Elements of a theory of simulation II: sequential dynamical systems

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1 Applied Mathematics and Computation 107 (2000) 121±136 Elements of a theory of simulation II: sequential dynamical systems C.L. Barrett *, H.S. Mortveit, C.M. Reidys Los Alamos National Laboratory, TSA/DO-SA, Mailstop TA-0, SM-1237, MS M997, Los Alamos 87545, New Mexico, USA Abstract We study a class of discrete dynamical systems that is motivated by the generic structure of simulations. The systems consist of the following data: (a) a nite graph Y with vertex set f1;...;ngwhere each vertex has a binary state, (b) functions F i : F n 2! Fn 2 and (c) an update ordering p. The functions F i update the binary state of vertex i as a function of the state of vertex i and its Y-neighbors and leave the states of all other vertices xed. The update ordering is a permutation of the Y-vertices. By composing the functions F i in the order given by p one obtains the sequential dynamical system (SDS): ; pš ˆYn F p i : F n 2!Fn 2 : iˆ1 We derive a decomposition result, characterize invertible SDS and study xed points. In particular we analyse how many di erent SDS that can be obtained by reordering a given multiset of update functions and give a criterion for when one can derive concentration results on this number. Finally, some speci c SDS are investigated. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Sequential dynamical systems; Fixed points; Structure; Orderings 1. Introduction This paper is the second of a series in which we intend to develop a basic theory of simulation. Here we build on the ideas presented in the rst paper * Corresponding author. barrett@lanl.gov /00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S (98)

2 122 C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136 [1] and introduce Sequential Dynamical Systems, (SDS), a new class of dynamical systems implied by the formalization of simulation as composed local maps. Intuitively, SDS are simply those dynamical systems produced by sequentially ordered compositions of local maps. The dynamical properties of SDS delimit the behavioral repertoire of simulations. An SDS basically consists of (i) a graph Y, (ii) local maps, i.e., Boolean functions indexed by the vertices and de ned on the states of the vertex itself and its corresponding nearest neighbors and (iii) a permutation of the vertices. As a particular example we have asynchronous cellular automata (sca). An sca consists of the following data: the circle graph on n labelled vertices, denoted by Circ n, a rule f : F 3 2! F 2 and a permutation p of the Circ n -vertices. Each vertex is assigned a binary state and the states of the vertices are updated by applying f in the order given by the permutation p. One usually writes an sca as a triple (Circ n, f, p). It may be viewed as a simulation in which the entities correspond to the Circ n -vertices, the support structure is the graph Circ n and the update schedule corresponds to the permutation p. The full update for the states of the entities gives a class of discrete dynamical systems which we will refer to as SDS [1]. Note that the mathematical constituents of SDS correspond to the essential elements of a computer simulation. Simulations typically are comprised of entities having state values and local rules governing state transitions, a spatial environment in which the entities act or interact, and some method with which to trigger an update of the state of each entity. Schedules for updates can be time stepped, event driven, scripted, etc., and result in the dynamical properties in state space that we call a ``simulated system''. As is seen above and in Ref. [1], the general form of the support structure for SDS is discrete. It is not that this theory is being constructed to apply only to simulations that represent space discretely. Rather, what is captured is that the idea of entity adjacency in the support structure is de ned by the causal dependency among local maps. That is, entities are adjacent in the support structure if and only if they can interact. This spatial representation (support structure), perhaps called ``interaction space'' or ``cause space'', is an inherently discrete (graph) structure having maps associated to vertices and dependency denoted by edges. The support structure is a transformation of the ``natural'' space that particular entities could be de ned with respect to and is, in that important sense, general and context free. This is obviously an essential issue for a truly general simulation theory. Locality, a property of a the maps, is de ned in terms of adjacency, a property of the support structure. The resulting interplay between the topological and algebraic properties of SDS is very interesting and seems to open new areas of purely mathematical investigation.

