Applied Mathematics and Computation 122 (2001) 325±340 www.elsevier.com/locate/amc

Elements of a theory of simulation III: equivalence of SDS C.L. Barrett a,*, H.S. Mortveit a,b, C.M. Reidys a a b

Los Alamos National Laboratory, TSA/DO-SA, Mailstop TA-0, SM-1237, MS M997, Los Alamos, NM 87545, USA Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7034 Trondheim, Norway

Abstract In the development of mathematical foundations for a theory of simulation, a certain class of discrete sequential dynamical systems (SDS) is of particular importance. These systems which we refer to as SDS consist of: (a) a graph Y with vertex set f1; 2; . . . ; ng; where each vertex has associated a binary state, (b) a vertex labeled set of functions Fi;Y : Fn2 ! Fn2 , and (c) a permutation p 2 Sn . The function Fi;Y update the state of vertex i as a function of the states of vertex i and its Y-neighbors and leaves all other states invariant. By composing these functions in the order given by p, we obtain the SDS ‰FY ; pŠ ˆ

n Y iˆ1

Fp…i†;Y : Fn2 ! Fn2 :

In this paper, we give a combinatorial upper bound for the number of non-equivalent SDS for a given graph, and we compute this bound explicitly for certain classes of graphs. We give a full characterization of invertible SDS, and analyze the set of ®xed points of sequential and parallel cellular automata. Further, we introduce the concept of Maj-type SDS and show that systems of this class only have ®xed points as attractors. Finally, we analyze SDS that are ®xed point free for arbitrary base graphs. Published by Elsevier Science Inc. Keywords: Sequential dynamical systems; Equivalence; Fixed points; Orbits

*

Corresponding author. E-mail addresses: [email protected] (C.L. Barrett), [email protected] (H.S. Mortveit), [email protected] (C.M. Reidys). 0096-3003/01/$ - see front matter. Published by Elsevier Science Inc. PII: S 0 0 9 6 - 3 0 0 3 ( 0 0 ) 0 0 0 4 2 - 4

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1. Introduction In this paper we continue the work on the special class of dynamical systems referred to as discrete sequential dynamical systems (SDS). The de®nition of these systems is motivated by the generic structure of computer simulations. In computer simulations we typically ®nd agents or entities with certain properties or states. The entities can retrieve information from other entities, and usually only from the ones in their own vicinity. Based on these states they may update their state. There will be some kind of scheduling that takes care of the update order of the entities. One possible interpretation of this is to have each entity as a vertex in a (dependency) graph, where two vertices are connected if the corresponding two entities can communicate. To each vertex or entity we associate a binary state. Finally, we ®x some ordering of the vertices that represent the update ordering of the entities. This construction will be made precise below. In a computer simulation one typically has ``perfect'' knowledge about each entity and which entities can communicate or exchange information. To retrieve information on the dynamics of this ``complex system'' one will have to run or simulate it. The character of the results in this paper is how to extract dynamical properties from known quantities, e.g., dependency graph, update rules, and without actually implementing and running the system on a computer. This is also the spirit of the earlier work on discrete SDS [1±4]. In a few places we have emphasized ideas and applications and have replaced proofs with the important ideas or an outline. We refer to [3,4] for the complete arguments. We start by reviewing the concepts and the setting needed for the de®nition of an SDS. Let Y be a loop-free undirected graph with vertex set v‰Y Š ˆ f1; . . . ; ng and edge set e‰Y Š. Let B0;Y …i† be the set of Y-vertices adjacent to vertex i and let di ˆ jB0;Y …i†j. We denote the increasing sequence of elements of the set B0;Y …i† [ fig by B1;Y …i† ˆ …j1 ; . . . ; i; . . . ; jdi †;

…1:1†

and set d ˆ max1 6 i 6 n di . To each vertex i there is associated a state xi 2 F2 , and for each k ˆ 1; . . . ; d ‡ 1 let fk : Fk2 ! F2 be a given symmetric function. For each vertex i 2 Nn ˆ f1; 2; . . . ; ng we introduce the map projY ‰iŠ : Fn2 ! F2di ‡1

…x1 ; . . . ; xn † 7! …xj1 ; . . . ; xi ; . . . ; xjdi †:

