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Complete Finite Automata Network. Graphs with Minimal Number of. Edges. P al D om osi. L. Kossuth University. Institute of Mathematics and Informatics.
Complete Finite Automata Network Graphs with Minimal Number of Edges Pal Domosi

L. Kossuth University Institute of Mathematics and Informatics 4032 Debrecen Egyetem ter 1 Hungary [email protected]

Chrystopher L. Nehaniv

School of Computer Science & Engineering University of Aizu Aizu-Wakamatsu City 965 Japan [email protected]

Turku Centre for Computer Science TUCS Technical Report No 185 June 1998 ISBN 952-12-0231-9 ISSN 1239-1891

Abstract An automata network graph is said to be n-complete (under projection) if every automata network having underlying graph with n vertices can be simulated (under projection) on it. In this paper n-complete automata network graphs with minimal number of edges are completely characterized.

TUCS Research Group

Mathematical Structures of Computer Science

DEDICATION THIS WORK IS DEDICATED TO PROFESSOR FERENC GE CSEG ON HIS 60TH BIRTHDAY

1 Basic Notions

Let f : X1  : : :  Xn ! X be a mapping having n variables for some positive integer n; moreover, let t 2 f1; : : : ; ng: f is said to really depend on its tth variable if there exist x1 2 X1; : : :; xt?1 2 Xt?1; xt; xt0 2 Xt; xt+1 2 Xt+1; : : :; xn 2 Xn having f (x1; : : :; xn) 6= f (x1; : : : ; xt?1; xt0; xt+1; : : :; xn): If f does not have this property then we also say that f is really independent of its tth variable. Moreover, if there is no danger of confusion then sometimes we omit the attribute \really". For a given non-empty set X and positive integer n denote by X n the th n cartesian power of X: Given a k-element subset H of f1; : : :; ng, H = fi1; : : :; ik g (i1 < : : : < ik ), the H -projection of X n is a mapping prH : X n ! X k de ned by prH (x1; : : : ; xn) = (xi1 ; : : : ; xik ), where (x1; : : : ; xn) 2 X n : The function prH (F ) : X k ! X k with prH (F (x1; : : : ; xn)) = prH (F )(prH (x1; : : :; xn)); (x1; : : :; xn) 2 X n is called the H -projection of F : X n ! X n (if it exists). If H = fhg for some h 2 f1; : : : ; ng, i.e., H is a singleton then sometimes we use the expression h-projection (of a vector or function) in the same sense as the concept \H -projection". (And in this case sometimes we use the notation prh instead of prfhg:) Moreover, for an arbitrary i 2 f1; : : : ; ng; we de ne the ith component of F : X n ! X n as the function cpi (F ) : X n ! X with cpi(F )((x1; : : :; xn)) = pri(F (x1; : : :; xn )) (x1; : : : ; xn) 2 X n : For any pair Fi : X n ! X n ; i = 1; 2; one denotes by F1  F2 : X n ! X n the function F1  F2(x1; : : :; xn) = F1(F2(x1; : : :; xn)); (x1; : : :; xn) 2 X n . A ( nite) directed graph (or, in short, a digraph) D = (V; E ) (of order n > 0) is a pair of the sets of vertices V = fv1; : : : ; vng and edges E  V  V: vi 2 V is an isolated vertex if (fvig  V [ V  fvig) \ E = ;: If (vi; vj ) 2 E and i = j then (vi; vj ) is called (self-) loop edge. The digraph D0 = (V 0; E 0) is a subdigraph of D if V 0 is a non-void subset of V; and E 0  E: D is said to be connected for vi 2 V if every vertex vj 2 V has a (directed) path from vi to vj : D is called strongly connected if it is connected for all of its vertices. Moreover, D is centralized if there exists a vi 2 V with V fvig  E (including (vi; vi) 2 E ). In addition, a digraph D = (V; E ) having a structure 1

V = fv1; : : :; vng; E = f(vi; vi+1(modn)) : i = 1; : : :; ng is called a cycle (with n length). We also say that a digraph D has a cycle (with n length) if there is a subdigraph of D which forms a cycle (with n length). A transformation F : X n ! X n is said to be compatible with a digraph D = (V; E ) (of order n) if F has the form F (x1; : : : ; xn) = (f1(x1; : : :; xn); : : : ; fn(x1; : : : ; xn)) ((x1; : : : ; xn) 2 X n ) and fi : X n ! X; i = 1; : : : ; n may depend only on xi and those xj for which (vj ; vi) 2 E (including the case i = j ). A word (over X ) is a nite sequence of elements of some nite non-empty set X: We call the set X an alphabet, the elements of X letters. If u and v are words over an alphabet X; then their catenation uv is also a word over X: Especially, for every word u over X; u = u = u; where  denotes the empty word having no letters. The length jwj of a word w is the number of letters in w, where each letter is counted as many times as it occurs. Thus jj = 0: By the free monoid X  generated by X we mean the set of all words (including the empty word ) having catenation as multiplication. We set X + = X  n fg; where the subsemigroup X + of X  is said to be free semigroup generated by X: By an automaton A = (A; X; ) we mean a nite automaton without outputs. Here A is the ( nite non-empty) state set, X is the input alphabet and  : A  X ! A is the transition function. We also use  in an extended sense, i.e., as a mapping  : A  X  ! A; where (a; ) = a (a 2 A) and (a; px) = ((a; p); x) (a 2 A; p 2 X ; x 2 X ): For a given word p 2 X ; the transition induced by p is the function p : A ! A that takes any state a 2 A to (a; p): If A = Z n for some jZ j  1 and n  1 (where jZ j denotes the cardinality, i.e., the number of elements in Z ) then we say that A is a nite statehomogenous automata network (of size n with respect to the basic local state set Z ). Then the underlying graph DA = (VA ; EA) of A is de ned by VA = f1; : : : ; ng; EA = f(i; j ) j 9x 2 X : cpj (x) really depends on its ith variableg. A is a D-network if D = (V; E ) is a digraph with V = VA and E  EA: In other words, A is a D-network if every mapping x : A ! A (x 2 X ) is compatible with D: Note that a size n automata network may be regarded as comprising n component automata Ai = (Z; Z n  X; i ), i 2 f1; : : :; ng, where the i are de ned by (z; x) = (1(z1; (z; x)); : : :; n(zn; (z; x))); for z = (z1; : : : ; zn) 2 Z n, x 2 X . One may of course suppress the components of Z n in the inputs to Ai upon which i does not really depend. If n = 1 or jZ j = 1 then we say that A = (Z n; X; ) is a trivial automata network. The purpose of this paper is to investigate the state-homogeneous 2

