In this paper we propose a probabilistic analysis of the fully asynchronous behavior (i.e., two cells are never simultaneously updated, as in a continuous time process) of elementary finite cellular automata (i.e., {0, 1} states, radius 1 and unidimensional) for which both states are quiescent (i.e., (0, 0, 0) ? 0 and (1, 1, 1) ? 1). It has been experimentally shown in previous works that introducing asynchronism in the global function of a cellular automaton may perturb its behavior, but as far as we know, only few theoretical work exist on the subject. The cellular automata we consider live on a ring of size n and asynchronism is introduced as follows: at each time step one cell is selected uniformly at random and the transition rule is applied to this cell while the others remain unchanged. Among the sixty-four cellular automata belonging to the class we consider, we show that fifty-five other converge almost surely to a random fixed point while nine of them diverge on all non-trivial configurations. We show that the convergence time of these fifty-five automata can only take the following values: either 0, ?(n ln n), ?(n2), ?(n3), or ?(n2n). Furthermore, the global behavior of each of these cellular automata can be guessed by simply reading its code.

The aim of this article is to analyze theoretically the asynchronous behavior of
unbounded finite cellular automata. During the last two decades, several
empirical studies [

Cellular automata are widely used to model systems involving a huge number
of interacting elements such as agents in economy, particles in physics, proteins
in biology, etc. In most of these applications, in particular in many real system
models, agents are not synchronous. Interestingly enough, in spite of this lack
of synchronism, real living systems are very resilient over time. One might then
expect the cellular automata used to model these systems to be robust to
asynchronism and other kind of failure as well (such as misreading the state of the
neighbors). Surprisingly enough, it turns out that the resilience to asynchronism
widely varies from one automata to another (e.g., [

As far as we know, the question of the importance of perfect synchrony on
the behavior of a cellular automaton is not yet understood theoretically. To our
knowledge, only G´acs shows in [

In this paper, we quantify the convergence time and describe the space-time
diagrams for a class of cellular automata under fully asynchronous updating,
where two cells are not updated simultaneously. This asynchronous regime, also
known as step-driven asynchronous dynamics [

Definitions and our main result are given in Section 2. In section 3, we present basic but useful properties of the automata we consider. Section 4 is a technical section that develops probabilistic tools used to analyze the automata. Section 5 finally analyzes in details the asynchronous behavior of each automaton. 2

In this paper, we consider two-state cellular automata on finite size configurations.

Definition 1. An Elementary Cellular Automata (ECA) is given by its transition function ? : {0, 1}3 ? {0, 1}. We denote by Q = {0, 1} the set of states. A state q is quiescent if ?(q, q, q) = q. An ECA is double-quiescent (DQECA) if both states 0 and 1 are quiescent.

We denote by U = Z/nZ the set of cells. A finite configuration with periodic boundary conditions x ? QU is a word indexed by U with letters in Q. For a given pattern w ? QU , we denote by |x|w = #{i ? U : xi+1 . . . xi+|w| = w} the number of occurrences of w in configuration x.

We consider two kinds of dynamics for ECAs: the synchronous dynamics and the fully asynchronous dynamics. The synchronous dynamics is the classic dynamics of cellular automata, where the transition function is applied at each (discrete) time step on each cell simultaneously.

Definition 2 (Synchronous Dynamics). The synchronous dynamics S? : QU ? QU of an ECA ?, associates to each configuration x the configuration y, such that for all i in U , yi = ?(xi?1, xi, xi+1).

The asynchronous regime studied here can be seen as the most extreme asynchronous regime as two cells are never updated simultaneously. Definition 3 (Fully Asynchronous Dynamics). The fully asynchronous dynamics AS? of an ECA ? associates to each configuration x a random configuration y, such that yj = xj for j = i, and yi = ?(xi?1, xi, xi+1), where i is uniformly chosen at random in U . AS? could equivalently be seen as a function with two arguments, the configuration x and the random index i ? U . For a given ECA ?, we denote by xt the random variable for the configuration obtained by t applications of the asynchronous dynamics function AS? on configuration x, i.e., xt = (AS?)t(x).

Definition 4 (Fixed point). We say that a configuration x is a fixed point for ? under fully asynchronous dynamics if AS?(x) = x whatever the choice of i (the cell to be updated) is. F? denotes the set of fixed points for ?.

The set of fixed points of the asynchronous dynamics is clearly identical to {x : S?(x) = x} the set of fixed points of the synchronous dynamics. Note that every DQECA admits two trivial fixed points, 0n and 1n.