Furthermore, let Sk with k 2 N denote the symmetric group on k letters. Let …fk †1 6 k 6 d…Y †‡1 be a multiset of symmetric functions fk : Fk2 ! F2 . Set x ˆ …x1 ; x2 ; . . . ; xn †. For each i 2 Nn there is a Y-local map Fi;Y given by

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yi ˆ fdi ‡1  projY ‰iŠ; Fi …x† ˆ …x1 ; . . . ; xi 1 ; yi …x†; xi‡1 ; . . . ; xn †: We refer to the multiset …Fi;Y †i as FY . It is clear that for each Y < Kn the multiset …fk †1 6 k 6 n induces a multiset FY , i.e., we have a map fY < Kn g ! Y g. Let QfF n p 2 Sn . Now de®ne the map ‰FY ; Š : Sn ! Func…Fn2 ; Fn2 † by ‰FY ; pŠ ˆ iˆ1 Fp…i†;Y . De®nition 1. The SDS over Y with respect to the ordering p is ‰FY ; pŠ. We call an SDS homogeneous if it is induced by a multiset of local functions of the form …fk †k ˆ …Bk †k , where B is a Boolean function. Example 1. Let Y ˆ Circn , the circle graph on n vertices. The graph Circ6 is shown in Fig. 1. For each vertex, we have a symmetric function on three arguments. To be speci®c we pick the parity function for each vertex. The parity function (3.2) returns the sum of its argument modulo 2. Thus, for the update schedule …1; 2; 3; 4; 5; 6† with initial state …1; 1; 1; 0; 0; 0† we get F1 …1; 1; 1; 0; 0; 0† ˆ …0; 1; 1; 0; 0; 0†; F2  F1 …1; 1; 1; 0; 0; 0† ˆ …0; 0; 1; 0; 0; 0†; F3  F2  F1 …1; 1; 1; 0; 0; 0† ˆ …0; 0; 1; 0; 0; 0†; F4  F3  F2  F1 …1; 1; 1; 0; 0; 0† ˆ …0; 0; 1; 1; 0; 0†; F5  F4  F3  F2  F1 …1; 1; 1; 0; 0; 0† ˆ …0; 0; 1; 1; 1; 0†; F6  F5  F4  F3  F2  F1 …1; 1; 1; 0; 0; 0† ˆ …0; 0; 1; 1; 1; 1†; and thus ‰FCirc6 ; …1; 2; 3; 4; 5; 6†Š…1; 1; 1; 0; 0; 0† ˆ …0; 0; 1; 1; 1; 1†. If we remove the restriction on having symmetric functions we obtain the sequential analog of Ulam and von Neumann's cellular automata. We call this class sequential cellular automata (sCA), and refer to the classical CA as parallel cellular automata (pCA). Formally, a pCA is a pair …f3 ; Circn † where f3 : F32 ! F2 is the rule used to update the states associated to the vertices of Circn in parallel. An sCA is a triple …f3 ; Circn ; p†. In this case the function f3

Fig. 1. The circle graph on six vertices and the wheel graph on seven vertices shown to the left and right, respectively.

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updates the states of Circn sequentially in the order given by p. Note that an sCA is an SDS only in the case where f3 is a symmetric function. We emphasize that the base graph of an SDS does not have to be a circle, but can be an arbitrary graph. Another example of a base graph is shown on the right in Fig. 1. We introduce the equivalence relation Y ;F on Sn  Sn by p Y ;F r i€ ‰FY ; pŠ ˆ ‰FY ; rŠ and let S…fk †k …Y † ˆ f‰FY ; pŠ j p 2 Sn g. The digraph C‰FY ; pŠ is the directed graph having vertex set Fn2 and edge set f…x; ‰FY ; pŠ…x†† j x 2 Fn2 g. For p ˆ …i1 ; . . . ; in † write i 1

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for any graph. Finally, in Section 6 we study phase space properties of a certain class of SDS, and we show that these systems are the only systems that will be ®xed point free for any dependency graph.

