Applied Mathematics and Computation 107 (2000) 121±136
www.elsevier.nl/locate/amc
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; . . . ; ng where each vertex has a binary state, (b) functions Fi : Fn2 ! Fn2
and (c) an update ordering p. The functions Fi 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 Fi in the order given by p one obtains the sequential dynamical system (SD S ):
‰FY ; pŠ ˆ
n
Y
iˆ1
Fp …i† : Fn2 ! Fn2 :
We derive a decomposition result, characterize invertible SD S and study ®xed points.
In particular we analyse how many di€erent SD S 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 SD S 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. E-mail: [email protected].
0096-3003/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.
PII: S 0 0 9 6 - 3 0 0 3 ( 9 8 ) 1 0 1 1 4 - 5
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C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136
[1] and introduce Sequential Dynamical Systems, (SD S ), a new class of dynamical systems implied by the formalization of simulation as composed local
maps.
Intuitively, SD S are simply those dynamical systems produced by sequentially ordered compositions of local maps. The dynamical properties of SD S
delimit the behavioral repertoire of simulations.
An SD S 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 Circn , a rule f : F32 ! F2 and a permutation p of the
Circn -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 (Circn , f, p). It may be viewed as a
simulation in which the entities correspond to the Circn -vertices, the support
structure is the graph Circn 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 SD S [1].
Note that the mathematical constituents of SD S 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
SD S 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 SD S is very interesting and seems to open
new areas of purely mathematical investigation.
C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136
123
2. Sequential dynamical systems
2.1. De®nition
We set Nn ˆ f1; 2; . . . ; ng. Let the set of Y-vertices adjacent to vertex i be
denoted by D1 …i† and set di ˆ jD1 …i†j. We denote the increasing sequence of
elements of the set B1 …i† ˆ D1 …i† [ fig by
B^1 …i† ˆ …j1 ; . . . ; i; . . . ; jdi †;
…1†
and set d ˆ max1 6 i 6 n di . Each vertex i has associated a binary state xi . Also, let
…fk †k with fk : Fk2 ! F2 where 1 6 k 6 d ‡ 1 be some given multiset of symmetric
functions. For each vertex i 2 Nn we de®ne the map
projY ‰iŠ : Fn2 ! F2di ‡1 ;
…x1 ; . . . ; xn † 7! …xj1 ; . . . ; xi ; . . . ; xjdi †:
Finally, let Sk with k 2 N denote the permutation group on k letters.
De®nition 1 (Y-local maps). Let …fk †1 6 k 6 d…Y †‡1 be a multiset of Sk -symmetric
functions fk : Fk2 ! F2 . For each i 2 Nn there is a Y -local map Fi;Y given by
yi ˆ fdi ‡1 projY ‰iŠ;
Fi;Y ……xj †j † ˆ …x1 ; . . . ; xiÿ1 ; yi …x†; xi‡1 ; . . . ; xn †:
Fi;Y is a map Fi;Y : Fn2 ! Fn2 that updates the state of vertex i as a function of the
states contained in B1 …i† and leaves all other vertex states ®xed. We refer to the
multiset …Fi;Y †i as FY .
In particular, let …fk †1 6 k 6 n be a ®xed multiset of Sk -symmetric functions as
de®ned above. Then 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 ! fFY g. Let p 2 Sn . The introduction of the
maps Fi;Y allows us to consider products of the form
‰FY ; pŠ ˆ
n
Y
Fp…i†;Y : Fn2 ! Fn2 :
…2†
iˆ1
De®nition 2 (SD S ). An SD S over a graph Y w.r.t. p is a product
‰FY ; pŠ ˆ
n
Y
Fp…i†;Y : Fn2 ! Fn2 :
…3†
iˆ1
In this paper we will be particularly interested in computing the number of
di€erent SD S , i.e.,
a…fk †k …Y † ˆ jf‰FY ; pŠjp 2 Sn gj
…4†
for a given multiset …fk †k and for a given graph Y. That is, how many di€erent
dynamical systems can be obtained by rescheduling.
