Cell-centric heuristics for the classification of cellular automata
Introduction
Cellular automata are discrete-time dynamical systems comprising finite-state units, called cells, whose states evolve in time as a result of the interactions with other cells. Since their introduction nearly five decades ago by von Neumann [1], cellular automata have acquired an ever more prominent status as a modeling tool in several research areas (cf., e.g., [2], [3] and references therein), and have even come to be regarded by some as a central abstraction in the modeling of nature’s fundamental processes [4].
For S = {0, … , s − 1} the set of possible states, and for t ⩾ 0 an integer, a cellular automaton with n cells evolves from time t to time t + 1 by synchronously updating all n states by the application of a deterministic mapping Ff from Sn to Sn. This mapping Ff is global in nature and depends on the local update rule f, which dictates how each individual state is to be updated given the cell’s state at time t as well as the states of those cells that lie within a neighborhood of size δ. The update rule f is then a mapping from S1+δ to S.
Normally a cell’s neighborhood in a cellular automaton is determined by an underlying multidimensional lattice according to several possible criteria. For example, a cell’s neighbors relative to a certain dimension of the lattice may be taken to be those cells that are r > 0 edges away along that dimension but no edges away along any other dimension, r being usually referred to as the radius of the update rule in that dimension. For unit radii in all dimensions, this characterizes what is known as the von Neumann neighborhood, but in this paper we employ the same denomination also for greater radii. Another example neighborhood comes from letting two cells be neighbors of each other whenever one can be reached from the other by treading no more edges along a certain dimension than the update rule’s radius along that dimension. For unit radii this is the Moore neighborhood, but once again we generalize and in this paper employ the same denomination under greater radii as well. When n is finite, it is customary to regard the lattice as having cylindrical boundaries, that is, as allowing every cell to have exactly two nearest neighbors along each dimension.
Finite cellular automata, those for which n is finite, are necessarily such that Ff eventually leads to a fixed point, or a limit cycle, of configurations in Sn, that is, either x such that Ff(x) = x or x0, … , xp−1, with p > 0, such that x0 = Ff(xp−1), x1 = Ff(x0), and so on [5]. The case of infinite cellular automata, on the other hand, is far more complicated and intriguing, since now n is formally infinite and no periodicity is guaranteed to emerge from the successive application of Ff.
Both in the finite and in the infinite cases, cellular automata have along the years been the subject of theoretical and experimental analyses. For a summary of key results, the reader is referred, for example, to [6], [7] and to their many references. One of the most appealing topics of investigation has been the classification of the update rule f, and consequently of the cellular automata based on it, regarding its “complexity”.
Interest in this question received its initial impetus from the study by Wolfram of infinite one-dimensional cellular automata [8], which resulted in the empirical finding that, nearly regardless of initial states, f consistently falls within one of four possible qualitative categories: (i) evolution leads to a homogeneous configuration, i.e., a configuration in which all cells have the same state; (ii) evolution leads to an inhomogeneous fixed point or to a limit cycle; (iii) evolution leads to a chaotic succession of configurations; or (iv) evolution leads to complex localized spatiotemporal structures that are “sometimes long-lived”. Although initially conceived for the one-dimensional case, there is in principle no reason why such a qualitative classification should not also be applicable to higher-dimensional cases. In fact, similar studies for the two-dimensional case have appeared as early as in [9].
Naturally, class (iv) update rules are intuitively associated with the realization of “complex” computations by the cellular automata that are built on them, that is, precisely those computations that underlie so much of the interest in cellular automata as modeling tools. Not surprisingly, then, considerable effort has been channeled into finding approaches to automatically categorize update rules into the classes (i)–(iv). Formally, all such efforts hover around the so-called limit set of an update rule f in the infinite case, which is the set of configurations that result from all possible initial configurations after the passage of arbitrarily long time. As it turns out, every nontrivial property of a limit set (i.e., a property that holds for at least one cellular automaton and does not hold for at least one other) can be proven undecidable through a reduction from the problem of whether a limit set is a singleton [10], itself known to be undecidable [11].
As a consequence of this inherent undecidability, every effective strategy for categorizing update rules must necessarily be of a heuristic nature or else eventually boil down to a heuristic if it is to have any practical use. Our interest in this paper is the study of heuristics that can be coupled with the parallel simulation of cellular automata in order to analyze the spatiotemporal patterns that emerge, aiming at categorizing the underlying update rule within Wolfram’s four classes. Efficiency in the form of minimal communications needs is then an essential requirement, leading to what we term cell-centric heuristics, that is, heuristics that depend as minimally as possible on the exchange of information among processors during the simulation of a cellular automaton.
We start in Section 2 with a review of some of the prominent heuristics that have been proposed for automatically classifying update rules, and proceed in Section 3 to a discussion of the so-called input-entropy measures. Our cell-centric heuristics are presented in Section 4, with results from computational experiments on one- and two-dimensional cellular automata given in Section 5. Further considerations on the computational results appear in Section 6 and concluding remarks come in Section 7.
Section snippets
Background
In broad terms, we distinguish two essential kernel classes of strategies for the categorization of update rules. The first class comprises those techniques that aim at extracting the update rules’ computational capabilities by solely considering the update rule itself, not simulations of cellular automata for examination of the resulting spatiotemporal patterns. Approaches of this type have concentrated on one-dimensional cellular automata, so a cell’s neighborhood size is in fact δ = 2r.
The
Input entropy
The concept of input entropy is due to Wuensche [21] and constitutes an attempt to merge together some of the key features of the two classes of strategies discussed in Section 2, those that seek to base update-rule classification on examining the update rule solely and those that rely on space-time signatures of evolving cellular automata. Given one of the X1 × ⋯ × Xd × T state blocks of that section, we start by considering the probability that, inside the block, each of the possible s1+δ inputs to
Cell-centric heuristics
Computing the mean and variance of the input entropy as indicated respectively in (8), (9) requires simulating the cellular automaton that is based on f for t+ time steps and accumulating the quantity given in (7) while an X1 × ⋯ × Xd × T block “window” is slid from an initial position that makes the block end at time t0 through a final position at which the block ends at time t+. When the simulation is performed in parallel, the X1 ⋯ Xd cells do not all reside at the same processor, so computing the
Computer experiments
An infinite cellular automaton cannot be simulated in its entirety, nor can a portion of it be simulated for an indefinitely long number of time steps. One crucial first decision when planning such a simulation is which contiguous cells to observe along each of the d dimensions and also the number of time steps t+ during which to perform the simulation. Choosing a finite number of cells to observe poses the question of how to handle the boundaries of the observed region, since those boundaries
Discussion
The data shown in Fig. 6, Fig. 7, respectively for one- and two-dimensional cellular automata, tend all to exhibit the following behavior regarding classes (i)–(iv). When plotted on doubly-logarithmic scales, they appear clustered roughly as a boomerang whose traversal from the low-mean, low-variance tip leads us through class (i) update rules, then class (ii), then class (iv) near the middle bend, and finally class (iii) past the bend. Entropy means increase at varying rates along the
Conclusions
We have in this paper addressed the automatic classification of the update rules of cellular automata. Our departing point has been the notion of input entropy, on which we built by the introduction of two novel entropy measures, both inspired by, and targeted at, the simulation of cellular automata by message-passing parallel machines. Our two new measures are the cell-centric input entropy and the cell-centric transition entropy. For both of them we provided extensive experimental results on
Acknowledgments
The authors acknowledge partial support from CNPq, CAPES, and a FAPERJ BBP grant.
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