automata networks having state sets of the form Z n; for a positive integer n > 1 and xed nite set Z of cardinality at least two. Therefore, by an automata network we shall mean a non-trivial nite state-homogeneous network. Let A = (Z n ; X; ); B = (Z m ; Y; 0) be networks (having the same basic set Z ). We say that B simulates A by projection if there exists an H  f1; : : :; mg such that every x : Z n ! Z n (x 2 X ) is an H -projection of a mapping p0 : Z m ! Z m (p 2 Y +): If there exists a D-network B which simulates a given network A by projection then it is said that A can be simulated on D by projection. A digraph D is called n-complete (with respect to simulation by projection) if every network of size n can be simulated on D by projection. The n-complete digraph D = (V; E ) has minimal number of edges if for every n-complete digraph D0 = (V 0; E 0); jV j = jV 0j implies jE j  jE 0j:

2 Preliminary results We start with the following technical result.

Lemma 2.1.(see [2]) Given a nite group G; a positive integer n > 1; let us de ne for every distinct i; j 2 f1; : : : ; ng the functions Fi;j(t) : Gn ! Gn(4); t = 1; 2; 3, Fj : Gn ! Gn ; and Ui;j : Gn ! Gn as follows.

Fi;j(1)(g1; : : : ; gn) = (g1; : : :; gj?1 ; gigj ; gj+1 ; : : :; gn ); Fi;j(2)(g1; : : :; gn ) = (g1 ; : : :; gj?1; gi ?1gj ; gj+1; : : :; gn ); Fi;j(3)(g1; : : : ; gn) = (g1; : : :; gj?1 ; gi; gj+1 ; : : :; gn ); Fj(4)(g1 ; : : :; gn ) = (g1; : : : ; gj?1; gj ?1; gj+1; : : : ; gn); Ui;j (g1; : : : ; gn) = (g1; : : :; gi?1 ; gj ; gi+1 ; : : :; gj?1 ; gi; gj+1; : : : ; gn): Then for arbitrary, pairwise distinct i; j; k 2 f1; : : :; ng we get (1)  F (1)  F (2); Fi;j(1) = Fi;k(2)  Fk;j i;k k;j (1)  F (2)  F (2); Fi;j(2) = Fi;k(1)  Fk;j i;k k;j

3

(1)  F (1)  F (2)  F (2)  F (2)  F (3); Fi;j(3) = Fk;j i;k k;j i;k k;j k;j (4) (1) (4) (4) (2) Ui;j = Fj  Fi;j  Fj  Fi  Fj;i  Fi;j(1):

2

Given a non-void set Y; a positive integer n, let TY denote the full transformation semigroup of all functions from Y to Y . In addition, for every subset H  TY ; let denote the subsemigroup of TY generated by H: Moreover, for any nite set X with jX j > 1 and positive integer n > 1, denote TX;n the subsemigroup of all transformations of TX n having the form F (x1; : : :; xn) = (xt(1); : : : xt(n)); (x1; : : : ; xn) 2 X n ; t : f1; : : : ; ng ! f1; : : : ; ng, and let ?X n = fF : X n ! X n j F (x1; : : : ; xn) = (x1; : : :; xi?1; f (xi; xj ); xi+1; : : : ; xn); where f : X 2 ! X; i; j 2 f1; : : : ; ng; (x1; : : :; xn ) 2 X n g; (It is understood that the case i = j is allowed in the above de nition of ?X n :) De ne the elementary collapsing tj;k : f1; : : :; ng ! f1; : : : ; ng for 1  j 6= k  n, ( i=k : tj;k (i) = ji ifotherwise Moreover, as usual we say that uj;k : f1; : : : ; ng ! f1; : : :; ng for 1  j 6= k  n is a transposition if 8 > < j if i = k uj;k (i) = > k if i = j : : i otherwise Let FX n?1fdg be the semigroup of functions fF 2 TX n : F (x1; : : :; xn) n ? X 1  fdg; x1; : : : ; xn 2 X; F is really independent of its last variableg:

2

Lemma 2.2. (see [2]) FX n?1fdg 6< ?X n > : Proof. Fix arbitrary c = 6 d 2 X and let (c1; : : : ; cn?1) 2 X n?1 ; (

(c1; : : :; cn?1; c); F(c1;:::;cn?1)(x1; : : :; xn) = ((xx1;; :: :: :: ;; xxn) ; d) ifif ((xx1;; :: :: :;:; xxn)) = 1 n?1 1 n 6= (c1 ; : : :; cn?1 ; c) ((x1; : : : ; xn) 2 X n): First we prove that F(c1;:::;cn?1 ) 2 < ?X n > : If n = 2, then our statement holds by de nition. Otherwise, n > 2 and for every b 2 X , de ne ( if xn?1 = b; xn = c; (0) Fb (x1; : : :; xn) = ((xx1;; :: :: :: ;; xxn?1;; dc)) otherwise ; 1 n?1 4