2. Equivalence of SDS In practice, one is interested in studying or prescribing a system with a given number of orbits and orbit sizes as well as a given transient behavior. Two dynamical systems can di€er as maps but nevertheless they may have the same dynamical properties. To be more precise let ‰FY ; pŠ and ‰FY ; p0 Š be two SDS. If there exists a bijection u : Fn2 ! Fn2 such that   …2:1† FY ; p0 ˆ u  ‰FY ; pŠ  u 1 ; we say that ‰FY ; pŠ and ‰FY ; p0 Š are dynamically equivalent SDS. Note that (2.1) implies, e.g., that the two dynamical systems have a 1±1 correspondence between ®xed points. For if x is ®xed under ‰FY ; pŠ we obtain ‰FY ; p0 Š…u…x†† ˆ u…x†, i.e., u…x† is a ®xed point for ‰FY ; p0 Š. For classical dynamical systems the de®nition of dynamically equivalent systems coincides with the concept of topologically conjugate systems when u is a homeomorphism. In light of all this, it is of interest to have estimates for the size of the set  RY ‰FY ; pŠ ˆ O 2 Acyc…Y † j C‰FY ; O# Š  C‰FY ; pŠ ; …2:2† where O# is the canonical permutation w.r.t. O [5]. In the following we will write S…Y † for S…fk †k …Y †. Proposition 1. Let Y < Kn and define the Sn -action on Fn2 by
q…x† ˆ xq 1 …1† ; . . . ; xq 1 …n† : The following holds: 1. The map Aut…Y †  Sn = Y ! Sn = Y defined by …c; ‰pŠY †7!‰c  pŠY is an Aut…Y †-action on Sn = Y . This action induces an Aut…Y †-action on Acyc…Y † given by fcOg…fi; kg† ˆ O…fc 1 …i†; c 1 …k†g†. 2. For all p 2 Sn and all c 2 Aut…Y † we have ‰FY ; cpŠ ˆ c  ‰FY ; pŠ  c 1 . 3. The map Aut…Y †  S…Y † ! S…Y † given by …c; ‰FY ; pŠ†7!‰FY ; c  pŠ is an Aut…Y †-action on S…Y † with the property ‰FY ; c  pŠ ˆ c  ‰FY ; pŠ  c 1 . In particular RY ‰FY ; pŠ is an Aut…Y †-set. For the proof of this proposition we refer to [3]. As a consequence of Proposition 1, we derive the following upper bound for the number of nonequivalent SDS.

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Corollary 1. Let Y < Kn . We have jfC‰FY ; pŠ j p 2 Sn gj 6

X 1 jFix…c†j; jAut…Y †j c2Aut…Y †

…2:3†

where Fix…c† ˆ fO 2 Acyc…Y † j c  O ˆ Og. The corollary is essentially a consequence of Burnside's theorem and the details P can be found in [3]. In the following we will set D…Y † ˆ c2Aut…Y † jFix…c†j=jAut…Y †j. A combinatorial interpretation of Fix…c† is given in [4]. It can be described as follows. Let G be a group and let Y be an undirected graph. We will denote Y-automorphisms by c. Now, G is said to act on Y if there exists a group homomorphism u : G ! Aut…Y †. De®nition 3. Assume G acts on Y < Kn . Then G n Y is the graph with v‰G n Y Š ˆ fG…i† j i 2 v‰Y Šg;

e‰G n Y Š ˆ fG…y† j y 2 e‰Y Šg

and pG is the surjective graph morphism given by pG : Y ! G n Y ;

i 7! G…i†:

Proposition 2. Let Y < Kn be an undirected graph. Then we have 1 X D…Y † ˆ ja…hci n Y †j: jCj c2C

…2:4†

Let …0 Y † denote the vertex join of 0 and Y. The map pG has the property pG …0 Y † ˆ 0 pG …Y †:

…2:5†

For the proof we refer to [4]. Some remarks are in order. To begin, take c 2 Aut…Y † and write it as a product of disjoint cycles where cycles of length 1 are also included, say c ˆ c1
c2


ck . The vertices in G n Y are in a 1±1 correspondence with the cycles c1 ; c2 ; . . . ; ck . However, one should note that G n Y is in general not a simple graph as it may contain loops. There are two main factors making the computations of a…hci n Y † relatively simple. The graph hci n Y is typically of a nature well suited for computing its number of acyclic orientations. The procedure is also simpli®ed by the fact that if hci n Y has loops every orientation is necessarily cyclic. Example 2 (The number of non-equivalent systems for the cube Q32 , see Fig. 2). Note that we use cycle notation for the permutations throughout the entire example. We have Aut…Q32 †  Z32 oS3 . Here we will simply use the fact that the automorphism group is generated by, e.g., q ˆ …0132†…4576†, l ˆ …0†…124†…365†…7† and s ˆ …04†…15†…26†…37†. The graph hsi n Q32 is shown to

