Abstract
Classification of asynchronous elementary cellular automata (AECAs) was explored in the first place by Fates et al. (Complex Systems, 2004) who employed the asymptotic density of cells as a key metric to measure their robustness to stochastic transitions. Unfortunately, the asymptotic density seems unable to distinguish the robustnesses of all AECAs. In this paper, we put forward a method that goes one step further via adopting a metric entropy (Martin, Complex Systems, 2000), with the aim of measuring the asymptotic mean entropy of local pattern distribution in the cell space of any AECA. Numerical experiments demonstrate that such an entropy-based measure can actually facilitate a complete classification of the robustnesses of all AECA models, even when all local patterns are restricted to length 1. To gain more insights into the complexity concerning the forward evolution of all AECAs, we consider another entropy defined in the form of Kolmogorov–Sinai entropy and conduct preliminary experiments on classifying their uncertainties measured in terms of the proposed entropy. The results reveal that AECAs with low uncertainty tend to converge remarkably faster than models with high uncertainty.
Highlights
A cellular automaton (CA) is a discrete dynamical system consisting of a huge number of locally interconnected cells [1]
The results show that all AECAs can possibly be classified into one of four qualitative sets according to their robustnesses, even when all binary patterns are restricted to length 1
CAs under asynchronous updating, this paper proposed two types of entropies, with one devoted to measure the robustness of AECAs against the randomness due to αasynchronism, and another one used for estimating the uncertainty in the forward evolutions
Summary
A cellular automaton (CA) is a discrete dynamical system consisting of a huge number of locally interconnected cells [1]. Our parameter depends on the distribution of local binary patterns of certain lengths in the cell space of an AECA, rather than the ratio of cells in state 1 To this end, we adopt a metric entropy [9] which was originally proposed to confirm the underlying dynamics of synchronous ECAs in accordance with the Wolfram’s empirical classification [2]. For AECAs, on the other hand, their global behavior may be sensitive to the initial configurations, and be seriously affected by the probability controlling the state transitions of each cell at any time This motives us to define another entropy in the form of the KS entropy, for the purpose to challenge the uncertainty measure problem of AECAs. Numerical experiments will be done to compute the entropies of each AECA model, which allow a preliminary classification of them in accordance with their estimated uncertainties.
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