3 2. Sequential dynamical systems 2.1. De nition C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121± We set N n ˆ f1; 2;... ; ng. Let the set of Y-vertices adjacent to vertex i be denoted by D 1 i and set d i ˆ jd 1 i j. We denote the increasing sequence of elements of the set B 1 i ˆ D 1 i [ fig by ^B 1 i ˆ j 1 ;... ; i;... ; j di ; 1 and set d ˆ max 1 6 i 6 n d i. Each vertex i has associated a binary state x i. Also, let f k k with f k : F k 2! F 2 where 1 6 k 6 d 1 be some given multiset of symmetric functions. For each vertex i 2 N n we de ne the map proj Y iš : F n 2! Fdi 1 2 ; x 1 ;... ; x n 7! x j1 ;... ; x i ;... ; x jdi : Finally, let S k with k 2 N denote the permutation group on k letters. De nition 1 (Y-local maps). Let f k 1 6 k 6 d Y 1 be a multiset of S k -symmetric functions f k : F k 2! F 2. For each i 2 N n there is a Y -local map F i;y given by y i ˆ f di 1 proj Y iš; F i;y x j j ˆ x 1 ;... ; x i 1 ; y i x ; x i 1 ;... ; x n : F i;y is a map F i;y : F n 2! Fn 2 that updates the state of vertex i as a function of the states contained in B 1 i and leaves all other vertex states xed. We refer to the multiset F i;y i as F Y. In particular, let f k 1 6 k 6 n be a xed multiset of S k -symmetric functions as de ned above. Then for each Y < K n the multiset f k 1 6 k 6 n induces a multiset F Y, i.e., we have a map fy < K n g! ff Y g. Let p 2 S n. The introduction of the maps F i;y allows us to consider products of the form ; pš ˆ Yn iˆ1 F p i ;Y : F n 2! Fn 2 : 2 De nition 2 (SDS). An SDS over a graph Y w.r.t. p is a product ; pš ˆ Yn iˆ1 F p i ;Y : F n 2! Fn 2 : 3 In this paper we will be particularly interested in computing the number of di erent SDS, i.e., a fk k Y ˆ jf ; pšjp 2 S n gj 4 for a given multiset f k k and for a given graph Y. That is, how many di erent dynamical systems can be obtained by rescheduling.

4 124 C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136 Sometimes the multiset f k k is induced by a single Boolean function B : F n 2! F 2. In this case we will say that the corresponding SDS is induced by B. The Boolean functions listed below have this property and will be studied later in some detail: NOR k : F k 2! F 2 x 1 ;... ; x k 7! x 1 x k 5 NAND k : F k 2! F 2 x 1 ;... ; x k 7! x 1 ^ ^ x k 6 PAR k : F k 2! F 2 x 1 ;... ; x k 7! Xk x i 7 iˆ1 MAJ k : F k 2! F 2 x 1 ;... ; x k 7! 1 iff jfx jjx j ˆ 1gj P jfx j jx j ˆ 0gj 0 else 8 MIN k : F k 2! F 2 x 1 ;... ; x k 7! 1 iff jfx jjx j ˆ 1gj < jfx j jx j ˆ 0gj 0 else 9 XOR k : F k 2! F 2 x 1 ;... ; x k 7! 1 iff jfx i ˆ 1gj ˆ else Although a slight abuse of terminology, we will simply write, e.g., a PAR, for these functions instead of using the full multiset f k k as index. Remark 1. Note that MAJ and MIN are complementary functions, i.e., MAJ k x ˆ MIN k x. Nevertheless, and as will be shown later, the corresponding two SDS for a given graph usually behave very di erently Combinatorial analysis The function a fk k Y is closely related to a combinatorial invariant of Y itself, namely the number of acyclic orientations of Y denoted by a(y). An acyclic orientation is a map that assigns a direction to each Y-edge such that the resulting directed graph is a forest. Some comments on this relation are in order. We will write a permutation p as an n-tuple i 1 ;... ; i n and when nothing else is stated the natural ordering 1;... ; n is assumed. Now SDS can be analysed from a purely combinatorial perspective [1]. This approach is based on the simple observation that if p ˆ i 1 ;... ; i n and p 0 ˆ i 0 1 ;... ; i0 n are two permutations di ering by a transposition of consecutive coordinates i k ; i k 1 where fi k ; i k 1 g 62 e[y], then independently of the choice of the maps F i;y we have ; pš ˆ ; p 0 Š. This leads to an analysis which is independent of the structure of the local maps, that is, it only considers formal dependencies and is thus determined by the underlying graph Y alone. It motivates the introduction of the update graph U(Y):