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the right in Fig. 2. The only automorphisms giving cycle free reduced graphs, and hence the only ones that contribute to D…Y †, are given in Fig. 3 along with their reduced graphs. Since we have a…Q32 † ˆ 1862 we get D…Q32 † ˆ 54. Thus, there are at most 54 non-equivalent SDS on the cube. De®ne the function Nork by Nork : Fk2 ! F2 ;

Nor…x1 ; . . . ; xk † ˆ x1 _


_ xk :

Fig. 2. The graphs Q32 and h…04†…15†…26†…37†i n Q32 .

Fig. 3. The elements of Aut…Q32 † with their reduced graphs.

…2:6†

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Numerical computations for the SDS induced by Nor show that this bound is sharp, i.e., jfC‰NorQ3 ; pŠ j p 2 Sn gj ˆ D…Q32 † ˆ 54: 2

No numerical computations so far have produced an example where equality does not hold for the SDS induced by Nor. In the following, we show how we can take full advantage of the results above when we apply it to families of graphs and vertex joins of such. As usual denote by Circn the graph with vertex set f1; 2; . . . ; ng and edge set ffi; i ‡ 1g j 1 6 i 6 n 1g [ ff1;ngg. Set Wn ˆ Circn 0 and Pn ˆ Wn …n ‡ 1†. De®ne Qn by v‰Qn Š ˆ v‰Pn Š and e‰Qn Š ˆ e‰Pn Š n ff0;n ‡ 1gg. Proposition 3. Let n > 2 and let / be the Euler /-function. Then we have D…Circn † ˆ D…Wn † ˆ

 1 X /…d† 2n=d 2n djn

 1 X /…d† 3n=d 2n djn

D…Pn † ˆ D…Qn †  1 X ˆ /…d† 2
4n=d 4n djn

2 3




4




n

‡ …1 ‡ … 1† †

‡ …1 ‡ … 1†n †




2n=2 ; 8

2
3n=2 ; 8

‡ …1 ‡ … 1†n †

3
4n=2 ; 8

…2:7† n 6ˆ 3;

…2:8†

n 6ˆ 4: …2:9†

Proof. Let Circin be the graph obtained from Circn by deleting the edge fi; i ‡ 1g. Similarly de®ne Wni , Pni and Qin as the graphs obtained from Wn , Pn and Qn , respectively, by deleting the edge fi; i ‡ 1g. Straightforward calculations show that a…Circin † ˆ 2n 1 ; a…Qin † ˆ a…Pni † a…Pn † ˆ 2
…4n

a…Wni † ˆ 2
3n 1 ;

a…Pni † ˆ 6
4n 1 ;

a…Wni †; a…Circn † ˆ 2n 2; a…Wn † ˆ 3n 4†; a…Qn † ˆ a…Pn † a…Wn †:

3;

Consider the graph Circn . Clearly Aut…Circn †  Dn , i.e., the dihedral group on 2n elements. Now Dn ˆ hsiohri where, using cycle notation, r ˆ …1; 2; . . . ; n† Qdn=2e and s ˆ iˆ2 …i; n i ‡ 2†. (i) If o…rk † ˆ n, then hrk i n Circn contains loops and consequently jFix…rk †j ˆ 0. Here o… † denotes order. (ii) If o…rk † ˆ n=2, then hrk i n Circn is a graph with two vertices connected by an edge and we obtain jFix…rk †j ˆ 2 ˆ 2n=…n=2† 2. (iii) In the case where rk has order n=d, d > 2 we have hrk i n Circn  Circd and thus jFix…rk †j ˆ 2d 2.