Definition 5 (Worst Expected Convergence Time). Given an ECA ? and a configuration x, we denote by T?(x) the random variable for the time to reach a fixed point from configuration x under fully asynchronous dynamics, i.e., T?(x) = min{t : xt ? F?}. The worst expected convergence time T? of ECA ? is : T? = max E[T?(x)].

x?QU We can now state our main theorem.

Theorem 1 (Main result). Under fully asynchronous dynamics, among the sixty-four DQECAs, ? fifty-five converge almost surely to a random fixed point on any initial configuration, and the worst expected convergence times of these fifty-five convergent DQECAs are 0, ?(n ln n), ?(n2), ?(n3), and ?(n2n); ? the nine others diverge almost surely on any initial configuration that is neither 0n, nor 1n nor, when n is even, (01)n/2.

Furthermore, the behaviors of the different DQECAs are similar within each class, and are obtained by simply reading its code as illustrated in Tab. 1. Figure 1 gives examples of the asynchronous space-time diagrams of a representative of each class (but Identity). It is interesting to notice that except for the first diagram (Fig. 1(a)), the asynchronous space-time diagrams (the larger ones) considerably differ from the corresponding synchronous ones (the smaller ones). 3

The transition function ? of an ECA is given by the set of its eight transitions
?(000), ?(001), . . . , ?(111), traditionally written ?(000000) , . . . , ?(111)
111 . The
following code describes each ECA by its differences to the Identity automaton. We
use this notation rather than the classic Wolfram?s one [

A B C D E F G H 000 001 100 101 010 011 110 111 1 1 1 1 0 0 0 0 We label each ECA by the set of its active transitions.

Note that with these notations, the DQECAs are exactly the ECAs having a label containing neither A nor H. By 0/1 and horizontal symmetries of configurations, we shall w.l.o.g. only consider the 24 DQECAs listed in Tab. 1 among the 64 DQECAs. For each of these 24 DQECAs, the number of the equivalent automata under symmetries is written within parentheses after their classic ECA code in the table.

From now on, we only consider the fully asynchronous dynamics (with uniform choice); this will be implicit in all the following propositions. Our results rely on the study of the evolution of the ?regions? in the space-time diagram (i.e., of the intervals of consecutive 0s or 1s in configuration xt). The key observation is that for DQECAs, under fully asynchronous dynamics, the number of regions is non-increasing since no new region can be created; furthermore, only regions of length one can disappear (see Fig. 1). We denote by Z(x) = |x|01 (= |x|10) the number of alternations from 0 to 1 in configuration x, which will be our counter for the number of regions.

Fact 2. For any DQECA, Z(xt) is a non-increasing function of time. Furthermore, Z(xt+1) < Z(xt) if and only if xt+1 is obtained from xt by applying a transition D or E at time t, and then Z(xt+1) = Z(xt) ? 1.

On the one hand, transitions D and E are thus responsible for decreasing the number of regions in the space-time diagram: D ?erases? the 1-regions and E the 0-regions. On the other hand, transitions B and F act on patterns 01. Intuitively, transition B moves a pattern 01 to the left, and transition F moves it to the right. In particular, patterns 01 perform a kind of random walk for DQECA with both transitions B and F. Similarly, transitions C and G act on patterns 10. Transition C moves a pattern 10 to the right, and transition G moves it to the left. The arrows in Tab. 1 represent the different behavior of the patterns: ? or ?, for left or right moves of the patterns 01 or 10; , for random walks of these patterns.

The following lemma characterizes the fixed points of a given DQECA according to its code.

? if ? contains transition B or C, then all 0s in x are isolated; ? if ? contains transition F or G, then all 1s in x are isolated; ? if ? contains transition D, then none of the 0s in x is isolated; ? if ? contains transition E, then none of the 1s in x is isolated.

The next section is devoted to analyzing particular random walk-like processes that will be used as tools to obtain our bounds on the convergence time.

Notation 2. For a given random sequence (Xt)t?N, we denote by (X? t)t>0 the random sequence X? t = Xt ? Xt?1.

Quadratic DQECA Toolbox. Consider > 0, a non-negative integer m, and (Xt)t?N a sequence of random variables with values in {0, . . . , m} given with a suitable filtration (Ft)t?N. In probability theory, Ft represents intuitively the ?-algebra (the ?set?) of the events that happened up to time t and is the formal tool to condition relatively to the past (see [7, Chap. 7]). In the sequel, Ft will either be the values of the previous random variables X0, . . . , Xt, or in some cases, the set of past configurations x0, . . . , xt. The following lemma bounds the convergence time of a random variable that decreases by a constant on expectation.

Lemma 4. Assume that if Xt > 0, then E[X? t+1|Ft] ? . Let T = min{t : Xt 0} denote the random variable for the first time t where Xt 0. Then, if X0 = x0,

E[T ] m + x0 .