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C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136
Sometimes the multiset …fk †k is induced by a single Boolean function
B : Fn2 ! F2 . In this case we will say that the corresponding SD S is induced by
B. The Boolean functions listed below have this property and will be studied
later in some detail:
NORk : Fk2 ! F2
NANDk :
Fk2
! F2
PARk : Fk2 ! F2
…x1 ; . . . ; xk † 7! x1 _ _ xk
…5†
…x1 ; . . . ; xk † 7! x1 ^ ^ xk
k
X
…x1 ; . . . ; xk † 7!
xi
…6†
MAJk :
Fk2
! F2
…x1 ; . . . ; xk † 7!
MINk : Fk2 ! F2
…x1 ; . . . ; xk † 7!
XORk : Fk2 ! F2
…7†
iˆ1
…x1 ; . . . ; xk † 7!
1
0
iff jfxj jxj ˆ 1gj P jfxj jxj ˆ 0gj
else
1
iff
…8†
jfxj jxj ˆ 1gj < jfxj jxj ˆ 0gj
0
else
…9†
1
iff
0
else
jfxi ˆ 1gj ˆ 1
…10†
Although a slight abuse of terminology, we will simply write, e.g., aPAR , for
these functions instead of using the full multiset …fk †k as index.
Remark 1. Note that MAJ and MIN are complementary functions, i.e.,
MAJk …x† ˆ MINk …x†. Nevertheless, and as will be shown later, the corresponding two SD S for a given graph usually behave very di€erently.
2.2. 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 …i1 ; . . . ; in † and when nothing
else is stated the natural ordering …1; . . . ; n† is assumed. Now SD S can be analysed from a purely combinatorial perspective [1]. This approach is based on
the simple observation that if p ˆ …i1 ; . . . ; in † and p0 ˆ …i01 ; . . . ; i0n † are two permutations di€ering by a transposition of consecutive coordinates …ik ; ik‡1 †
where fik ; ik‡1 g 62 e[Y], then independently of the choice of the maps Fi;Y we
have ‰FY ; pŠ ˆ ‰FY ; p0 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):
C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136
125
De®nition 3. Let U(Y) be the graph having vertex set Sn and in which two
di€erent vertices …i1 ; . . . ; in †; …h1 ; . . . ; hn † are adjacent i€ (a) i` ˆ h` ; ` 6ˆ k; k ‡ 1
and (b) fik ; ik‡1 g 62 e[Y].
Write p Y p0 i€ p and p0 occur in a U(Y)-path and set ‰pŠ ˆ fp0 jp0 Y pg.
Then for p; p0 2 ‰rŠY we have ‰FY ; pŠ ˆ ‰FY ; p0 Š. That is, U(Y)-components do
independently of the maps Fi;Y represent equivalence classes of SD S . As shown
in Ref. [2] the combinatorial analysis allows us to interprete an equivalence
class ‰pŠY as an acyclic orientation of Y. That is, there is a bijection
f …Y ; † : ‰Sn =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 ‰FY ; pŠ.
2.3. 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 …ÿ1†jv‰Y Šj vY …ÿ1†;
…12†
where vY …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; v0 gjv 2 v‰Y Š ^ v0 2 v‰Y 0 Šg. Then we have [6]
vY Kn …X † ˆ
k ÿ1
Y
…X ÿ j†vY …X ÿ k†:
…13†
jˆ0
Eqs. (13) and (12) imply that being able to compute …ÿ1†jv‰Y Šj vY …ÿ1† 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 Gn;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].
126
C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136
probability space of graphs with vertex set Nn obtained by selecting each Kn edge with independent probability p, is used. In Ref. [2] to prove
"
#ÿ1
n
Y
i
ÿ1
hn
‰1 ÿ …1 ÿ p† Š pn n! 6 a…Gn;p † a:s:;
…14†
iˆ1
where hn tends to 1 arbitrarily slowly. Since the map
h…fk †k : ‰Sn =Y Š ! f‰FY ; pŠjp 2 Sn g;
h…fk †k …‰pŠY † ˆ ‰FY ; 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 SD S as a result of the
speci®c structure of the Y-local maps. For instance, for the MAJ-function one
has in general aMAJ …Y † < a…Y † while for the NOR-function one always has
aNOR …Y † ˆ a…Y † [10], see Lemma 1.