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(iv) Finally it is seen that the only case in which hsrk i n Circn does not contain loops is when n; k  0 mod 2, and in this case hsrk i n Circn  Circin=2‡1 and jFix…srk †j ˆ 2n=2 for all k. Thus 1 D…Y † ˆ 2n

X

1 ˆ 2n

X

k

jFix…r †j ‡

k

djn

X

! k

jFix…sr †j

k

 /…d† 2n=d



1 2 ‡ …1 ‡ … 2

! n n=2 1† †

2 : 2 n

Now consider Wn . Clearly we also have Aut…Wn †  Dn . The calculation of D…Wn † now follows e€ortlessly from what we did above. To be speci®c: (i) By the same argument we have jFix…rk †j ˆ 0 whenever rk has order n. (ii) Since pG …0 Y † ˆ 0 G…Y † we have hrk i n Wn  Circ3 when rk has order n=2 and thus jFix…rk †j ˆ 6 ˆ 3n=…n=2† 3. (iii) When o…rk † ˆ n=d, d > 2 we have obtain hrk i n Wn  Wd and we get jFix…rk †j ˆ 3d 3. i (iv) Using the property of pG again we obtain hsrk i n Wn  Wn=2‡1 when k n=2 n; k  0 mod 2 and consequently jFix…sr †j ˆ 2
3 . Adding up produces the given formula. The graphs Pn and Qn can be dealt with similarly. The only di€erence is that now we have Aut…Pn † ˆ Aut…Qn † ˆ hdio…hsiohri†, where d ˆ …0; n ‡ 1†.  Remark 1. This process can be generalized. Let Pm;0 ˆ Circm and de®ne Pm;n by Pm;n ˆ Pm;n 1 …m ‡ n†. Clearly, Pm;n ˆ Circm Kn and we have Aut…Pm;n †  Dm  Sn . Similarly de®ne Qm;n ˆ Circm En , where En is the empty graph on n vertices. Also we have Aut…Qm;n † ˆ Aut…Pm;n †.

3. Invertibility In this section, we give a complete characterization of invertible SDS. To begin we recall some of the structure of symmetric functions. De®ne Hk : Fk2 ! N by Hk …x† ˆ jfxi j xi ˆ 1gj, and the equivalence relation H on Fk2  Fk2 by x H y i€ Hk …x† ˆ Hk …y†. The symmetric functions f : Fk2 ! F2 are precisely the functions that are constant on the equivalence classes of H . The equivalence classes will also be referred to as Hamming classes. As a consequence we note that there are 2k‡1 such functions.

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In [3] we derived the following result on invertibility: Proposition 4. Let Y < Kn , let …fk †k be a multi-set fk : Fk2 ! F2 and let id; inv : F2 ! F2 be the maps defined by id…x† ˆ x and inv…x† ˆ x. An SDS ‰FY ; pŠ is bijective if and only if for each 1 6 i 6 n and fixed coordinates x1 ; . . . ; xi 1 ; xi‡1 ; . . . ; xn the map gi ˆ fdi‡1 ;Y  projY ‰iŠ…x1 ; . . . ; xi 1 ; ; xi‡1 ; . . . ; xn † : F2 ! F2 …3:1† has the property gi 2 fid; invg. Furthermore, let p ˆ …i1 ; . . . ; in 1 ; in † 2 Sn , p ˆ …in ; in 1 ; . . . ; i1 † and ‰FY ; pŠ be a bijective SDS. Then we have ‰FY ; pŠ

1

ˆ ‰FY ; p Š:

Consequently we have the interesting fact that the inverse of an invertible SDS is again an SDS. De®ne the two functions Park and Park by k X xi ; …3:2† Park : Fk2 ! F2 ; Par…x1 ; . . . ; xk † ˆ iˆ1

and Park :

Fk2

! F2 by Park …x1 ; . . . ; xk † ˆ 1

Park …x1 ; . . . ; xk †.

Theorem 1. Let Y be a graph and let ‰FY ; pŠ be an invertible SDS over Y. Then FY ˆ …Fi;Y †i , where Fi;Y ˆ Pari;Y or Fi;Y ˆ Pari;Y . Proof. Let i be a vertex with degree di ˆ k 1 and associated symmetric local function fk . We show by induction on the Hamming class that the value of fk on …0† determines fk completely. Induction basis: The value fk …0† determines the value of fk on Hamming class 1. Assume fk …0† ˆ y0 . Then by using (3.1) with i ˆ 1 and representative …1; 0; 0; . . . ; 0† from Hamming class 1, we obtain fk …0; 0; . . . ; 0† ˆ y0 ;

fk …1; 0; . . . ; 0† ˆ y0 :

