Computation properties of spatial dynamics simulation by probabilistic cellular automata

A method is proposed, which is intended for constructing a probabilistic cellular automaton (CA), whose evolution simulates a spatially distributed process, given by a PDE. The heart of the method is the transformation of a real spatial function into a Boolean array whose averaged form approximates the given function. Two parts of a given PDE (a differential operator and a function) are approximated by a combination of their Boolean counterparts. The resulting CA transition function has a basic (standard) part, modeling the differential operator and the updating part modifying it according to the function value. Special attention is paid to the reaction-diffusion type of PDE. Some experimental results of simple processes simulation are given and perspectives of the proposed method application are discussed.

Accuracy and stability properties of fine-grained parallel computations, based on modeling spatial dynamics by cellular automata (CA) evolution, are studied. The problem arises when phenomena under simulation are represented as a composition of a CA and a function given in real numbers, and the whole computation process is transferred into a Boolean domain. To approach the problem accuracy of real spatial functions approximation by Boolean arrays, as well as of some operations on cellular arrays with different data types are determined and approximation errors are assessed. Some methods of providing admissible accuracy are proposed. Stability is shown to depend only of the nonlinear terms in hybrid methods, the use of CA-diffusion instead of Laplace operator having no effect on it. Some experimental results supporting the theoretical conclusions are presented.

This special issue contains a selection of papers presented at the ‘‘Third International Workshop on Asynchronous Cellular Automata and Asynchronous Discrete Models’’ (ACA 2014), held as a satellite workshop of the 11th International Conference on Cellular Automata for Research and Industry (ACRI 2014) in Krakow (Poland) in September 2014. Six papers were selected and, after an additional review process, five of them have been included in this special issue. They are now presented in an extended and improved form with respect to the already refereed workshop version that appeared in the proceedings of the ACRI 2014 conference. The ACA workshop is devoted to the theme of asynchrony, a hot topic, inside Cellular Automata and other Discrete Models as, for instance, Boolean Networks. Cellular Automata are a well-known formal tool for modeling complex systems; they are considered in many scientific fields and industrial applications. Synchronicity is one of the main features of Cellular Automata evolutions. Indeed, in the most common Cellular Automata framework, all cells are updated simultaneously at each discrete time step by means of a same rule. Recent trends consider the modeling of asynchronous systems based on local and possibly non-uniform interactions. The aim of this workshop is to bring together researchers dealing with the theme of the asynchrony inside Cellular Automata and Discrete Models. Typical, but not exclusive, topics of the workshop are dynamics, complexity and computational issues, emergent properties, models of parallelism and distributed systems, and models of real phenomena. The paper ‘‘Local structure approximation as a predictor of second-order phase transitions in asynchronous cellular automata’’ by Henryk Fukś and Nazim Fates considers aasynchronous elementary cellular automata, that is elementary cellular automata in which each cell independently updates with probability a. By means of an extension of the mean-field approximation technique, the authors study the phase transitions in such automata, i.e., the changes of the dynamical behavior which may occur when the parameter a varies. In the paper ‘‘Supercritical probabilistic cellular automata: How effective is the synchronous updating?’’, PierreYves Louis deals with the issue of quantifying the effectiveness of the parallel updating in probabilistic cellular automata, i.e., cellular automata where the local rule is defined by means of a probability. Two interesting classes of probabilistic cellular automata are considered. An analysis of simulation is presented and shows that the behavior of these classes is nearly asynchronous when transition phase phenomena occur. Boolean Networks model the dynamical interaction of components which take a binary state. They have been & Alberto Dennunzio dennunzio@disco.unimib.it

CRITICAL BEHAVIOR OF A PROBABILISTIC LOCAL AND NONLOCAL SITE-EXCHANGE CELLULAR AUTOMATON

In a probabilistic cellular automaton (PCA), the cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. A PCA can be viewed as a Markov chain whose ergodicity is investigated. A classical cellular automaton (CA) is a particular case of PCA. For a 1-dimensional CA, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to be also a PCA.

Reversibility is an important phenomena in nature as well as in Computer Science. Obtaining plaintext back from ciphertext can be modeled as one kind of reversibility. Image restoration problem can be modeled as another kind of reversibility. Cellular automata (CA) are lattices and that are used as computation tools for modeling diverse complex dynamical systems. The CA evolve from one configuration to another over iterations using local transition rules. Number of cells that are allowed to undergo the local transition or update function in every time step varies from one kind of CA to another. In probabilistic CA (PCA), cells are selected randomly for update. Reversibility is one important issue in CA. Reversible CA are those CA which comes back to the initial state for any given inital state after some time steps. In this paper, we have studied the reversibility of a PCA where maximum two cells are selected randomly for possible updates in every time step. We have introduced a new tool, reachable state graph to understand the PCA reversibility dynamics and proposed a deterministic algorithm to find if a rule is reversible for PCA of arbitrary size.