5 C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121± De nition 3. Let U(Y) be the graph having vertex set S n and in which two di erent vertices i 1 ;... ; i n ; h 1 ;... ; h n are adjacent i (a) i` ˆ h`; ` 6ˆ k; k 1 and (b) fi k ; i k 1 g 62 e[y]. Write p Y p 0 i p and p 0 occur in a U(Y)-path and set pš ˆ fp 0 jp 0 Y pg. Then for p; p 0 2 rš Y we have ; pš ˆ ; p 0 Š. That is, U(Y)-components do independently of the maps F i;y represent equivalence classes of SDS. As shown in Ref. [2] the combinatorial analysis allows us to interprete an equivalence class pš Y as an acyclic orientation of Y. That is, there is a bijection f Y ; : S n = Y Š! Acyc Y ; 11 where Acyc(Y) is the set of all acyclic orientations 1 of Y. We set a Y ˆ jacyc Y j. The bijection given in Eq. (11) shows that each U Y - component corresponds uniquely to an acyclic orientation of Y, and consequently we obtain an upper bound on the number of dynamical systems of the form ; pš Acyclic orientations In Ref. [4] Linial shows that the computation of a Y is a hard problem. To prove this we combine the following two results: The rst one is due to Stanley [5] and provides an interpretation of a Y in terms of the chromatic polynomial as follows jacyc Y j ˆ 1 jv Y Šj v Y 1 ; 12 where v Y X is the chromatic polynomial of Y. The second result is the following property of the chromatic polynomial: Suppose Y ; Y 0 are undirected graphs and let Y Y 0 be the graph with vertex set v Y Š [ v Y 0 Š and edge set e Y Š [ e Y 0 Š [ ffv; v 0 gjv 2 v Y Š ^ v 0 2 v Y 0 Šg. Then we have [6] v Y Kn X ˆ Yk 1 X j v Y X k : jˆ0 13 Eqs. (13) and (12) imply that being able to compute 1 jv Y Šj v Y 1 for any graph allows one to determine the chromatic polynomial of any graph Y [4]. Hence the computation of a(y) is at least NP-hard. Linial's hardness result motivates the construction of estimates for a(y), and various upper and lower bounds have been derived, see Ref. [7±9]. The random graph G n;p, i.e., the 1 The number of acyclic orientations is of independent interest in theoretical computer science, since it provides lower bounds on the computational complexity of various decision and sorting problems [3].

6 126 C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136 probability space of graphs with vertex set N n obtained by selecting each K n - edge with independent probability p, is used. In Ref. [2] to prove " 1 Y n 1 1 p Š# i p n n! 6 a G n;p a:s:; 14 h 1 n iˆ1 where h n tends to 1 arbitrarily slowly. Since the map h fk k : S n = Y Š! f ; pšjp 2 S n g; h fk k pš Y ˆ ; pš 15 is clearly surjective we have a fk k Y 6 a Y. Another way of stating this is that some components in the update graph U(Y) give the same SDS as a result of the speci c structure of the Y-local maps. For instance, for the MAJ-function one has in general a MAJ Y < a Y while for the NOR-function one always has a NOR Y ˆ a Y [10], see Lemma Structure of this paper SDS have so far only been studied from a purely combinatorial point of view [1,2], that is, all results are formulated, w.r.t. the underlying graph Y and are independent of local maps. In this paper we will extend the combinatorial analysis by taking into account the structure of the Y-local maps. In Section 3 we derive general structural results on SDS. All results that are not presented with full proofs can be found in Refs. [10,14,15]. In Proposition 2 we analyse xed points of SDS, and we show that if x is a xed point for an SDS ; pš then x is also a xed point for every other SDS of the form ; rš. In Proposition 3 we characterize bijective SDS. In particular it can be applied to determine all invertible sca (see Corollary 2). Further we will consider SDS over the random graph G n;p i.e., the probability space consisting of all K n -subgraphs where each edge is selected with independent probability p. We will show that the r.v. log 2 a fk k (see Eq. (4)) is for certain multisets f k k, sharply concentrated at its mean (see Corollary 3). 3. Fixed points, bijectivity and a concentration result In the following we will write x ˆ x 1 ;... ; x n. We begin by showing that an SDS over the graph Y is a direct product of SDS over the Y-components. Proposition 1. Let Y be a graph, ; pš an SDS, C a Y-component, n C ˆ jcj and F C ; p C Š ˆ Qi F C 1 <ic 2 <<ic n pc i C C j ;Y. Then we have ; pš ˆ Y F C ; p C Š; 16 C<Y where p C denotes the restriction of the bijective map p to the elements j 2 v CŠ.