2.4. Structure of this paper
SD S 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 SD S . 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 SD S , and we show that if x is a ®xed point for an SD S ‰FY ; pŠ then
x is also a ®xed point for every other SD S of the form ‰FY ; rŠ. In Proposition 3
we characterize bijective SD S . In particular it can be applied to determine all
invertible sCA (see Corollary 2). Further we will consider SD S over the random
graph Gn;p i.e., the probability space consisting of all Kn -subgraphs where each
edge is selected with independent probability p. We will show that the r.v.
log2 a…fk †k (see Eq. (4)) is for certain multisets …fk †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 …x1 ; . . . ; xn †. We begin by showing that an
SD S over the graph Y is a direct product of SD S over the Y-components.
PropositionQ1. Let Y be a graph, ‰FY ; pŠ an SD S , C a Y-component, nC ˆ jCj and
‰FC ; pC Š ˆ iC <iC <<iCn FpC …iCj †;Y . Then we have
1
2
C
Y
‰FC ; pC Š;
…16†
‰FY ; pŠ ˆ
C<Y
where pC denotes the restriction of the bijective map p to the elements j 2 v‰CŠ.
C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136
Proof. For the SD S ‰FY ; pŠ we immediately observe that
2
3
Y
Y
4
FpC …iCj †;Y 5:
‰FY ; pŠ ˆ
C<Y
127
…17†
iC
<iC
<<iC
nc
1
2
In fact (Eq. (17)) is well de®ned since for any two components C1 ; C2 , we have
h
i
‰FC1 ; pC1 Š; ‰FC2 ; pC2 Š ˆ 0;
where ‰ ; Š denotes the commutator of two maps.
Proposition 2. Let Y be a graph and ‰FY ; pŠ an SD S over Y. Denote by
Fix…‰FY ; pŠ† the set fxj‰FY ; pŠ…x† ˆ xg. Then we have
8r 2 Sn :
Fix…‰FY ; rŠ† ˆ Fix…‰FY ; pŠ†:
…18†
Proof. Suppose x is a ®xed point of ‰FY ; pŠ and let us compute ‰FY ; rŠ…x†. We
immediately observe (using induction on ` 6 n)
8` 6 n:
Ỳ
Fr…i†;Y …x† ˆ x;
iˆ1
whence the proposition.
We will now give a characterization of bijective SD S .
Proposition 3 [2]. Let Y < Kn , let …fk † be a multiset fk : Fk2 ! F2 and let id, inv :
F2 ! F2 be the maps de®ned by id(x) ˆ x and inv(x) ˆ x. Then an SD S ‰FY ; pŠ is
bijective if and only if for each 1 6 i 6 n and ®xed coordinates
x1 ; . . . ; xiÿ1 ; xi‡1 ; . . . ; xn the map
fdi;Y ‡1 projY ‰iŠ…x1 ; . . . ; xiÿ1 ; xi‡1 ; . . . ; xn † : F2 ! F2
has the property fdi;Y ‡1 projY ‰iŠ…x1 ; . . . ; xiÿ1 ; xi‡1 ; . . . ; xn † 2 fid; invg: Furthermore, let p ˆ …i1 ; . . . ; inÿ1 ; in † 2 Sn ; let p ˆ …in ; inÿ1 ; . . . ; i1 † and let ‰FY ; pŠ be a
bijective SD S . Then we have
‰FY ; pŠ
ÿ1
ˆ ‰FY ; p Š:
Remark 2. Obviously, the bijectivity of one particular SD S ‰FY ; pŠ implies that
any SD S ‰FY ; rŠ is bijective.
In particular we have the following Corollary.
Corollary 1. Let …PARk †1 6 k 6 n be the multiset of maps de®ned in Eq. (7). Then
for arbitrary Y < Kn all SD S induced by …PARk †1 6 k 6 n are invertible.
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C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136
P
P
Proof. Obviously, if j2D1 …i† xj ˆ 0, then xi 7! xi and if j2D1 …i† xj ˆ 1, we derive
xi 7! xi . The corollary now follows from Proposition 3. Proposition 3 immediately allows one to determine all bijective sCA
2
[1].
2
Corollary 2. There are, independent of n, exactly 2…2 † ˆ 16 di€erent bijective
sCA.
