:M N ) of M such that 
 
 (i) (?i)(l*i*H) [ ^(1^) = B] 
 
 N N 
 
 is a minimum. 
 
 (ii) H H (iog o n(0. S\ 
 i=l .1=1 2 ^ 
 
 We first refer to page 60 for the definition of 
 ©ji> l-i,j-N. We remark that if one is interested, one may add 
 an input term. In such a case, we may define two inputs 
 t-j_>t 2 CT as equivalent in an automaton M. as follows: 
 (t 1 W i t 2 )^=i> [(Vse S)ff (s,t i )0 ii f (s,t 2 )J] . Then clearly ti ± is 
 a reflexive and symmetric relation on T. To find n(60) is 
 again a covering problem. If it is needed, one may add the term 
 
65 
 We first considered decomposing a given automaton 
 M = <S,T,f> into a pair of automata [Chapters I, II, III]. 
 We then extended our theory to the multiple case and showed 
 that any multiple decomposition can be obtained from succes- 
 sive pairwise decompositions [Chapter IV ]. Therefore, in or- 
 der to obtain an algorithm to find the desired GNDs, we need 
 only study the properties of pair decomposition* 
 
 In Chapter I, we introduced the definitions of GPD 
 of M and GPDT of M. We showed that any GPDT of M is of the 
 type <E^, Fy 3 » K f> where E^, E^ are index sets of the ^-covers 
 Ot,-fc of S respectively and [dC 9 fe) has ADDP. We studied 
 the properties of GPD of M in Chapter II. We showed that in 
 discussing the properties of a pair decomposition, it suffices 
 to assume that all ^-covers are of the partition type. We 
 deduced that to find the number of input configurations (thus 
 the number of connecting lines) is a covering problem. We 
 then introduced the definition of a congruence relation of 
 degree k on the automaton M. We showed that each congruence 
 relation of degree k on M induces a set of pair decompositions 
 [M-, :M.-, ? ] such that M-, has as its states the equivalence classes 
 under 9. Furthermore, we proved that k is the lower bound on 
 the number of input configurations from M-, ? to M-, in any in- 
 duced decomposition [M -M^] of M under 9. For any GPD [M-^M^ 
 of M, it is considerably more involved for solving the covering 
 problem to get the exact number of input configurations than 
 
 N 
 
 2_ (log 9 n(tO„)| to formula (ii), and require their sum to be 
 i=l * x 
 
 a minimum. 
 
66 
 
 determining the degree k of 9 which induces [iI-jiI-'O. Thus 
 in the desired algorithm we may first use the lower bound to 
 check whether the complexity for a component automaton already 
 exceeds the bound. Hence it eliminates much unnecessary search- 
 inging for the desired GNDs. Suppose there exists a non-trivial 
 congurence relation 9, , of degree k-, on M. Then we showed that 
 there exists a non-trivial equivalence relation 9 22 on S such 
 that if £m, :M 2 ) is the GPD induced by 9-,-,, 9 22 , then the number 
 of input configurations from ML to M, is exactly k-,. 
 
 In Chapter III, we showed that our results include 
 the known results for series and parallel decompositions of 
 an automaton as special cases. We then raised the question 
 of the sufficiency of set systems for considering decomposi- 
 tions for automata. 
 
 We conclude by listing two interesting yet open 
 questions. 
 (1) At present, we can only give an exhaustive algorithm 
 
 for finding all the desired decompositions. Yet any ex- 
 haustive search procedure will not be satisfactory, since 
 it is too slow. Thus, we have the following question. 
 
 Given an automaton M, does there exist a finite 
 subcollection F(M) of the collection E(M) of all GPDs 
 of M satisfying the following condition: For any 
 CM 1 :M 2 ]6 E(M), there exists [M,» iM,*) € F(M) such that 
 M^ realizes M 1 t and M 2 realizes M»? 
 
 This result will give a bound on the number of de- 
 compositions that an algorithm has to go through. 
 
 Partly to answer this question, the author has ob- 
 n algorithm for generating all decompositions 
 
67 
 
 without duplication C 193 for those GPDs (M-, :M ? J of M, 
 where V> is an isomorphism in its GPDT (S-,,S 2 ,f). 
 (2) Find an example of an automaton M for which there exists 
 a GND Cm.] -i^^m obtained by using ^-covers such that 
 
 (i) (¥i)(tf(M i )<B) 
 (ii) The number of interconnections is less than that of 
 any decomposition of M which is obtained by using 
 set systems on S, and which has the same upper bound 
 B for the complexity of each component. 
 This result will indeed show that it is not sufficient 
 to consider just set systems for decomposition of an autom- 
 aton. 
 
63 
 
 BIBLIOGRAPHY 
 
 1. Bayer, Rudolf. "Series-parallel decomposition of sequen- 
 tial machines." Department of Computer Science, File No. 
 677, University of Illinois, Aug. 1965. 
 