7 Proof. For the SDS ; pš we immediately observe that 2 3 ; pš ˆ Y Y 4 5: 17 C<Y i C 1 <ic 2 <<ic nc F pc i C j ;Y In fact (Eq. (17)) is well de ned since for any two components C 1 ; C 2, we have h i F C1 ; p C1 Š; F C2 ; p C2 Š ˆ 0; where ; Š denotes the commutator of two maps. Proposition 2. Let Y be a graph and ; pš an SDS Fix ; pš the set fxj ; pš x ˆ xg. Then we have over Y. Denote by 8r 2 S n : Fix ; rš ˆ Fix ; pš : 18 Proof. Suppose x is a xed point of ; pš and let us compute ; rš x. We immediately observe (using induction on ` 6 n) 8` 6 n: Ỳ F r i ;Y x ˆ x; iˆ1 whence the proposition. We will now give a characterization of bijective SDS. Proposition 3 [2]. Let Y < K n, let f k be a multiset f k : F k 2! F 2 and let id, inv : F 2! F 2 be the maps de ned by id(x) ˆ x and inv(x) ˆ x. Then an SDS ; pš is bijective if and only if for each 1 6 i 6 n and xed coordinates x 1 ;... ; x i 1 ; x i 1 ;... ; x n the map f di;y 1 proj Y iš x 1 ;... ; x i 1 ; x i 1 ;... ; x n : F 2! F 2 has the property f di;y 1 proj Y iš x 1 ;... ; x i 1 ; x i 1 ;... ; x n 2 fid; invg: Furthermore, let p ˆ i 1 ;... ; i n 1 ; i n 2 S n ; let p ˆ i n ; i n 1 ;... ; i 1 and let ; pš be a bijective SDS. Then we have ; pš 1 ˆ ; p Š: Remark 2. Obviously, the bijectivity of one particular SDS ; pš implies that DS ; rš is bijective. any SDS C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121± In particular we have the following Corollary. Corollary 1. Let PAR k 1 6 k 6 n be the multiset of maps de ned in Eq. (7). Then for arbitrary Y < K n all SDS DS induced by PAR k 1 6 k 6 n are invertible.

8 128 C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136 Proof. Obviously, if P j2d 1 i x j ˆ 0, then x i 7! x i and if P j2d 1 i x j ˆ 1, we derive x i 7! x i. The corollary now follows from Proposition 3. Proposition 3 immediately allows one to determine all bijective sca 2 [1]. Corollary 2. There are, independent of n, exactly 2 22 ˆ 16 di erent bijective sca. Proof. An sca is an SDS over the base graph Y ˆ Circ n, i.e., the cycle graph on n vertices. Obviously the corresponding multiset f k k consists of the single map f 3 : F 3 2! F 2 and Proposition 3 implies that either or f 3 x i 1 ; x i ; x i 1 ˆ x i f 3 x i 1 ; x i ; x i 1 ˆ x i where x i 1 and x i 1 are arbitray and i 1; i; i 1 2 Z=nZ, proving the Corollary. In contrast to this characterization, bijectivity of parallely updated CA (pca) does in fact depend on the number of cells. For example, CA-rule 150 is not bijective for n ˆ 6 and bijective for n ˆ 7 [11,12]. In applications it is often important to obtain knowledge on the structure of the periodic orbits and xed points of the system. A result on this is obtained as a consequence of Proposition 3. It states that an SDS induced by MAJ only has xed points. Proposition 4 [1]. Let Y < K n and let p 2 S n. The SDS [MAJ Y ;p ] has no periodic points of period p P 2. We next consider SDS over the random graph G n;p, i.e., the probability space consisting of all K n -subgraphs where each edge is selected with independent probability p. We will study a fk k as a random variable w.r.t. the probability space G n;p and prove a concentration result for log 2 a NOR G n;p. The existence of a concentration result for log 2 a fk k G n;p can be interpreted as follows: the number of di erent SDS depends only on the number of edges of Y and not on the particular choice of Y itself. Insofar it can be viewed as a generic property. To begin we will de ne a key property of real valued G n;p random variables. 2 Here we will assume closed boundary conditions and nearest neighbor rules.