where (x1; : : :; xn) 2 X n : For every i 2 f1; : : : ; n ? 1g; (ci; : : :; cn?1 ) 2 X n?i ; let ( (xi; : : :; xn) = (ci; : : :; cn?1 ; c); F(ci;:::;cn?1 )(x1; : : : ; xn) = ((xx1;; :: :: :;:; xxn?1;; cd)) ifotherwise ; 1 n?1 where x = (x1; : : :; xn) 2 X n ): It is clear that F(cn?1) = Fc(0) n?1 : On the other hand, for every i 2 f2; : : :; n ? 1g; F(ci?1;:::;cn?1 ) = F(ci;:::;cn?1 )  Ui?1;n?1  Fc(0) i?1  (0) Ui?1;n?1: Simultaneously, we have by de nition that Fci?1 2 ?X n holds for every i 2 f2; : : : ; n ? 1g: Moreover, using Lemma 2.1, it can be shown easily Ui;j 2 < ?X n > : Thus we get our statement by induction. Now we consider a pair (c1; : : :; cn?1 ); (d1; : : : ; dn?1) 2 X n?1 ; d 2 X; ((c1; : : : ; cn?1) = (d1; : : :; dn?1 ) is allowed), and de ne 8 > (c1; : : :; cn?1; d) if (x1; : : :; xn?1) = (d1; : : :; > > dn?1 ); < ( d ; : : :; d ; d ) if (x1; : : :; xn?1) = (c1; : : :; F((1) ( x ) = 1 n?1 c1 ;:::;cn?1 );(d1 ;:::;dn?1 ) > cn?1 ); > > : (x1; : : : ; xn?1 ; d) otherwise ; 8 > < (c1 ; : : :; cn?1 ; d) if (x1 ; : : :; xn?1 ) = (d1 ; : : :; (2) dn?1 ); F(c1;:::;cn?1 );(d1;:::;dn?1)(x) = > : (x1; : : : ; xn?1 ; d) otherwise ; n where x = (x1; : : : ; xn) 2 X : Next we show F((ci1);:::;cn?1);(d1;:::;dn?1) 2 < ?X n >; i = 1; 2: We have c 2 X arbitrary with c 6= d and set Fc(3)(x) = (x1; : : :; xn?1; c), (3) Fd (x) = (x1; : : :; xn?1; d), and 8 > < (c1 ; : : : ; cn?1 ; c) if (x1; : : : ; xn ) = (d1 ; : : : ; (4) F(c1;:::;cn?1);(d1;:::;dn?1 )(x) = > dn?1; c); : (x1 ; : : :; xn?1 ; d) otherwise where x = (x1; : : :; xn) 2 X n ; c; d 2 X; c 6= d; moreover, consider F(c1;:::;cn?1 ) as before. In addition, let 8 > < (x1 ; : : :; xn?1 ; c) if xn = d; (5) F (x1; : : :; xn) = > (x1; : : :; xn?1; d) if xn = c; : (x1 ; : : :; xn?1 ; xn ) otherwise; and let for every a 2 X; ( if xn = c; (6) Fa (x1; : : : ; xn) = ((xx1;; :: :: :;:; xxn?2 ;; xa; x) n) otherwise 1 n?1 n 5

((x1; : : : ; xn) 2 X n): It is clear that Fc(3); Fd(3); F (5); Fa(6) 2 ?X n : Next we show that F((4) c1 ;:::;cn?1 );(d1 ;:::;dn?1 ) 2< ?X n > : Indeed, by an easy computation we get (4) (6) F(c1;:::;cn?1);(d1;:::;dn?1 ) = Un?2;n?1 : : :U2;n?1 U1;n?1Fc(6) 1 U1;n?1 Fc2 U2;n?1  (6) : : :  Un?3;n?1  Fc(6) n?2  Un?2;n?1  Fcn?1  F(d1 ;:::;dn?1 ): On the other hand, by Lemma 2.1 we can see easily Ui;j 2 < ?X n > : But then F((2) c1 ;:::;cn?1 );(d1 ;:::;dn?1 ) = (3) (4) (2) (3) Fd  F(c1;:::;cn?1 );(d1;:::;dn?1)  Fc implies F(c1;:::;cn?1);(d1;:::;dn?1) 2< ?X n > : It remains to prove that F((1) c1 ;:::;cn?1 );(d1 ;:::;dn?1 ) 2 < ?X n >. This connection, (3) completing the proof, comes from F((1) c1 ;:::;cn?1 );(d1;:::;dn?1 ) = Fd  (4) (5) (3) F((4) d1 ;:::;dn?1 );(c1 ;:::;cn?1 )  F  F(c1 ;:::;cn?1 );(d1;:::;dn?1 )  Fc : Finally, for every pair (c1; : : :; cn?1); (d1; : : : ; dn?1) 2 X n?1 ; let us con(2) sider the mappings F((1) c1 ;:::;cn?1 );(d1;:::;dn?1 ) ; F(c1 ;:::;cn?1 );(d1;:::;dn?1 ) de ned before. Observe that F((1) c1 ;:::;cn?1 );(d1;:::;dn?1 ) acts as a transposition in the permutation group over the set X n?1 fdg; while F((2) c1;:::;cn?1 );(d1 ;:::;dn?1 ) acts as an elementary collapsing in the transformation semigroup over the set X n?1 fdg. We have already proved that all of these transpositions and elementary collapsings are in < ?X n >. Moreover, it is well-known that the set of all transpositions and elementary collapsings on a set generates all mappings on that set, so any map taking X n?1  fdg to itself may be written as the restriction to X n?1  fdg of a composite of the the above functions. A moment's re ec(2) tions shows that the set of all these F((1) c1 ;:::;cn?1 );(d1;:::;dn?1 ) ; F(c1 ;:::;cn?1 );(d1;:::;dn?1 ) in fact generates all of FX n?1fdg, since a function in the latter is uniquely determined by its behaviour on X n?1  fdg. In addition, it is clear that 2 ?X n n FX n?1fdg is non-void. This completes the proof. Next we show

Lemma 2.3. Given a nite group G; a pair of relatively prime integers m; n with 1  m < n; let us de ne for every ` 2 f1; : : :; ng; the transformations Ti(0) : Gn ! Gn ; Ti(k) : Gn ! Gn ; k = 1; 2; 3; 4 as follows. T (0)(g1; : : :; gn ) = (gn ; g1; : : : ; gn?1 ); T`(1)(g1; : : :; gn ) = (gn; g1; : : : ; g`?2; g`?m?1( mod n)g`?1 ; g` ; : : :; gn?1 ); T`(2)(g1; : : :; gn ) = (gn; g1; : : : ; g`?2; g`??1m?1( mod n)g`?1 ; g` ; : : :; gn?1 ); 6