Induction step: The value of fk on Hamming class l determines the value of fk on Hamming class l ‡ 1. Let xl ˆ …0; 1; 1; . . . ; 1; 0; 0; . . . ; 0† 2 Hl and assume fk …xl † ˆ yl . Then as in the induction basis we derive with xl‡1 ˆ …1; 1; . . . ; 1; 0; 0; . . . ; 0† fk …0; 1; 1; . . . ; 1; 0; 0; . . . ; 0† ˆ yl fk …1; 1; 1; . . . ; 1; 0; 0; . . . ; 0† ˆ yl ; completing the induction step. If y0 ˆ 0 we obtain fk ˆ Park and if y0 ˆ 1 we obtain fk ˆ Park , and the proof is complete.  Corollary 2. The only homogeneous invertible SDS are the ones induced by Par and Par. Thus there are 2n invertible SDS for a given graph Y < Kn .

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4. Fixed points of sCA and pCA In this section we give a characterization of the ®xed points of pCA and sCA, and we show how to derive recursion relations for the number of ®xed points as a function of n. The structure result is then used to show that the only homogeneous SDS on Circn that are ®xed point free for any n are the ones induced by Nor and Nand, where the function Nandk is given by Nandk : Fk2 ! F2 ;

Nandk …x1 ; . . . ; xk † ˆ x1 ^


^ xk :

…4:1†

In [3] we have shown that the ®xed points of an SDS are invariant with respect to the update ordering. An immediate consequence of this is that the ®xed points of sequentially and parallely updated systems can be dealt with in the same way. De®nition 4. Let x; x0 2 F32 . We say that x is compatible with x0 if x2 ˆ x01 and x3 ˆ x02 and we write this as x . x0 . The multiset C ˆ …xi 2 F32 j 1 6 i 6 n† is a compatible covering of Circn if x1 . x2 .


. xn . x1 . De®nition 5. Let f : F32 ! F2 be some function. A compatible covering C is called a compatible ®xed point covering if f …xi † ˆ xi2 , 1 6 i 6 n. The set of all compatible ®xed point coverings of Circn with respect to f is denoted Cnf . Let UCircn : Func…F32 ; F2 † ! DiGraph be the map assigning to f 2 Func…F32 ; F2 † the graph Cf with [   …x; x0 † j x; x0 2 F32 ; x . x0 : v‰Cf Š ˆ x 2 F32 j f …x† ˆ x2 ; e‰Cf Š ˆ x2v‰Cf Š

Thus a cycle of length n in Cf corresponds to a compatible ®xed point covering of Circn . Clearly, UCircn …f † has at most eight vertices. Let f 2 Func…F32 ; F†. Each C 2 Cnf corresponds uniquely to a ®xed point of the induced pCA/sCA. To be precise de®ne


…4:2† u : Cnf ! Fix…f ; Circn †; u x1 ; x2 ; . . . ; xn ˆ x12 ; x22 ; . . . ; xn2 : The map u is 1±1 by construction. Theorem 2. Let f 2 Func…F32 ; F†. The number of fixed points Ln of the induced sCA (or pCA) equals jCnf j. Let A be the adjacency matrix of Cf . We have …4:3† Ln ˆ Tr An : Pk Let vA …l† ˆ iˆ0 ai lk i be the characteristic polynomial of A. The number of fixed points Ln satisfies the recursion relation

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ai L n

i

ˆ 0:

…4:4†

iˆ0

Proof. The ®rst statement follows from the fact that u is 1±1 and that ‰An Šii is the number of cycles starting at vertex i. Now, (4.3) can be rewritten as Ln ˆ Tr An ˆ

k X iˆ0

ei An eTi ;

where ei denotes the ith unit vector. The LHS of (4.4) now becomes ! k k k X X X n i T ai Ln i ˆ ai ej A ej iˆ0

iˆ0

ˆ ˆ

jˆ0

k X

k X

jˆ0

iˆ0

k X

!

ej ai An i eTj

e j a 0 An ‡ a 1 An

1

‡


‡ a k An

jˆ0

ˆ

k X jˆ0

k




eTj

ej vA …A†An k eTj ˆ 0;

where the last equality follows from the Hamilton±Cayley theorem.