9 C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121± De nition 4. Let g n;p : G n;p! R be a random variable (r.v.). Then g n;p is called Lipschitz if and only if for any two graphs Y ; Y 0 < K n that di er by the alteration of exactly one edge one has jg n;p Y g n;p Y 0 j 6 1: 19 In particular we will be interested in multisets f k k for which the r.v. log 2 a fk k is Lipschitz, i.e., jlog 2 a fk k Y log 2 a fk k Y 0 j 6 1: 20 Lemma 1. Let Y < K n be an arbitrary graph. Then the following assertions hold (i) log 2 a NOR : fy < K n g! N is Lipschitz (ii) log 2 a NAND : fy < K n g! N is Lipschitz (iii) log 2 a XOR : fy < K n g! N is Lipschitz (iv) log 2 a PAR : fy < K n g! N is not Lipschitz (v) log 2 a MAJ : fy < K n g! N is not Lipschitz Proof. A detailed Proof of (i)±(iii) can be found in Ref. [10] and can be sketched as follows: rst one proves that h fk k : S n = Y Š! f ; pš j 2 S n g; pš Y 7! ; pš 21 is injective for NOR k k. Second one considers the bijection in Eq. (11) and uses the fact that log 2 a Y is Lipschitz. To prove (iv) we consider the graphs in Fig. 1. Let Y be the graph displayed to the left and Y 0 the graph obtained from Y by adding the edge f1; 5g. Then one has log 2 a PAR Y 0 log 2 a PAR Y < 1:2. Similarly one obtains with Y equal the graph displayed to the right log 2 a PAR Y 0 log 2 a PAR Y > 1:6, where Y 0 is obtained from Y by adding the edge f1; 2g. Finally, to prove (v) we take Y to be the graph with e Y Š ˆ e K n Šnfy 0 g and Y 0 ˆ K n. There is exactly one SDS over K n induced by MAJ and in case of K 6 the number of di erent SDS drops from 78 to 1 when one passes from Y to Y 0. Fig. 1. Graphs demonstrating the nonbijectivity of h PAR.

10 130 C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136 Remark 3. The above Proposition implies that h PAR is not bijective. A simple example demonstrating this is provided by the graph Y ˆ Circ 4. For instance, the two permutations p 1 ˆ 2134 and p 2 ˆ 4132 satisfy p 1 Y p 2, that is, they are contained in di erent components of U Circ 4. However, because of the structure of the PAR-function these two components give the same SDS. The number of di erent SDS is 11 whereas the number of acyclic orientations is 14. Theorem 1. Let g n;p Lipschitz. Then for G n;p and arbitrary probability p one has p l n;p fjg n;p G n;p E g n;p G n;p Šj > k n n 1 =2g < 2e k2 =2 : 22 In particular, if for some Boolean function B the map h B (see Eq. (21)) is bijective, we have n log 2 n log 2 e log 2 p o 1 Š 6 E log 2 a B G n;p Š: 23 The rst assertion of Theorem 1 is a consequence of a general result of Milman and Schechtman [14]. It is proved by (a) constructing a nite martingale X i i that converges to the r.v. g n;p G n;p ; (b) showing that g n;p being Lipschitz implies jx i 1 X i j 6 1 and (c) by applying Azuma's inequality. Theorem 2. Let X 0 ;... ; X m be a martingale with the property jx i 1 X i j 6 1; 0 6 i 6 m. Then we have p 8k > 0: Prob fjx m X 0 j > k m g < 2e k 2 =2 : 24 The second assertion of Theorem 1 follows from Theorem 2 of [2]. In particular we have the following theorem. Corollary 3. For the random graph G n;p ; B 2 fnor; NAND; XORg and arbitrary probability p one has p l n;p fjlog 2 a B G n;p E log 2 a B G n;p Šj > k n n 1 =2g < 2e k2 =2 ; 25 and we have n log 2 n log 2 e log 2 p o 1 Š 6 E log 2 a B G n;p Š. 4. Analysis of some special systems In this section we will present some results on SDS induced by the Boolean functions NOR, PAR, MAJ, MIN as listed in Eqs. (5)±(9). As will be shown below the dynamics for the complete graph and the empty graph is well understood. To convey information on what one can expect for a graph Y < K n we make use of random graph theory. Denote by G n;p the probability space consisting of all Y < K n where edges are chosen indepen-