 2. Beatty, J> and Miller, R.E. "Some theorems for incompletely 
 specified sequential machines with applications to state 
 minimization." AIEE Proceedings of the Third Annual Sympo - 
 sium on Switching Circuit Theory and Logical Design, pp. 123- 
 136, September 1962. 
 
 3. Birkhoff, G. Lattice Theory . AMS Colloquium Publication, 
 23, 1943. 
 
 4. Gill, A. "Cascade finite-state machines." Trans. IRE , 
 EC- 10, pp. 366-370 (1961). 
 
 5. Ginsburg, S. "Synthesis of minimal state machine." Trans . 
 IRE , pp. 441-449, Dec. 1959. 
 
 6. Ginsburg, S. An Introduction to Mathematical Machine Theory . 
 Addison-Wesley, 1962. 
 
 7. Hartmanis, J. "Symbolic analyses of a decomposition of 
 information processing machines." Information and Control , 
 3, pp. 154-173, June I960. 
 
 3. Hartmanis, J. "On the state assignment problem for sequen- 
 tial machine I." Trans. IRE , EC-10, pp. 157-165, June 1961. 
 
 9. Hartmanis, J. "Loop-free structure of sequential machine." 
 Information and Control, 5, PP- 25-43, March 1962. 
 
 10. Hartmanis, J. and Stearns, R. E. "On the state assignment 
 problem of sequential machine II." Trans. IRE , EC-10, 
 
 pp. 593-603, Dec. 1961. 
 
 11. Hartmanis, J. and Stearns, R. E. "Some danger in state 
 reduction of sequential machine." Information and Control , 
 5, pp. 252-260, Sept. 1962. 
 
 12. Hartmanis, J. and Stearns, R. E. "Pair algebra and its 
 application to automata theory." Information and Control , 
 7, pp. 435-507, Dec. 1964o 
 
 L3« Karp, R. "Some techniques of state assignment for synchronous 
 sequential machines." IBM Report RC-933, May 1963. 
 
 14. Kohavi, Z. Secondary state assignment for sequential machine." 
 Trans. IRE, EC-13, June 1964. 
 
69 
 
 15. Krohn, K. and Rhodes, J. L. "Algebraic theory of machines 
 I, Prime decomposition theorems for finite semigroups and 
 machines." Seminar notes, University of California, 
 Berkeley, 1963. 
 
 16. Miller, R. E. "State reduction for sequential machines." 
 IBM Research Report RC-121, June 1959. 
 
 17. Muller, D. E. "A treatment of sequential machines." De- 
 partment of Computer Science, File No. 567, University of 
 Illinois, June 1963. 
 
 Id. Paull, M. C. and Unger, S. H. "Minimizing the number of 
 states in incompletely specified sequential switching 
 functions." Trans. IRE , EC-10, pp. 356-367, Sept. 1959. 
 
 19. Pu, A. "Some theorems and algorithms on a new form of 
 decomposition for automata." Department of Computer 
 Science Report No. 174* University of Illinois, April 1965* 
 
 20. Rabin, M. 0. and Scott, D. "Finite automata and their 
 decision problems." IBM Journal of Research and Develop - 
 ment , 3, pp. 114-125, 1959. 
 
 21. Shanon, C. E. and McCarthy, J., editors. Automata studies . 
 Princeton University Press, 1956. 
 
 22. Yoeli, M. "The cascade decomposition of sequential machines." 
 Trans. IRE , Ec-10 pp. 537-592, Dec. 1961. 
 
 23. Yoeli, M. "Cascade-parallel decomposition of sequential 
 machines." Trans. IRE , EC-12, pp. 322-323, June 1963. 
 
 24. Zeiger, H. P. "Cascade synthesis of finite state machines." 
 1965 IEEE Conference Record on Switching Theory and Logical 
 De"sign 9 p. 45-51* Oct. 1965» 
 
70 
 
 VITA 
 
 The author was born in Peiping, China on June 27, 
 1937. In September, 1956, he entered the Hamline University 
 at St. Paul, Minnesota, where he was a awarded a full scholar- 
 ship. In September, 1957* he transferred to the University of 
 Illinois and received the B. S. degree in Electrical Engineering 
 in June, I960. He then entered the Graduate School of the 
 Michigan State University majoring in Mathematics. He received 
 the M. S. degree in Mathematics in February, 1962. Beginning 
 June, 1962, he attended the Graduate College of the University 
 of Illinois. He was a graduate assistant in the Department of 
 Mathematics during the time he was at M. S. U. From 1962 to 1963, 
 he was a research assistant in the Biological Computed Group 
 of the Electrical Engineering Research Laboratory of the Univer- 
 sity of Illinois. Since September, 1963, he has been a research 
 assistant in the Department of Computer Science. 
 
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