T`(3)(g1; : : :; gn ) = (gn; g1; : : : ; g`?2 ; g`?m?1( mod n); g` ; : : :; gn?1 ); T`(4)(g1; : : : ; gn) = (gn ; g1; : : :; g`?2 ; g`??11 ; g`; : : : ; gn?1 ): Then for any xed ` 2 f1; : : :; ng; TG;n 6< fT (0); T`(k) : k = 1; 2; 3; 4g > : Proof. For every i 2 f1; : : :; ng; k 2 f1; : : :; 4g; Ti(k) = (T (0))n+i?`  T`(k)  (T (0))n+`?i : Thus we shall show only TG;n 6< fT (0); T`(k) : ` 2 f1; : : : ; ng; k = 1; 2; 3; 4 > : It is clear that using the notions in Lemma 2.1, by the simple fact that every permutation is a composite of transpositions, and moreover, transformations can be generated by permutations and elementary collaps(3) ings, we obtain TG;n < fFi;j ; Ui;j : i; j 2 f1; : : :; ng > : On the other hand, < fT (0); T`(k) : k = 1; 2; 3; 4; ` = 1; : : : ; ng > nTG;n 6= ; is clear. Thus, it is enough to prove that for every i; j 2 f1; : : : ; ng; Fi;j(3); Ui;j 2< fT (0); T`(k) : k = 1; 2; 3; 4; ` = 1; : : : ; ng > : Using Fi(+(d)j?1)m?1( mod n);i+jm?1( mod n) = ) (T (0))n?1  Ti(+djm ( mod n) ; d = 1; 2; 3; i 2 f1; : : :; ng; j = 0; 1; : : : ; by an inductive application of Lemma 2.1, we have Fi(?dm) ?1( mod n);i+jm?1( mod n) 2< fT (0); T`(k) : k = 1; 2; 3; 4; ` = 1; : : : ; ng > (i 2 f1; : : : ; n; g; j = 0; 1 : : :): Therefore, because m and n are relatively prime, we receive Fi;j(d) 2< fT (0); T`(k) : k = 1; 2; 3; 4; ` = 1; : : : ; ng > (d = 1; 2; 3; i; j 2 f1; : : : ; ng): (4) (0) n?1 Moreover, we also have Fi(4) ?1 = (T )  Ti ; i 2 f1; : : :; ng: Hence, applying Lemma 2.1 again, we obtain Ui;j 2< fT (0); T`(k) : k = 1; : : :; 4; ` = 1; : : :; ng >; i; j 2 f1; : : : ; ng and thus, having Fi;j(3) < fT (0); T`(k) : k = 1; 2; 3; 4g > (i; j 2 f1; : : : ; ng); the proof is complete. 2

We shall use the following

Lemma 2.4. (see [1], [2]) Given a positive integer n; let G = denote a nite non-trivial cyclic group with a generator g 2 G: There exists an arrangement a1 ; : : :; am (m = jGjn ) of the elements in the nth direct power Gn of G such that for every i = 1; : : : ; m ? 1 there is a j 2 f1; : : :; ng with ai+1 2 f(g1; : : : ; gj?1; gj g?1; gj+1 ; : : :; gn ); (g1; : : :; gj?1 ; gj g; gj+1; : : :; gn )g; whenever ai = (g1; : : :; gn ) (2 Gn ):2 Now we are ready to prove the following key lemma. 7

Lemma 2.5. For any xed ` 2 f1; : : : ; ng; TX n is generated by the union of < fT (0); T`(k) : k = 1; : : : ; 4g > and the set of all functions F : X n ! X n

having the form F (x1; : : :; xn) = (x1; : : :; x`?1 ; f (x1; : : : ; xn); x`+1 ; : : :; xn); f : X n ! X , where x1; : : :; xn 2 X . Proof. We can take out of consideration the trivial case jX j = 1: Thus we assume jX j > 1: It is clear that without loss of generality we may suppose ` = 1: (On the other hand, using Lemma 2.3, fUi;j : i; j 2 f1; : : :; ngg 6< fT (0); T` k) : k = 1; : : : ; 4 > : Thus it is enough to prove that the union of fUi;j : i; j 2 f1; : : : ; ngg and the set of all functions F : X n ! X n having the form F (x1; : : :; xn) = (f (x1; : : : ; xn); x2; : : :; xn); generates TX n : For every pair i 2 f1; : : : ; ng, f : X n ! X , de ne the function Fi;f : X n ! X n with Fi;f (x1; : : : ; xn) = (x1; : : : ; xi?1; f (x1; : : :; xn); xi+1; : : :; xn ) (x1; : : : ; xn 2 X ): Thus, by letting f 0 = f  Ui;j , we have Fj;f = Ui;j  Fi;f 0  Ui;j . So for every pair i 2 f1; : : :; ng, f : X n ! X , Fi;f 2< TX;n [ fF : X n ! X n j F (x1; : : :; xn) = (f (x1; : : : ; xn); x2; x3; : : : ; xn), f : X n ! X; x1; : : : ; xn 2

Xg > : Let us identify X with a non-trivial nite cyclic group with generating element g 2 X . Thus we also have that for any c1; : : :; cn 2 X; F (1);j;(c1;:::;cn); F (2);j;(c1 ;:::;cn) 2 < TX;n [ fF : X n ! X n j F (x1; : : : ; xn) = (f (x1; : : :; xn); x2; x3; : : : ; xn); f : X n ! X; x = (x1; : : :; xn) 2 X n g >; whenever  2 f1; ?1g; F (1);j;(c1 ;:::;cn)(x) =

8 > if x = (c1; : : :; cj?1 ; cj g; cj+1; : : : ; cn); < (c1 ; : : :; cn ) (c ; : : :; cj?1; cj g ; cj+1; : : :; cn ) if x = (c1; : : :; cn ); > : x1 otherwise; ( 

if x = (c1; : : :; cn ); F (2);j;(c1 ;:::;cn)(x) = x(c1; : : : ; cj?1; cj g ; cj+1 ; : : :; cn) otherwise ; where x = (x1; : : : ; xn) 2 X n : On the other hand, by Lemma 2.4, there exists an arrangement a1; : : :; am of X n ; such that for every k = 1; : : : ; m ? 1; pk 2 fF (1);j;(c1;:::;cn) :  2 f?1; 1g; j 2 f1; : : : ; ng; c1; : : : ; cn 2 X g; tk 2 fF (2);j;(c1 ;:::;cn) :  2 f?1; 1g; j 2 f1; : : :; ng; c1; : : :; cn 2 X g; where 8 > < ak+1 if ` = k; pk (a`) = > ak if ` = k + 1; : a` otherwise;