11 C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121± dently with probability p. Then we have l n;p Y ˆ p m q N m where m ˆ jv Y Šj; q ˆ 1 p and N ˆ n. For an SDS 2 ; pš we denote by m F Y ; p and c F Y ; p the number of di erent periodic orbits and the size of a largest periodic orbit respectively. In the following we will study the random variables Fix fk k Y ˆ jfix F Y j; N fk k Y ˆ maxfm F p2s Y ; p g; n C fk k Y ˆ maxfc F p2s Y ; p g; n a fk k Y ˆ jf ; pšjp 2 S n gj for the functions in Eqs. (5)±(9). Remark 4. Proposition 2 shows that Eq. (26) is well de ned. In general c F Y ; p and m F Y ; p depend on the particular choice of the ordering. By taking the maximum over all orderings in Eq. (27) and Eq. (28) one ensures that the corresponding random variables are well de ned. Obviously, l n;p converges for n! 1 to the uniform measure on graphs with p n edges. However, for small n the deviations between the uniform measure 2 and l n;p are signi cant. Accordingly, we will use an adapted version of the measure l n;p for the following computer experiments: for xed n 2 N and a given set of graphs, Exp ˆ fy 1 ;... ; Y m g; M 2 N, we obtain the multiset of probabilities P l Y 1 ˆ p 1 ;... ; l Y M ˆ p M. Now we take b E 2 R such that b E M iˆ1 p i ˆ 1 and de ne l E : Exp! R by l E ˆ b E l n;p. We will denote expectation value and variance w.r.t. the measure l E by E E Š and V E Š. Figs. 2±5 show expectations and variances for basic properties of SDS induced by the Boolean functions mentioned above. Fig. 2. The number of di erent SDS, the number of orbits and the size of a largest orbit for the NOR-function. From the left: E E a NOR Š, E E N NOR Š and E E C NOR Š with error bars showing the standard deviation with respect to the measure l E. Here n ˆ 7 with sample size 50.

12 132 C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136 Fig. 3. The number of xed points, the number of di erent SDS, the number of orbits and the size of a largest orbit for the PAR function. From the left: E E Fix PAR Š, E E a PAR Š, E E N PAR Š and E E C PAR Š with error bars showing the standard deviation with respect to the measure l E. Here n ˆ 7 with sample size 50. Fig. 4. The number of xed points, the number of di erent SDS, the number of orbits and the size of a largest orbit for the MIN-function. From the left: E E Fix MIN Š, E E a MIN Š, E E N MIN Š and E E C MIN Š with error bars showing the standard deviation with respect to the measure l E. Here n ˆ 7 with sample size 50. Fig. 5. The number of xed points, the number of di erent SDS, the number of orbits and the size of a largest orbit for the MAJ-function. From the left: E E Fix MAJ Š, E E a MAJ Š, E E N MAJ Š and E E C MAJ Š with error bars showing the standard deviation with respect to the measure l E. Here n ˆ 7 with sample size 50. De nition 5. Let ; pš be an SDS. The digraph C ; pš has vertex set F n 2 and its directed edges are all pairs of the form x; ; pš x. Clearly, C ; pš-cycles correspond to periodic orbits of the SDS ; pš. Remark 5. Obviously, C ; pš is a unicyclic digraph and typically for ; pš 6ˆ ; rš we have C ; pš À C ; rš. However, for Y ˆ K n and for any r; p 2