Proof. An sCA is an SD S over the base graph Y ˆ Circn , i.e., the cycle graph on
n vertices. Obviously the corresponding multiset …fk †k consists of the single map
f3 : F32 ! F2 and Proposition 3 implies that either
f3 …xiÿ1 ; xi ; xi‡1 † ˆ xi
or
f3 …xiÿ1 ; xi ; xi‡1 † ˆ xi
where xiÿ1 and xi‡1 are arbitray and i ÿ 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 SD S induced by MAJ only has
®xed points.
Proposition 4 [1]. Let Y < Kn and let p 2 Sn . The SD S [MAJY ;p ] has no periodic
points of period p P 2.
We next consider SD S over the random graph Gn;p , i.e., the probability space
consisting of all Kn -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 Gn;p and prove a concentration result for log2 aNOR …Gn;p †. The existence of
a concentration result for log2 a…fk †k …Gn;p † can be interpreted as follows: the
number of di€erent SD S 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 Gn;p random variables.
2
Here we will assume closed boundary conditions and nearest neighbor rules.
C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136
129
De®nition 4. Let gn;p : Gn;p ! R be a random variable (r.v.). Then gn;p is called
Lipschitz if and only if for any two graphs Y ; Y 0 < Kn that di€er by the
alteration of exactly one edge one has
jgn;p …Y † ÿ gn;p …Y 0 † j 6 1:
…19†
In particular we will be interested in multisets …fk †k for which the r.v. log2 a…fk †k
is Lipschitz, i.e.,
j log2 a…fk †k …Y † ÿ log2 a…fk †k …Y 0 †j 6 1:
…20†
Lemma 1. Let Y < Kn be an arbitrary graph. Then the following assertions hold
(i) log2 aNOR : fY < Kn g ! N is Lipschitz
(ii) log2 aNAND : fY < Kn g ! N is Lipschitz
(iii) log2 aXOR : fY < Kn g ! N is Lipschitz
(iv) log2 aPAR : fY < Kn g ! N is not Lipschitz
(v) log2 aMAJ : fY < Kn 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 : ‰Sn =Y Š ! f‰FY ; pŠ j 2 Sn g;
‰pŠY 7! ‰FY ; pŠ
…21†
is injective for …NORk †k . Second one considers the bijection in Eq. (11) and
uses the fact that log2 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 log2 aPAR …Y 0 † ÿ log2 aPAR …Y † < ÿ1:2. Similarly one obtains with Y
equal the graph displayed to the right log2 aPAR …Y 0 † ÿ log2 aPAR …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‰Kn Šnfy0 g and Y 0 ˆ Kn . There is exactly one SD S
over Kn induced by MAJ and in case of K6 the number of di€erent SD S drops
from 78 to 1 when one passes from Y to Y 0 . Fig. 1. Graphs demonstrating the nonbijectivity of hPAR .
130
C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136
Remark 3. The above Proposition implies that hPAR is not bijective. A simple
example demonstrating this is provided by the graph Y ˆ Circ4 . For instance,
the two permutations p1 ˆ …2134† and p2 ˆ …4132† satisfy p1 ¿ Y p2 , that is, they
are contained in di€erent components of U …Circ4 †. However, because of the
structure of the PAR-function these two components give the same SD S . The
number of di€erent SD S is 11 whereas the number of acyclic orientations is 14.
Theorem 1. Let gn;p Lipschitz. Then for Gn;p and arbitrary probability p one has
p
2
ln;p …fjgn;p …Gn;p † ÿ E‰gn;p …Gn;p †Šj > k n…n ÿ 1†=2g† < 2eÿk =2 :
…22†
In particular, if for some Boolean function B the map hB (see Eq. (21)) is bijective, we have
n‰ log2 …n† ÿ log2 e ÿ log2 p ÿ o…1†Š 6 E‰ log2 aB …Gn;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 …Xi †i that converges to the r.v. gn;p …Gn;p †; (b) showing that gn;p being
Lipschitz implies jXi‡1 ÿ Xi j 6 1 and (c) by applying Azuma's inequality.
Theorem 2. Let X0 ; . . . ; Xm be a
jXi‡1 ÿ Xi j 6 1; 0 6 i 6 m. Then we have
8k > 0:
martingale
with
the
p
2
Prob…fjXm ÿ X0 j > k mg† < 2eÿk =2 :
property
…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 Gn;p ; B 2 fNOR; NAND; XORg and
arbitrary probability p one has
p
2
ln;p …fj log2 aB …Gn;p † ÿ E‰ log2 aB …Gn;p †Šj > k n…n ÿ 1†=2g† < 2eÿk =2 ; …25†
and we have n‰ log2 …n† ÿ log2 e ÿ log2 p ÿ o…1†Š 6 E‰ log2 aB …Gn;p †Š.