8

(

` = k; tk (a`) = aak+1 ifotherwise : ` But then p1; : : :; pm?1 is a set of transpositions such that fp1; : : : ; pm?1 g generates all permutations over X n : And simultaneously, t1; : : : ; tm?1 is a set of elementary collapsings over X n : Thus by the well-known fact that for every j = 1; : : : ; m ? 1; fp1; : : : ; pm?1; tj ; g generates all transformations over X n; the proof is complete. 2

3 Main Results First we show the next statement.

Theorem 3.1. Given a positive integer n > 1; D = (V; E ) with V = f1; : : :; ng is an n-complete digraph with minimal number of edges if and only if there exists a permutation p : f1; : : : ; ng ! f1; : : : ; ng such that E = f(p(i); p(j )) : i; j 2 f1; : : : ; ng; p(j ) = p(i + 1 mod n)g [ f(p(i); p(1)) : i 2 f1; : : :; ngg: Proof. We may assume without loss of generality that the permutation p is the identity. Then it is clear that for an arbitrary m 2 f1; : : : ; ng; the functions T (0); T`(k) : k = 1; 2; 3; 4 de ned in Lemma 2.3 are compatible with D: Suppose that m is choosen such that it is relatively prime to n: Then the suciency of this statement is a direct consequence of Lemma 2.5. To the necessity rst we show the existence of j 2 V with f(i; j ) : i 2 V g  E; whenever D is n-complete. Let T : X n ! X n such that jfT (x1; : : : ; xn) : x1; : : : ; xn 2 X gj = jX n j?1: First we show that for every F1; : : :; Fm 2 TX n ; T = F1  : : :  Fm implies the existence of an index i preversing the property jfFi(x1; : : :; xn) : x1; : : :; xn 2 X gj = jX n j ? 1: Of course, if F1; : : : ; Fm are injective then T = F1  : : :  Fm should be also injective, a contradiction. On the other hand, T = F1  : : :  Fm implies jfF (x1; : : :; xn) : x1; : : : ; xn 2 X gj  minfjfFi(x1; : : :; xn) : x1; : : :; xn 2 X gj : i = 1; : : :; mg: Therefore, we obtain our assumption regarding the existence of an index i preversing the property jfFi(x1; : : : ; xn) : x1; : : : ; xn 2 X gj = jX n j ? 1: Now we identify the elements of X in a xed but arbitrary way with the elements of f1; : : : ; jX jg and consider X n as a subset of the nth directPpower of integers. For every (a1;1; : : :;Pa1;n); : : : ; (am;P1; : : :; am;n) 2 X n ; let f(ai;1; : : :; ai;n) : i = 1; : : : ; mg = ( mi=1 ai;1; : : : ; mi=1 ai;n): Let a = (a1; : : :; an); b = (b1; : : : ; bn) 2 X n denote distinct elements with jFi?1(a)j = 0 and jFi?1 (b)j = 2: And let j 2 f1; : : : ; ng be an index with aj 6= bj :

9

Prove that jXPj does not divide prj (PfFi(x1; : : :; xn) : x1P; : : : ; xn 2 X g): Indeed, then prj ( fFi(x1; : : :; xn) : x1; : :P:; xn 2 X g) = prj ( f(x1; : : :; xn) : x1; : : : ; xn 2 X g) + bj ? aj ) = jX n?1 j( kjX=0j?1 k) + bj ?Paj : Of course, by this equality we received that jX j does not divide prj ( fFi(x1; : : : ; xn) : x1; : : : ; xn 2 X g): Suppose that for every j 2 V there exists an i 2 V with (i; j ) 2= E: Consider the set DX of all functions of the form X n ! X n which are compatible with D: Now we show that for every F 2 DX ; jX j divides prj (PfF (x1; : : :; xn) : x1; : : : ; xn 2 X g); implying Fi 2= DX : By F 2 DX we have that for an appropriate ` 2 f1; : : : ; ng; prj (F (x1; : : :; xn)) = prj (F (x1; : : :; x`?1 ; x0`; x`+1 ; : : :; xn)) ((x1; : : :; xn) 2PX n ; x0` 2 X; ` = j is allowed). Therefore, forPan arbitrary xed c 2 X; prj ( fF (x1; : : :; xn) : x1; : : : ; xn 2 X g) = jX jprj ( fF (x1; : : : ; x`?1;Pc; x`+1; : : : ; xn)) : x1; : : :; x`?1; x`+1; : : : ; xn 2 X g: But then jX j divides prj ( fF (x1; : : : ; xn) : x1; : : : ; xn 2 X g) for every j = 1; : : :; n: Hence we get Fi 2= DX : Consequently, there exists a T 2 TX n whith T 2= < DX > : This ends the proof of the existence of j 2 V with f(i; j ) : i 2 V g  E; whenever D is n-complete. Then we are ready if we can prove the existence of a permutation p : f1; : : : ; ng ! f1; : : : ; ng having f(p(i); p(j )) : i; j 2 f1; : : :; ng; p(j ) = p(i) + 1(mod n)g  E: Consider the mapping T (0) : X n ! X n de ned by T (0)(x1; : : : ; xn) = (xn; x1; : : :; xn?1 ) (x1; : : :; xn 2 X ): To complete the proof of our theorem, we will show T (0) 2= DX if there exists no such a permutation p: It is also clear that an n-complete digraph D; having n vertices, should be strongly connected. Therefore, all vertices have (non-loop) incoming edges. Thus, by the minimality of jE j; we get jE n f(i; j ) : i 2 V gj = n ? 1: Simultaneously, the strongly connectivity of D implies fj g  (V n fj g) \ E 6= ; (where j 2 V with f(i; j ) : i 2 V g  E ). On the other hand, if there exists no permutation p having the above discussed property, then by the strongly connectivity of D; V  fj g  E and jE n f(i; j ) : i 2 V gj = n ? 1; we can prove jfj g  (V n fj g) \ E j  2; implying the existence of two distinct vertices i1; i2 2 V with f(`; ir ) : r = 1; 2; ` 2 V g \E = f(j; i1); (j; i2)g: It is enough to prove that in this case T (0) 2= DX : Clearly, F1 2 DX implies the existence of functions fk : X ! X; k = 1; 2 with prik (F1(x1; : : : ; xn)) = fk (xj ): Therefore, the cardinality of f(y1; y2) : yk = prik (F1(x1; : : :; xn)); k = 1; 2; x1; : : :; xn 2 X g is not greater than jX j: In a similar way, for every F1; : : :; Fm 2 DX ; m > 1 there exist functions fk : X ! X; k = 1; 2 such that prik (F1  : : :  Fm(x1; : : : ; xn)) = fk (prj (F2  : : :  Fm(x1; : : :; xm))) implying that the cardinality of f(y1; y2) : yk = prik (F1  : : :  Fm(x1; : : :; xn)); k = 1; 2; x1; : : : ; xn 2 10