13 C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121± S n we have the isomorphism of digraphs C F Kn ; pš C F Kn ; rš. Accordingly, for an analysis of SDS over K n it su ces to restrict ourselves to study F Kn ; idš. Let F Kn ; idš be an SDS induced by the Boolean function f n : F n 2! F 2. Let further O be an orbit of F Kn ; idš and denote by res O f n the restriction of f n to O. Suppose (a) res O f n satis es / x 1 ;... ; x n 1 ; / x 1 ;... ; x n ˆ x n 30 and (b) that we have the commutative diagram 31 where proj x 1 ;... ; x n ; x n 1 ˆ x 1 ;... ; x n ; i f x 1 ;... ; x n ˆ x 1 ;... ; x n ; f x 1 ;... ; x n ; r n 1 x 1 ; x 2 ;... ; x n 1 ˆ x n 1 ; x 1 ;... ; x n : Then we have n 1 0 (mod joj). Clearly, the commutative diagram implies ; idš ` ˆ proj r n 1 i f ` and from the functional Eq. (30) we conclude proj i f ˆ id, whence ; idš ` ˆ proj r ` n 1 i f : In particular, for ` ˆ n 1 one has ; idš n 1 ˆ proj i f ˆ id. Lemma 2. Let F Kn ; idš be the SDS induced by the symmetric function f n. Let A k ˆ f x 1 ;... ; x n j jfx i ˆ 1gj ˆ kg and let O be an orbit of the system. Suppose that for x 2 O Y l iˆ1 F i;kn x 2 A k [ A k 1 ; 1 6 l 6 n; and that there is at least one l 1 such that Q l 1 iˆ1 F i;k n x 2 A k and at least one l 2 such that Q 1 2 iˆ1 F i;k n x 2 A A 1. Then n 1 0 (mod joj). Proof. First note that the conditions imply f n x 1 ;... ; x n ˆ 1 for x 2 O \ A k and f n x 1 ;... ; x n ˆ 0 for x 2 O \ A k 1. Now the lemma follows from the following two observations. First, for x 2 O one has f x 1 ;... ; x n 1 ; f x 1 ;... ; x n ˆ x n ; and secondly, 81 6 k 6 n 1 F Kn ; idš x k 1 ˆ x k From this we conclude that Eq. (31) commutes, and the lemma follows.

14 134 C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136 In the following we present some results on SDS induced by the functions NOR, PAR, MAJ and MIN. An SDS induced by MAJ and PAR over an empty graph only has xed points, or equivalently, the corresponding digraph (see De nition 5) has an empty edge set. For SDS induced by NOR and MIN all points are contained in a period 2 cycle. Accordingly there are 2 n xed points in the former case and 2 n 1 period 2 orbits in the latter case. Let now e k be the kth unit vector and hx; yi be the standard inner product of x and y. Proposition 5 (NOR). Let F Kn ; idš be the SDS induced by NOR. The states x for which hx; e n i ˆ 1 are mapped to zero. If hx; e n i 6ˆ 1 then x is mapped to e k where k ˆ 1 max i fx i ˆ 1g. The set L ˆ f0; e 1 ; e 2 ;... ; e n g is the unique limit cycle of F Kn ; idš. Moreover, for arbitrary dependency graph Y the SDS induced by NOR has no xed points. Proof. Clearly, all points are mapped into L. Also, 0 is mapped to e 1 ; e k is mapped to e k 1 for 1 6 k 6 n 1 and e n is mapped to 0. For the second part of the proposition it is clear that x ˆ (0) is the only candidate for a xed point. But x ˆ (0) is clearly not xed. Proposition 6 (PAR). Let F Kn ; idš be the system induced by PAR. Then all points are contained in a periodic orbit O and we have n 1 0 (mod joj). Proof. For arbitrary graphs, an SDS induced by PAR is bijective, whence all states are periodic. It is clear that any orbit which contains at least two points satis es the conditions in Lemma 2, for some odd k, and the last statement follows. Proposition 7 (MIN). Let F Kn ; idš be the SDS induced by MIN. For any periodic orbit O one has n 1 0 (mod joj). Proof. A periodic orbit for this system satis es the conditions in Lemma 2 for k ˆ bn=2c, whence the proposition. Proposition 8 (MAJ). Let F Kn ; idš be the SDS induced by MAJ. Every state is xed or eventually xed, that is C F Kn ; idš is cycle free. The xed points are 0; 0;... ; 0 and 1; 1;... ; 1. Proof. Obviously, 0; 0;... ; 0 and 1; 1;... ; 1 are xed points. By de nition, application of MAJ n yields 1 for a state x containing an equal number of 1's and 0's whence x is mapped to 1; 1;... ; 1 Clearly, any other point will be mapped to 0; 0;... ; 0 or 1; 1;... ; 1.

