A tutorial on Elementary cellular automata with fully asynchronous updating - General properties and convergence dynamics

International audienceWe present a picture of the convergence properties of the 256 Elementary Cellular Automata under the fully asynchronous updating , that is, when only one cell is updated at each time step. We regroup here the results which have been presented in different articles and expose a full analysis of the behaviour of finite systems with periodic boundary conditions. Our classification relies on the scaling properties of the average convergence time to a fixed point. We observe that different scaling laws can be found, which fall in one of the following classes: logarithmic, linear, quadratic, exponential and non-converging. The techniques for quantifying this behaviour rely mainly on Markov chain theory and martingales. Most behaviours can be studied analytically but there are still many rules for which obtaining a formal characterisation of their convergence properties is still an open problem

An Experimental Study of Noise and Asynchrony in Elementary Cellular Automata with Sampling Compensation

<p><strong>An Experimental Study of Noise and Asynchrony in Elementary Cellular Automata with Sampling Compensation </strong>by Fernando Silva, and Luís Correia</p> <p> </p> <p> <strong>Abstract: </strong>This article focuses on the set of 32 legal Elementary Cellular Automata. We perform an exhaustive study of the systems' response under: (i) $\alpha$-asynchronous dynamics, from full asynchronism to perfect synchrony, (ii) $\kappa$ asynchrony, which extends $\alpha$-asynchrony to compensate for less cell activity, and (iii) $\phi$ noise scheme, a perturbation that affects the local transition function and causes a cell to probabilistically miscalculate the new state when it is updated. We propose a new classification in three classes under asynchronous conditions: $\alpha$ invariant, $\alpha$-robust, and $\alpha$-dependent. We classify the 32 legal ECA according to the degree of behavioural modification, and we show that our classifying scheme provides results coherent with the density-based classification. We also show that $\kappa$-asynchrony provides results comparable to synchronous systems, both quantitatively and qualitatively. Subsequently, we analyse the effects of including different levels of noise in synchronous systems. We identify different responses to noise, including systems that are robust to asynchrony and susceptible to noise. To conclude, we investigate the behavioural changes caused by simultaneous asynchrony and noise in models tolerant to both perturbations. We describe a number of effects caused by the interplay of noise and asynchrony, thus further reinforcing that both aspects are pertinent for future studies.</p> <p> </p> <p><strong>Description of the dataset:</strong> </p> <p>The dataset contains a number of results and data with respect to our experimental study of noise and asynchrony in Elementary Cellular Automata. The dataset is divided into three folders, namely:</p> <p> </p> <p>1 - folder "Asynchrony", in which we provide a number of results related to the classification of the 32 ECA in three classes, $\alpha$-invariant, $\alpha$-robust, and $\alpha$-dependent, according to the degree of behavioural modification under asynchronous conditions. We also analyse and compare the effects of $\kappa$-asynchrony and $\alpha$-asynchrony in CA evolution.</p> <p>  </p> <p>2 - folder "Noise" </p> <p>Stochastic noise in the local transition function consists of a perturbation to a cell's state when it is updated. We examined the impact of noise in the 32 legal rules. CA are subject to noise and updated according to a synchronous scheme in order to distinguish the effects of noise and the effects of asynchronous updating. Analysis is conducted with respect to the different classes of response to $\alpha$-asynchrony. We define 4 levels of tolerance to noise, coherent with our proposed classification according to the degree of behavioural modification under asynchronous conditions. The four degrees for classifying systems subject to noise are: (i) $\phi$ invariant as models that instantaneously forget perturbations due to noise, (ii) $\phi$-MR when the asymptotic inter-CA correlation >= 0.5, (iii) $\phi$ LR as models where perturbations are contained in the neighbourhood but the asymptotic inter-CA correlation < 0.5, and (iv) $\phi$-dependent as models highly susceptible to noise in which a single perturbation causes significant changes in behaviour.</p> <p> </p> <p>3 - Folder "Noise and Asynchrony"</p> <p>We analyse the degree of behavioural modification when systems are simultaneously subject to asynchrony and noise. We investigate how models robust or invariant to asynchrony and noise, separately, respond when both aspects are present. We concentrate our study in two sets of CA: (i) $\alpha$-invariant and $\phi$-invariant, i.e., ECA 0, 32, 128, 160, 250, and 254, and (ii) $\alpha$ invariant and $\phi$-MR, namely ECA 4, 36, 72, 104, 164, and 218. Remaining systems part of the 32 legal rules are sensitive to the presence of noise and/or asynchrony. Expectedly, these systems exhibit low robustness to simultaneous perturbations. We represented the asymptotic inter-CA correlation between $\alpha$-asynchronous and synchronous systems, both of which are subject to noise. The set of values for different synchrony rates and noise rates is represented in a three dimensional space, which is projected on a two dimensional sampling surface.</p

Transient time for robust elementary cellular automata

<p>Transient time, in number of time steps, for distinct Elementary Cellular Automata when subject to distinct synchrony rates.</p> <p>The plots show the transient time for ECA 4, 36, 72, 76, 94, 104, 108, 132, 164, 218, 222, and 232.</p> <p>Updating scheme:</p> <p>1 - Perfect synchrony -- all cells are updated simultaneously and in parallel.</p> <p>2 - $\alpha$-asynchrony -- at each time step, every cell is updated with an independent and equal probability alpha.</p> <p> </p> <p>3 - $\kappa$-asynchrony -- Given a synchrony rate $\alpha$-, at each time step, $\kappa$ = 1/$\alpha$ updates are performed. For non-integer $\kappa$ = 1/$\alpha$, decimal values are probabilistically used for deciding between performing n or n + 1 intermediate updates. The resulting conguration is considered as the automaton's next stage.</p> <p> </p> <p>See "An Experimental Study of Noise and Asynchrony in Elementary Cellular Automata with Sampling Compensation".</p> <p> </p