X g is not greater than jX j: On the other side, the cardinality of f(y1; y2) : yk = prik (T (0)(x1; : : : ; xn)); k = 1; 2; x1; : : :; xn 2 X g is jX j2 yielding to T (0) 2= DX : The proof is complete.2 Now we prove the following characterization.

Theorem 3.2. Given a positive integer n > 1; D = (V; E ) with V = f1; : : :; mg;

m > n is an n-complete digraph with minimal number of edges if and only if there exists a permutation p : f1; : : : ; mg 7! f1; : : : ; mg such that E = f(p(i); p(j )) : p(i); p(j ) 2 f1; : : :; n + 1g; p(j ) = p(i + 1 mod n + 1)g [ f(p(i0); p(j 0))g; where i0; j 0 2 f1; : : : ; n +1g; jj 0 ? i0j 6= 1; moreover, jj 0 ? i0j? 1 and n + 1 are relatively prime. NB: The case i0 = j 0 is not excluded. Moreover, if there are more than n +1 vertices then all except for n +1 are isolated. Proof. To the suciency it is enough to prove for any n > 2 the ncompleteness of D = (f1; : : : ; n + 1g; f(i; i + 1(modn + 1)) : i 2 f1; : : : ; n + 1gg [ f(1; r)g; where r 2 f1; : : : ; n + 1g; r 6= 2; and in addition, r ? 2 and n + 1 are relative primes. Consider the set DX of all functions of the form X n(+1 ! X n+1 which are k ) (0) compatible with D. By de nition, we obtain fT ; T` : k = 1; : : : ; 4; g 6 DX ; where T (0); T`(k); k = 1; : : : ; 4 are de ned as in Lemma 2.3 (taking m of the lemma to be r ? 2). Identifying X with a nite group and using Lemma 2.3, then we get TX;n 6< DX >; too. On the other hand, we have by de nition fF : X n+1 ! X n+1 j F (x1; : : :; xn+1 ) = (xn+1; x1; : : : ; xr?1; f (x1; xr?1( mod n+1)); xr+1; : : :; xn ); f : X 2 ! X; i 2 f1; : : :; n +1g; (x1; : : : ; xn+1) 2 X n+1 g 2 DX : But then fF : X n+1 ! X n+1 j F (x1; : : : ; xn+1) = (x1; : : :; xi?1; f (xi; xi+1( mod n+1)); xi+1; : : :; xn+1); f : X 2 ! X; i 2 f1; : : : ; n +1g; (x1; : : :; xn+1) 2 X n+1 g[TX;n+1  DX resulting ?X n  DX : Applying Lemma 2.2, this shows the n-completeness of

D:

Using the obvious fact that n-complete digraph should have a strongly connected n-complete subdigraph, by our minimality conditions, we will consider digraphs which have a strongly connected subdigraph and all vertices outside of this digraph are isolated. Thus, the suciency of our statement implies that by our minimality conditions, we can restrict our investigations to the strongly connected n-complete digraphs having not more than n+2 edges. (We can take out of consideration the isolated vertices.) If we have n +1 vertices and fewer than n + 1 edges then our digraph is not strongly connected. On the other hand, if we consider a strongly connected digraph D with n +1 11

vertices and n + 1 edges, i.e., a cycle having n + 1 length, then for every F 2< DX >; there exist k 2 f1; : : :; n +1g; fi : X ! X; i = 1; : : : ; n +1 with F (x1; : : :; xn) = (f1(xk ); f2(xk+1( mod (n+1)); : : :; fn+1 (xn+k( mod n+1)) (x1; : : :; xn 2 X ): Therefore, for any 1  i1 < i2 < : : : < im  n+1; pri1 ;:::;im (F (x1; : : : ; xn+1)) = (fi1 (xi1+k( mod n+1) ); : : :; fim (xim+k( mod n+1))); (x1; : : :; xn+1 2 X ) which obviously shows that this type of digraphs can not be n-complete. Therefore, to the necessity of our statement, we can consider only strongly connected digraphs having n + 1 vertices and n + 2 edges. By the strongly connectivity of D we may suppose that D = (V; E ); with jV j = n + 1; jE j = n + 2; has a cycle C = (V 0; E 0) with k length for some 1  k  n + 1; where V 0 = fv1; : : : ; vk g(6 V ); E 0 = f(vi; vi+1( mod k)) j i = 1; : : : ; kg(6 E ): Using the strongly connectivity of D again, for every V 0 6 V there are distinct (vi; vj ); (vs; vt) 2 E with vi; vt 2 V 0; vj ; vs 2 V n V 0: Therefore, by an induction we get the structure of D in the following manner. If k < n+1 then V = fv1; : : :; vk ; vk+1; : : :; vn+1g; E = E 0[f(vk+i?1; vk+i) j i = 1; : : : ; n ? k + 1g [ f(vn+1; v`)g; where ` 2 f1; : : : ; kg is arbitrarily xed. If k = n + 1 then, of course, V = V 0; and E = E 0 [ f(vn+1; v`)g for some ` 2 f2; : : : ; n + 1g: To complete the case k = n + 1; rst we study digraphs having the form D = (fv1; : : :; vn+1g; f(vi; vi+1( mod n+1)) : i 2 f1; : : : ; n + 1gg [ f(v1; v`)g; where ` 2 f1; : : : ; n +1g; ` 6= 2; such that ` ? 2(mod n +1) and n +1 are not relative primes. Then n+1 has a divisor d > 1 such that for any mapping F 2 DX ; F (x1; : : :; xn+1) = (f1(xi1;1 ; : : : ; xi1;j1 ); : : : ; fn+1(xin+1;1 ; : : : ; xin+1;jn+1 ); where for every w 2 f1; : : :; n + 1g; u; v 2 f1; : : : ; jw g; iw;u  iw;v (mod d); iw;u  w ? 1(mod d) (x1; : : :; xn 2 X ): These hold for compatible maps, i.e. if w 6= r then fw depends only on xw?1; otherwise w = r and fw depends only on xr?1 and x1: It is also clear that every composition of such functions preserves this property. Therefore, for every F 2< DX > and i 2 f1; : : : ; n + 1g; pri (F ) depends on proper divisor of n + 1 many variables which is fewer than n: Therefore, digraphs having this like structures are not n-complete. It is remained to study the case k < n +1: Then V = fv1; : : :; vk ; vk+1; : : :; vn+1g; E = E 0 [ f(vk+i?1 ; vk+i) : i = 1; : : : ; n ? k + 1g [ f(vn+1; v`)g; where ` 2 f1; : : : ; kg is arbitrarily xed. Of course, if k = 1 or ` = 1 then we have one of the cases discussed previously. Thus we assume k; ` 6= 1: Given a set X with jX j  2; let MX = fF : X n ! X n : jX n j ? 1  jfF (x1; : : :; xn ) : (x1; : : :; xn ) 2 X n gj( jX n j)g: Clearly, then for every F : X n ! X n ; F 2< MX > : To complete our proof, now we show that there exists a network D0 = 12