15 C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121± Proposition 9. Let F Km;n ; pš be the SDS over K m;n that is induced by the MAJ function. Then all states are xed or eventually xed. More precisely there are m n 2 xed points. bm=2c bn=2c Proof. Note that MAJ returns 1 when applied to an x containing an equal number of 0's and 1's. Let the vertex classes of K m;n be V m and V n. Call a state x balanced if the states contained in V m has exactly dm=2e zeroes and the states contained in V n has exactly dn=2e zeroes. Clearly, all balanced states are xed and all other points eventually map to 0; 0;... ; 0 or 1; 1;... ; 1. Obviously a balanced state has no preimage apart from itself. Thus, the dynamics of this system is fully understood. Remark 6. Note that for the system K m;n with n ˆ 2 one has states with a minority of zeros that is mapped to 0; 0;... ; 0 for some orderings and to 1; 1;... 1 for other orderings. Acknowledgements We gratefully acknowledge the proofreading of W.Y.C. Chen and Q.H. Hou. 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. Comp. 98 (1999) 241±259. [2] C.M. Reidys, Acyclic orientations of random graphs, Adv. Appl. Math. 21 (1998) 181±192. [3] W. Goddard, C. Kenyon, V. King, L.J. Schulman, Optimal randomized algorithms for local sorting and set-maxima, SIAM J. Comp. 22 (1993) 272±283. [4] N. Linial, Hard enumeration problems in geometry and combinatorics, SIAM J. Alg. Disc. Meth. 7 (2) (1986) 331±335. [5] R. Stanley, Acyclic orientations of graphs, Discrete Math. 5 (1973) 171±178. [6] W.T. Tutte, Graph Theory, Addison-Wesley, Reading, MA, [7] N. Kahale, L.J. Schulman, Bounds on the chromatic polynomial and the number of acyclic orientations of a graph, Combinatorica 16 (1996) 383±397. [8] U. Manber, M. Tompa, The e ect of number of Hamiltonian paths on the complexity of a vertex-coloring problem, SIAM J. Comp. 13 (1984) 109±115. [9] R. Graham, F. Yao, A. Yao, Information bounds are weak in the shortest distance problem, J. ACM 27 (1980) 428±444. [10] H.S. Mortveit, C.M. Reidys, On a certain class of discrete dynamical systems, Discrete Math.

16 136 C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136 [11] A. Wuensche, M.J. Lesser, in: The Global Dynamics of Cellular Automata; An Atlas of Basin of Attraction Fields of One-Dimensional Cellular Automata. Santa Fe Institute Studies in the Sciences of Complexity, Addison-Wesley, Reading, MA, [12] C.L. Barrett, W.Y. Chen, C.M. Reidys, preprint. [13] D. Gorenstein, Finite Groups. Harper & Row, New York, [14] V.D. Milman, G. Schechtman, Asymptotic theory of nite dimensional normed spaces. Lecture Notes in Mathematics, Springer (1200) [15] C.L., Barrett, H.S. Mortveit, and C.M. Reidys, Elements of a theory of simulation III: Equivalence of SDS, in progress. [16] H.S. Mortveit, and C.M. Reidys, Discrete dynamical system. Discrete Mathematics. Submitted.