4. Analysis of some special systems
In this section we will present some results on SD S 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 < Kn we make use of random graph theory. Denote by Gn;p the
probability space consisting of all Y < Kn where edges are chosen indepen-
C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136
131
dently with probability p. Then we have ln;p …Y † ˆ pm qN ÿm where m ˆ jv‰Y Šj; q ˆ
1 ÿ p and N ˆ …n2†. For an SD S ‰FY ; pŠ we denote by m…FY ; p† and c…FY ; 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…FY †j;
…26†
N…fk †k …Y † ˆ maxfm…FY ; p†g;
…27†
C…fk †k …Y † ˆ maxfc…FY ; p†g;
…28†
a…fk †k …Y † ˆ jf‰FY ; pŠjp 2 Sn gj
…29†
p2Sn
p2Sn
for the functions in Eqs. (5)±(9).
Remark 4. Proposition 2 shows that Eq. (26) is well de®ned. In general c…FY ; p†
and m…FY ; 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, ln;p converges for n ! 1 to the uniform measure on graphs with
p…n2† edges. However, for small n the deviations between the uniform measure
and ln;p are signi®cant. Accordingly, we will use an adapted version of the
measure ln;p for the following computer experiments: for ®xed n 2 N and a
given set of graphs, Exp ˆ fY1 ; . . . ; Ym g; M 2 N, we obtain the multiset of
probabilities
l…Y1 † ˆ p1 ; . . . ; l…YM † ˆ pM . Now we take bE 2 R such that bE PM
iˆ1 pi ˆ 1 and de®ne lE : Exp ! R by lE ˆ bE ln;p . We will denote expectation value and variance w.r.t. the measure lE by EE ‰ Š and VE ‰ Š. Figs. 2±5 show
expectations and variances for basic properties of SD S induced by the Boolean
functions mentioned above.
Fig. 2. The number of di€erent SD S , the number of orbits and the size of a largest orbit for the
NOR-function. From the left: EE ‰aNOR Š, EE ‰NNOR Š and EE ‰CNOR Š with error bars showing the
standard deviation with respect to the measure lE . Here n ˆ 7 with sample size 50.
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C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136
Fig. 3. The number of ®xed points, the number of di€erent SD S , the number of orbits and the size
of a largest orbit for the PAR function. From the left: EE ‰FixPAR Š, EE ‰aPAR Š, EE ‰NPAR Š and EE ‰CPAR Š
with error bars showing the standard deviation with respect to the measure lE . Here n ˆ 7 with
sample size 50.
Fig. 4. The number of ®xed points, the number of di€erent SD S , the number of orbits and the size
of a largest orbit for the MIN-function. From the left: EE ‰FixMIN Š, EE ‰aMIN Š, EE ‰NMIN Š and EE ‰CMIN Š
with error bars showing the standard deviation with respect to the measure lE . 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: EE ‰FixMAJ Š, EE ‰aMAJ Š, EE ‰NMAJ Š and EE ‰CMAJ Š
with error bars showing the standard deviation with respect to the measure lE . Here n ˆ 7 with
sample size 50.
De®nition 5. Let ‰FY ; pŠ be an SD S . The digraph C‰FY ; pŠ has vertex set Fn2 and
its directed edges are all pairs of the form …x; ‰FY ; pŠ…x††.
Clearly, C‰FY ; pŠ-cycles correspond to periodic orbits of the SD S ‰FY ; pŠ.
Remark 5. Obviously, C‰FY ; pŠ is a unicyclic digraph and typically for ‰FY ; pŠ
6ˆ ‰FY ; rŠ we have C‰FY ; pŠ À C‰FY ; rŠ. However, for Y ˆ Kn and for any r; p 2
C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136
133
Sn we have the isomorphism of digraphs C‰FKn ; pŠ  C‰FKn ; rŠ. Accordingly, for
an analysis of SD S over Kn it suces to restrict ourselves to study ‰FKn ; idŠ.