(V 0; E 0) with jV 0j = n; E 0 = V 0  V 0 n f(vi; vi) : vi 2 V 0g such that for every pair F 2< DX >; H 6 f1; : : : ; n + 1g; jH j = n; the existence of prH (F ) implies prH (F ) 2< D0X > whenever prH (F ) 2 MX (where D0 X denotes the set of all functions of the form F : X n 7! X n to be compatible with D0 ). Observe that for every Fm 2 DX there are fj : X ! X; j = 1; : : : ; `?1; `+ 1; : : :; n+1; f` : X 2 ! X with Fm(x1; : : :; xn) = (f1(xk ); f2(x1); : : :; f`?1 (x`?2); f`(x`?1; xn+1); f`+1 (x`); : : :; fn+1 (xn))((x1; : : : ; xn+1) 2 X n+1 ): Therefore, H = f1; : : :; n + 1g n fig; i 2 f1; : : : ; n + 1g n f` ? 1; n + 1g and F = F1  : : :  Fm; F1; : : :; Fm 2 DX implies jfprH (F )(x1; : : : ; xn) : (x1; : : :; xn) 2 X ngj  jX n?1 j: Hence, in this case prH (F ) 2= MX : Thus we may assume H = f1; : : : ; n +1gnfig; i 2 f` ? 1; n +1g: In addition, it is clear that by the structure of D, for every T 2 DX ; cp1(T ) and cpk+1 (T ) may really depend only on the same kth variable of F: Let F = F1  : : :  Fm with F1; : : :; Fm 2 DX ; such that, prH (F ) 2 MX exists for a suitable H = f1; : : : ; i ? 1; i + 1; : : : ; n + 1g; i 2 f` ? 1; n + 1g: First we suppose m = 1: Consider functions fj : X ! X; j 2 f1; : : : ; ` ? 1; ` + 1; : : : ; n + 1g; f` : X 2 ! X with F (x1; : : : ; xn+1) = (f1(xk ); f2(x1); : : :; f`?1(x`?2); f` (x`?1; xn+1); f`+1 (x`); : : :; fn+1 (xn)) (x1; : : : ; xn+1) 2 X n+1 ): Clearly, then i 2 f1; k + 1g also holds provided prH (F ) 2 MX : Suppose i = 1: Then in consequence of i 2 f` ? 1; n + 1g; we have ` = 2: Clearly, then f2 really may not depend on its rst variable, i.e. there exists a g : X ! X with f2(x1; x2) = g(x2) (x1; x2 2 X ): Construct the function T : X n ! X n with T (x1; : : :; xn) = (g(xn); f3(x1); : : : ; fn+1(xn?1 )) ((x1; : : :; xn) 2 X n ): Then we get prH (F ) = T: On the other side, T 2 D0 X is also obvious. Suppose i = k +1: By i 2 f` ? 1; n +1g and `  k; this implies k = n: On the other side, then f` really may not depend on its second variable, i.e. there exists a g : X ! X with f` (x1; x2) = g(x1) (x1; x2 2 X ): Let T : X n ! X n with T (x1; : : :; xn) = (f1(xn ); f2(x1); : : :; f`?1(x`?2 ); g(x`?1); f`+1(x` ); : : :; fn(xn?1) ((x1; : : :; xn) 2 X n): It is obvious that T 2 D0 X and prH (F ) = T: Now we turn to the case m > 1: Then rst we de ne the mappings 1 m 2 DX in the following way. For every r = 1; : : :; m; de ne functions fr : X 7! X; gr : X 7! X with fr (x) = pr1 (Fr(x1; : : :; xk?1; x; xk+1; : : : ; xn+1)); gr (x) = prk+1 (Fr(x1; : : :; xk?1; x; xk+1; : : :; xn+1 )); x1; : : :; xk?1; x; xk+1; : : :; xn+1 2 X: (Fr 2 DX implies that fr and gr are well-de ned.)