Cubic DQECA Toolbox. Let > 0 and (Xt)t?N a sequence of random variables with values in {0, . . . , m}, given with a suitable filtration (Ft)t?N. Definition 6. The following two types of process will be extensively used in the next section: ? We say that (Xt)t?N is of type I if for all t: ? E[Xt+1|Ft] = Xt (i.e., (Xt) is a martingale), and ? if 0 < Xt < m, then Pr{X? t+1 1} = Pr{X? t+1 ?1} . ? We say that (Xt)t?N is of type II if for all t: ? if Xt < m, then E[Xt+1] = Xt (i.e., (Xt) behaves as a martingale when

Xt < m), and ? if 0 < Xt < m, then Pr{X? t+1 1} = Pr{X? t+1 ?1} , and ? if Xt = m, then Pr{Xt+1 m ? 1} (i.e., Xt ?bounces on m?).

Note that when (Xt) is of type I, if for some t, Xt ? {0, m}, then Xt = Xt for all t t, because (Xt) is a martingale bounded between 0 and m. Thus, {0, m} are the (only) fixed points of any type I sequence. When (Xt) is of type II, if for some t, Xt = 0, then Xt = Xt for all t t, because (Xt) is a martingale lower bounded by 0. Thus, 0 is the (only) fixed point of any type II sequence. Definition 7. The convergence time of a type I sequence (Xt) is defined as the random variable T = min{t : Xt ? {0, m}}. The convergence time of a type II sequence (Xt) is similarly defined as the random variable T = min{t : Xt = 0}.

The following lemmas bound the convergence time of these two types of random processes. Lemma 5. For sequence (Xt), if X0 = x0, the expectation of T satisfies: In this section, we evaluate the worst expected convergence time for each of the twenty-four representative automata in Tab. 1. Our results rely on studying the evolution of quantities computed on the random configurations (xt), whose convergence implies the convergence of the automaton. The upper bounds on the convergence time of these quantities are obtained by coupling them with one of the integer random processes analyzed in the previous section. The lower bounds are obtained by analyzing the exact expected convergence time for a particular initial configuration (most of the time, a configuration with a single 0region and a single 1-region). This involves building suitable variants measuring progress towards fixed points. One of the main difficulties is to handle correctly the mergings of the regions, i.e., the applications of transitions D and E.

We introduce the following convenient functions that simplify the evaluation of the quantities that are used to bound the convergence time. These function will spare us tedious parsings of the patterns in the configurations. For a given configuration x, we denote by a(x), . . . , h(x) the number of cells where transitions A, . . . , H are applicable, i.e.: a(x) = |x|000, b(x) = |x|001, c(x) = |x|100, d(x) = |x|101, e(x) = |x|010, f (x) = |x|011, g(x) = |x|110, h(x) = |x|111.

For instance, consider rule BCG. For convenience, we denote by p = 1/n the probability that a given cell is updated under fully asynchronous dynamics. Applying the transitions A, . . . , D increases the number of 1s by one and applying E, . . . , H decreases it by one. The expected variation of the number of 1s for configuration x in one step is then immediately p ˇ (b(x) + c(x) ? g(x)). When the context is clear, the argument x will be omitted. Clearly, parsing properly configuration x gives the following useful relationships.

Fact 6. For all configurations x ? QU , the following equalities hold: |x|01 = b + d = e + f = c + d = e + g = |x|10, |x|001 = b = c = |x|100, |x|011 = f = g = |x|110.

Let us now analyze the worst expected convergence time for DQECAs. Due
to space constraints, most of the proofs are omitted and can be found in [

?Coupon collector? DQECAs
The behavior of the DQECAs in this class (see Fig. 1(a)) is similar to the classic
Coupon Collector random process (e.g., [

Theorem 7. Under fully asynchronous dynamics, DQECAs E and DE converge
a.s. to a fixed point on any initial configuration. Their worst expected convergence
time is ?(n ln n). The fixed points for E and DE respectively are the configurations
without isolated 1 and the configurations without isolated 0 and 1.
Proof. These rules simply erase either isolated 0s, isolated 1s or both. They
never create any of them (by Fact 2), and reach a fixed point as soon as no more
0 or 1 are isolated (by Fact 3). These processes are then similar to a coupon
collector process that has to collect all the isolated 0s or 1s, by drawing at each
time step a random location uniformly in {1, . . . , n} (see e.g., [

Finally, configuration (010) n/3 0n mod 3, which is a proper coupon collector process, provides a lower bound of ? (n ln n) for both rules. 5.2