Let ‰FKn ; idŠ be an SD S induced by the Boolean function fn : Fn2 ! F2 . Let
further O be an orbit of ‰FKn ; idŠ and denote by resO fn the restriction of fn to O.
Suppose (a) resO fn satis®es
/…x1 ; . . . ; xnÿ1 ; /…x1 ; . . . ; xn †† ˆ xn
…30†
and (b) that we have the commutative diagram
…31†
where
proj…x1 ; . . . ; xn ; xn‡1 † ˆ …x1 ; . . . ; xn †;
if …x1 ; . . . ; xn † ˆ …x1 ; . . . ; xn ; f …x1 ; . . . ; xn ††;
rn‡1 …x1 ; x2 ; . . . ; xn‡1 † ˆ …xn‡1 ; x1 ; . . . ; xn †:
Then we have n ‡ 1 0 (mod jOj).
Clearly, the commutative diagram implies ‰FY ; idŠ…`† ˆ …proj rn‡1 if †…`†
and from the functional Eq. (30) we conclude proj if ˆ id, whence
…`†
‰FY ; idŠ
…`†
ˆ proj rn‡1 if :
…n‡1†
In particular, for ` ˆ n ‡ 1 one has ‰FY ; idŠ
ˆ proj if ˆ id.
Lemma 2. Let ‰FKn ; idŠ be the SD S induced by the symmetric function fn . Let
Ak ˆ f…x1 ; . . . ; xn †j jfxi ˆ 1gj ˆ kg and let O be an orbit of the system. Suppose
that for x 2 O
l
Y
Fi;Kn …x† 2 Ak [ Ak‡1 ;
1 6 l 6 n;
iˆ1
Q l1
and that there
Q12 is at least one l1 such that iˆ1 Fi;Kn …x† 2 Ak and at least one l2
such that iˆ1 Fi;Kn …x† 2 AA‡1 . Then n ‡ 1 0 (mod jOj).
Proof. First note that the conditions imply fn …x1 ; . . . ; xn † ˆ 1 for x 2 O \ Ak and
fn …x1 ; . . . ; xn † ˆ 0 for x 2 O \ Ak‡1 . Now the lemma follows from the following
two observations. First, for x 2 O one has
f …x1 ; . . . ; xnÿ1 ; f …x1 ; . . . ; xn †† ˆ xn ;
and secondly,
81 6 k 6 n ÿ 1
…‰FKn ; idŠ…x††k‡1 ˆ …x†k
From this we conclude that Eq. (31) commutes, and the lemma follows.
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C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136
In the following we present some results on SD S induced by the functions
NOR, PAR, MAJ and MIN. An SD S 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 SD S induced by NOR
and MIN all points are contained in a period 2 cycle. Accordingly there are
2n ®xed points in the former case and 2nÿ1 period 2 orbits in the latter case.
Let now ek be the kth unit vector and hx; yi be the standard inner product of
x and y.
Proposition 5 (NOR). Let ‰FKn ; idŠ be the SD S induced by NOR. The states x for
which hx; en i ˆ 1 are mapped to zero. If hx; en i 6ˆ 1 then x is mapped to ek where
k ˆ 1 ‡ maxi fxi ˆ 1g. The set L ˆ f0; e1 ; e2 ; . . . ; en g is the unique limit cycle of
‰FKn ; idŠ. Moreover, for arbitrary dependency graph Y the SD S induced by NOR
has no ®xed points.
Proof. Clearly, all points are mapped into L. Also, 0 is mapped to e1 ; ek is
mapped to ek‡1 for 1 6 k 6 n ÿ 1 and en 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 ‰FKn ; 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 SD S 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 ‰FKn ; idŠ be the SD S 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 ‰FKn ; idŠ be the SD S induced by MAJ. Every state is
®xed or eventually ®xed, that is C‰FKn ; 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 MAJn 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†. C.L. Barrett et al. / Appl. Math. Comput. 107 (2000) 121±136
135
Proposition 9. Let ‰FKm;n ; pŠ be the SD S over Km;n that is induced by the MAJ
function. Then all states are ®xed or eventually ®xed. More precisely there are
m
n
†…bn=2c
† ‡ 2 ®xed points.
…bm=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 Km;n be Vm and Vn . Call a state x
balanced if the states contained in Vm has exactly dm=2e zeroes and the states
contained in Vn 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 Km;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.
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