F 0; : : :; F 0

13

In addition, let for every r = 1; : : : ; m; prj (Fr0(x1; : : : ; xn+1)) = 8 > > > > > > < > > > > > > :

f1(xk ) g1(xk ) xk prj (Fm(x1; : : :; xn+1))

if r = 1 and j = 1; if r = 1 and j = k + 1; if r > 1 and j 2 f1; k + 1g; if r = m and j 2 f2; : : :; k; k + 2; : : : ; n + 1g;

prj (Fr(fr+1(x1); x2; : : :; xk ; gr+1(xk+1); xk+2; : : : ; xn+1)) otherwise

(x1; : : : ; xn+1 2 X ): By an easy computation we get F1 : : :Fm = F10 : : :Fm0 : De ne for a xed c 2 X; m = 2; prj (Fr00(x1; : : : ; xn)) (

if 1  j < i; j (Fr0(x1 ; : : :; xi?1 ; c; xi; : : : ; xn )) = pr 0 prj+1 (Fr(x1; : : :; xi?1; c; xi; : : : ; xn)) if i  j  n (x1; : : : ; xn+1 2 X; r = 1; 2): Similarly, for a xed c 2 X and m = 3; let prj (Fr00(x1; : : :; xn)) 8 > > > > > > > > > > > > > < => > > > > > > > > > > > > :

prj (F10(xk ; x2; : : :; x`?2; xn; x`?1; : : :; xn?1; x1))

if i = ` ? 1; r = 1 and 1  j < ` ? 1;

prj+1(F10 (xk ; x2; : : :; x`?2; xn; x`?1; : : :; xn?1; x1))

if i = ` ? 1; r = 1 and ` ? 1  j  n; 0 prj (F1(xk+1 ; x2; : : :; xn)) if i = n + 1; r = 1 and 1  j  n; prn+1 (F20(x1; : : : ; xi?1; c; xi; : : :; xn)) if r = 2 and j = 1; prj (F20(x1; : : :; xi?1; c; xi; : : :; xn)) if r = 2 and 1 < j < i; prj+1(F20 (x1; : : :; xi?1; c; xi; : : :; xn)) if r = 2 and i  j  n; prj (F30(x1; : : :; xi?1; c; xi; : : :; xn)) if r = 3 and 1  j < i; 0 prj+1(F3(x1; : : :; xi?1; c; xi; : : :; xn)) if r = 3 and i  j  n;

(x1; : : : ; xn+1 2 X; r = 1; 2; 3): 14

In addition, let for a xed c 2 X and m > 3; prj (Fr00(x1; : : :; xn)) 8 > > > > > > > > > > > > > > > > > > > > > > < => > > > > > > > > > > > > > > > > > > > > > :

prj (Fr0(xk ; x2; : : : ; x`?2; xn; x`?1; : : : ; xn?1; x1))

if i = ` ? 1; r = 1; 1  j < ` ? 1; or i = ` ? 1; 1 < r  m ? 2 and 1 < j < ` ? 1;

prj+1 (Fr0(xk ; x2; : : :; x`?2; xn; x`?1; : : : ; xn?1; x1))

if i = ` ? 1; r = 1; ` ? 1  j  n; or i = ` ? 1; 1 < r  m ? 2 and ` ? 1  j  n;

prn+1 (Fr0(xk ; x2; : : :; x`?2; xn; x`?1; : : : ; xn?1; x1))

if i = ` ? 1; 1 < r  m ? 2 and j = 1; prn+1 (Fr0(xk+1 ; x2; : : :; xn)) if i = n + 1; 1 < r  m ? 2 and j = 1; 0 prj (Fr(xk+1; x2; : : :; xn)) if i = n + 1; r = 1; 1  j  n; or i = n + 1; 1 < r  m ? 2 and 1 < j  n; 0 prn+1 (Fm?1(x1; : : : ; xi?1; c; xi; : : :; xn)) if r = m ? 1 and j = 1; prj (Fm0 ?1(x1; : : :; xi?1; c; xi; : : :; xn)) if r = m ? 1 and 1 < j < i; 0 prj+1 (Fm?1(x1; : : :; xi?1; c; xi; : : :; xn)) if r = m ? 1 and i  j  n; prj (Fm0 (x1; : : :; xi?1; c; xi; : : :; xn)) if r = m and 1  j < i; 0 if r = m and i  j  n; prj+1 (Fm(x1; : : :; xi?1; c; xi; : : :; xn))

(x1; : : :; xn+1 2 X; r 2 f1; : : : ; ng)): We remark that, of course, for every j = 2; : : :; m; the value of Fj00  : : :  Fm00 (x1; : : :; xm) (x1; : : :; xn 2 X ) may depend of the value of (the above xed) c 2 X: But the value of F100  : : :  Fm00 (x1; : : :; xm) (x1; : : : ; xn 2 X ) may not depend on the value of c 2 X in question, because FH = F100  : : :  Fm00 by de nition. (Remember that the existence of prH (F ) (= prH (F1  : : :  Fm); m > 1) is supposed with H = f1; : : : ; i ? 1; i + 1; : : : ; n + 1g for a xed i 2 f` ? 1; n + 1g:) By an elementary computation we can prove F100; : : :; Fm00 2 D0 X : Applying Theorem 3.1, D0 may not be n-complete because it is not centralized. Therefore, there exists a T 2 MX with T 2= < D0X > : But then for every F 2< DX >; H = f1; : : : ; n + 1g; jH j = n; prH (F ) 6= T: Therefore, D can not be n-complete. 15

This ends the proof. 2

Acknowledgement. This work has been supported by grants of the Univer-

sity of Aizu \Algebra & Computation" and \Automata Networks" projects (R-10-1, R-10-4), the \Automata & Formal Languages" project of the Hungarian Academy of Sciences and Japanese Society for Promotion of Science (No. 15), the Academy of Finland (No 137358), the Hungarian National Foundation for Scienti c Research (OTKA T019392), and the Higher Education Research Foundation of the Hungarian Ministry of Education (No. 222). The authors express their gratitude for the supports.

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References [1] Domosi, P. and Kovacs, B., Simulation on nite networks of automata. Words, Languages and Combinatorics, Ed. by M. Ito (Kyoto, 1990), 131138, World Sci. Publishing, River Edge, NJ, 1992. [2] Domosi, P., Nehaniv, C. L., Some Results and Problems on Finite Homogeneous Automata Networks, Proc. Japanese Association of Mathematical Sciences Annual Meeting on \Languages, Computation and Algebra", Kobe University, August 27-28, 1997 (in press). [3] Tchuente, M., Computation on Finite Networks of Automata, In: Automata Networks, Ed. by C. Cho rut (Argeles-Village, France, 1986), 53-67, Lecture Notes in Computer Science 316, 1988.

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