Lemma 8. Given an initial configuration x, for each DQECA B, BC, BDE, BCDE, BCDEG, BE, EF, BCE, EFG, BCEFG, and BEFG, there exists a sequence (Xt) of random variables with values in {0, . . . , n} (the variant), such that: (a) if Xt = 0, then xt is a fixed point. (b) for all t such that xt is not a fixed point, E[X? t+1|Xt] ?p. Proof. Rules B and BC. Set Xt = |xt|0 the number of 0s in xt. (a) is clear since Xt = 0 implies that xt = 1n. We obtain (b) by noticing that each application of transitions B or C decreases Xt by one, and that for any non fixed-point configuration, an active transition is performed with probability greater or equal to p. Similarly, Xt = |xt|1 is suitable for rules EF and EFG. Remaining Rules. We need to take into account the presence of isolated 0s and 1s. We set Xt = |xt|0 + Z(xt) for rules BDE, BCDE, BE, BCE, and BCDEG; and Xt = |xt|1 + Z(xt) for rule BEFG. Consider automaton BEFG. Clearly, Xt ? {0, . . . , n}, and we have (a) Xt = 0 implies that xt = 0n. For this rule,

E[X? t+1|xt] = p ˇ (b ? e ? f ? g)(xt) ? p ˇ e(xt), since only transition E acts on Z(xt). By Fact 6, one can rewrite

E[X? t+1|xt] = ?p ˇ (d + e + g)(xt). Second, if x is not a fixed point, then (b + e + f + g)(x) > 0. But by Fact 6, if d + e = 0, then b = f = g. Thus, b + e + f + g > 0 implies d + e + g > 0. We conclude that if xt is not a fixed point, we have (b). The proof is similar for all the remaining automata. We can now state the theorem.

Theorem 9. Under fully asynchronous dynamics, DQECAs B, BC, BDE, BCDE, BCDEG, BE, EF, BCE, EFG, BCEFG, and BEFG converge almost surely to a fixed point on any initial configuration. Their worst expected convergence time is ?(n2). Only the DQECAs B, BC, BE, and BCE have non-trivial fixed points, which are the configurations where all the 0s are isolated. Proof. The property on the fixed points is a direct application of Fact 3. Consider now one of the rules. Let Xt be the variant given by Lemma 8. Xt does not exactly verify the hypotheses of Lemma 4: Xt needs to be extended beyond a fixed point if it is reached before Xt = 0. We consider the random sequence Xt defined as follow: Xt = Xt if xt is not a fixed point, and Xt = 0 otherwise. Thus, Xt = 0 if and only if xt is a fixed point, and we can now apply Lemma 4 with m = n and = p and we obtain E[T ] X0/p = O(n2).

The lower bound ? (n2) on the convergence time is simply given by
considering the following initial configuration x = 0 n/2 1 n/2 . Note that Xt = |xt|1
works for all the rules on initial configuration x and its exact expected
convergence time is straightforward to compute by first step analysis (see [

Observe that we can divide this class into two subcategories: the automata that are monotonic, for which the variant is a non-increasing function of time, and the non-monotonic, for which the variant follows a biased random walk (see Tab. 1). Interestingly enough, this distinction is observed on the space-time diagrams. 5.3

? for each DQECA BDEF, BDEG, and BCDEFG, there exists an integer m 2n and a random integer sequence (Xt) of type I (see section 4) with values in

{0, . . . , m}, such that: for all t, if Xt = 0 or Xt = m, then xt is a fixed point. ? for each DQECA BEF, BEG, and BCEFG, there exists an integer m 2n and a random integer sequence (Xt) of type II (see section 4) with values in {0, . . . , m}, such that for all t, if Xt = 0, then xt is a fixed point. Theorem 11. Under fully asynchronous dynamics, DQECAs BDEF, BDEG, BCDEFG, BEF, BEG, and BCEFG converge almost surely to a fixed point on any initial configuration. Their worst expected convergence time is ?(n3). All of them admit only 0n and 1n as fixed point.

For DQECAs BDEF, BDEG, and BCDEFG, the fixed points 0n and 1n can be reached from any configuration (respectively distinct from 1n and 0n). For DQECAs BEF, BEG, and BCEFG, any configuration distinct from 1n converges almost surely to 0n.

Theorem 12. The fixed points of DQECA BCEF are 0n and 1n. From any nonfixed point initial configuration, DQECA BCEF converges almost surely to 0n and its expected convergence time is exactly ?(n2n).

Theorem 13. Under fully asynchronous dynamics, the DQECAs BF, BG, BCF, and BCFG diverge almost surely on any initial configuration that is not one of the three following fixed points 0n, 1n and, if n is even, (01)n/2. Furthermore, given an initial configuration, all reachable configurations are accessed an infinite number of times almost surely.