Corollary 3. Let Y ˆ Circn . The only homogeneous SDS that are fixed point free for any n are ‰NorCircn ; pŠ and ‰NandCircn ; pŠ. Proof. Let ‰FCircn ; pŠ be the SDS induced by the single function f. From Theorem 2 we have that the non-existence of ®xed points is equivalent Cf having no loops. Set ai ˆ f …x†, where H3 …x† ˆ i. Clearly, a0 ˆ 1 and a3 ˆ 0. Now, a1 ˆ 1 implies a2 ˆ 1. Likewise, a1 ˆ 0 implies a2 ˆ 0. In the latter case we see that f ˆ Nor3 . In the former case f ˆ Nand3 , and the proof is complete.  Remark 2. In [4] it is shown that Nor and Nand are indeed the only local functions with the property that there are no graphs for which the induced SDS has ®xed points. A Garden-of-Eden (GOE) point of a dynamical system is a point having no preimage. Similarly, we de®ne a GOE ®xed point of a dynamical system to be a ®xed point that has no preimage apart from itself. We call a ®xed point that has no preimage apart from itself for arbitrary schedule a global GOE ®xed point.

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Corollary 4. The following two statements are equivalent: (i) ‰MajCircn ; pŠ has global GOE ®xed points. (ii) n  0 mod 4. Proof. From the graph UCircn …Maj3 † we see that the ®xed points of ‰MajCircn ; pŠ are all points without isolated states, i.e., if xi ˆ 0, then xi 1 ˆ 0 or xi‡1 ˆ 0 and similarly if xi ˆ 1. If a ®xed point x has three or more consecutive zeros we can easily ®nd a permutation such that there is a preimage of x di€erent from x itself. To be explicit assume xi 1 ˆ xi ˆ xi‡1 ˆ 0. Pick r 2 Sn such that i 1. There is an index i which is maximal with respect the ordering p such that ‰FY ; pŠ…x†k ˆ xk ; ‰FY ; pŠ…x†i ˆ 1

k >p i; xi ;

and, Fi;Y  ‰FY ; pŠ…x†i ˆ xi :

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Proof. By assumption x is periodic with period p > 1, so there is at least one index l such that ‰FY ; pŠ…x†l 6ˆ xl , and thus there is a maximal (w.r.t. p) index i such that ‰FY ; pŠ…x†k ˆ xk for k >p i. The last statement follows from the fact that restricted to an orbit ‰FY ; pŠ is invertible with inverse ‰FY ; p Š and using ‰FY ; p Š  ‰FY ; pŠ ˆ id on such an orbit.  Proposition 5. An SDS of Maj-type has no periodic points of period p P 2. Proof. Let ‰FY ; pŠ be an SDS of Maj-type and assume that x is a periodic point of period p > 1. Clearly, property 3 of Lemma 1 does not hold for Maj-type system yielding the desired contradiction.  Proposition 6. For a given m there is a graph Y with jv‰Y Šj 6 2m ordering p such that ‰Maj; pŠ has transient length m.

1 and an

Proof. Let Y be the graph with vertex set v‰Y Š ˆ f1; 2; . . . ; 2m 1g and edge set e‰Y Š ˆ ff1; 2g; . . . fm; m ‡ 1g; f2; 2 ‡ mg; . . . ; fm; 2mgg. Let x be the initial state with xi ˆ 0 for 1 6 i 6 m and xi ˆ 1 for m ‡ 1 6 i 6 2m 1 and let p ˆ …1; 2; . . . ; 2m 1†. By direct calculation is clear that ‰MajY ; pŠ…l† …x† 6ˆ …1; 1; . . . ; 1† for 1 6 l < m and ‰MajY ; pŠ…m† …x† ˆ …1; 1; . . . ; 1†.  6. Fixed point free SDS It is fairly straightforward to show that any SDS induced by Nor or Nand cannot have ®xed points. A more interesting fact is that for any other multiset of local functions than …Nork †k and …Nandk †k there will always be a graph such that the induced SDS has at least one ®xed point. The precise statement is given in the theorem below. Recall that an independence set of a graph Y is a subset I of v‰Y Š such that no two elements in I are connected by an edge in Y. We denote the set of independence sets of a graph Y with I…Y †. We can now state the following: Theorem 3 [4]. Let p 2 Sn , and let …fk †k be a multiset of local functions. Assume that the induced SDS ‰FY ; pŠ is fixed point free for any graph Y. Then we have ‰FY ; pŠ  ‰Nor; pŠ; and the following assertions hold: (a) There exists a bijective mapping i : Per‰FY ; pŠ ! I…Y †:

…6:1†

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(b) One of the following alternatives hold: (i) each C‰FY ; pŠ vertex is contained in a C‰FY ; pŠ cycle or (ii) it has indegree 0. Proof. We give an outline of the proof. If one of the local functions fk is neither Nandk nor Nork we know that there are Hamming classes l; m 6ˆ 0; k such that the value of fk on class l is 1 and 0 on class m. From this fact one can construct in a generic way a graph Y such that the induced SDS would have a ®xed point. The full details can be found in [4]. Here it is also shown that a system having local functions both Nor and Nand would also have ®xed points. Statement (a) follows from the fact that the set CY ˆ f…nj † 2 Fn2 j 8j 2 Nn : nj ˆ 1 ) 8i 2 B0;Y …j†: ni ˆ 0g

…6:2†

precisely the periodic points of ‰Nor; pŠ and CY can be brought in a 1±1 correspondence with the independence sets of Y. It is easy to see that any point maps into CY upon one application of ‰Nor; pŠ, and hence every point is either periodic or has indegree 0 in C‰NorY ; pŠ. The full details can be found in [4].  Let Linen be the graph with vertex set f1; 2; . . . ; ng and edge set ffi; i ‡ 1g j 1 6 i 6 n 1g. We ®nally count the number of periodic points of SDS induced by Nor on Linen and Circn . p Proposition 7. Let / be the golden ratio i.e., / ˆ …1 ‡ 5†=2. The number of periodic points for an SDS induced by Nor on Linen is Fn‡1 ; where Fn denotes the nth Fibonacci number (F0 ˆ 1, F1 ˆ 1 and Fn ˆ Fn 1 ‡ Fn 2 ; n P 2). The number of periodic points of an SDS induced by Nor on Circn is /n ‡ n … 1=/† . Proof. This is clear from Theorem 3 and the following argument: Let Bn be the number of periodic points of ‰NorLinen ; pŠ. For periodic points with x1 ˆ 1 one must have x2 ˆ 0. Clearly, for the remaining coordinates there are as many choices as there are periodic points for ‰NorLinen 2 ; pŠ. Thus the number of periodic points of ‰NorLinen ; pŠ with x1 ˆ 1 is Bn 2 . Similarly we get that the number of periodic points of ‰NorLinen ; pŠ having x1 ˆ 0 is equal to Bn 1 . Thus we have Bn ˆ Bn 1 ‡ Bn 2 , n P 3, where B1 ˆ 2, B2 ˆ 3, and we derive Bn ˆ Fn‡1 as claimed. Denote by An the number of periodic points of ‰NorCircn ; pŠ. The number of periodic points with x1 ˆ 1 is clearly equal to Bn 3 while the number of periodic points with x1 ˆ 0 equals Bn 1 , and thus An ˆ Bn 1 ‡ Bn 3 ˆ Fn ‡ Fn 2 , n P 4. Using the equations for the nth Fibonacci number gives An ˆ /n ‡ … 1=/†n , n P 4. The equation also holds for n ˆ 3 and we are done. 

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Acknowledgements Special thanks and gratitude to D. Morgeson, for his continuous support. This research is supported by Laboratory Directed Research and Development under DOE contract W-7405-ENG-36 to the University of California for the operation of the Los Alamos National Laboratory. References [1] C.L. Barrett, C.M. Reidys, Elements of a theory of simulation I: sequential CA over random graphs, Appl. Math. Comput. 98 (1999) 241±259. [2] C.L. Barrett, H.S. Mortveit, C.M. Reidys, Elements of a theory of simulation II: sequential dynamical systems, Appl. Math. Comp. 107 (2000) 121±136. [3] H.S. Mortveit, C.M. Reidys, Discrete sequential dynamical systems, Discrete Math. 226 (2001) 281±295. [4] C.M. Reidys, On acyclic orientations and SDS Adv. Appl. Math., submitted. [5] C.M. Reidys, Acyclic orientations of random graphs, Adv. Appl. Math. 21 (1998) 181±192.