A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov chain. A classical cellular automaton is a particular case of PCA. For a one-dimensional cellular automaton, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to also be a PCA. Last, we focus on the PCA majority, whose asymptotic behavior is unknown, and perform numerical experiments using the perfect sampling procedure.

A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov chain. A classical cellular automaton is a particular case of PCA. For a one-dimensional cellular automaton, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to also be a PCA. Last, we focus on the PCA majority, whose asymptotic behavior is unknown, and perform numerical experiments using the perfect sampling procedure.

We propose some conjectures on the asymptotic distribution of the probabilistic Burgers cellular automaton (PBCA), which is defined by a simple rule of particle motion with a probabilistic parameter. Asymptotic distribution of configurations converges to a unique steady state for PBCA. We propose a new and widely-applicable approach to analyze probabilistic particle systems and apply it concretely to PBCA and its extensions. We introduce a conjecture on the distribution and derive the asymptotic probability expressed by the GKZ hypergeometric function. If the space size goes into infinity, we can evaluate the relationship between the density and flux of particles for infinite space. Moreover, we propose two extended systems of PBCA and analyze their asymptotic behavior.

The combination of the notions of a cellular automaton and a probabilistic automaton, called a probabilistic cellular automaton, was proposed in 1973 by the first author of this paper as a more adequate model of any system with unreliable components that operate in a parallel manner. In this paper a language-acceptor type of such a probabilistic cellular automaton, called aprobabilistic bounded cellular acceptor (PBCA), is defined and studied. It is shown that the class of all languages accepted by one-dimensional PBCAs includes both the class of all languages accepted by bounded cellular acceptors (BCAs) and the class of all languages accepted by probabilistic acceptors (PAs). Also, it is shown that every language accepted by ad-dimensional PBCA at a given cut point is accepted by ad-dimensional PBCA at an arbitrary nonzero cut point. The class of all languages accepted by one-dimensional PBCAs at cut point 0 is shown to be precisely the class of all context-sensitive languages. Several decision problems for PBCAs are shown to be recursively unsolvable. Finally, various open problems concerning PBCAs and PBCLs are discussed.

OPINION article Front. Comput. Neurosci., 01 October 2013 Volume 7 - 2013 | https://doi.org/10.3389/fncom.2013.00130

Numerical models of the global atmosphere have spatial resolutions that are much too coarse to resolve clouds and convection processes explicitly. Because these processes play an important role in the atmosphere and climate system, they are included in numerical models by means of simplified representations, so-called parameterizations. Traditional parameterization schemes for atmospheric convection are deterministic. To overcome the limitations of these deterministic schemes, stochastic parameterizations are being developed. The use of probabilistic cellular automata (PCA) for this application is very new and can provide a way to generate spatial patterns of convection as observed in the atmosphere. It is approached from two directions, both briefly reviewed here. In one approach, convection and other sub-grid-scale processes are represented with deterministic CA. In recent work, this is extended to PCA. In the other approach, convection is represented by means of discrete stochastic processes (finite state Markov chains) on a lattice. In most studies in this direction, there is no direct coupling between neighboring lattice nodes, however recently such couplings are considered as well. To illustrate the concept of parameterization, a frequently used test model (the L96 model) is discussed as well in this chapter. Parameterization of atmospheric convection and clouds with PCA has several interesting mathematical aspects. One is the interactive (two-way) coupling of the PCA to a partial differential equation for large-scale atmospheric flow. The state of the PCA couples to the time evolution of the flow, and in turn the PCA rules (transition probabilities) depend on the flow state. Furthermore, for convection it is natural to consider N-state PCAs with \(N > 2\) rather than a binary (\(N = 2\)) PCA. Finally, statistical inference can be a fruitful approach to construct the PCA rules or transition probabilities for convection. The PCA dependence on the time-evolving atmospheric flow and the large number of configurations for PCAs with \(N > 2\) provide interesting challenges for such inference.

Cellular automata are both seen as a model of computation and as tools to model real life systems. Historically they were studied under synchronous dynamics where all the cells of the system are updated at each time step. Meanwhile the question of probabilistic dynamics emerges: on the one hand, to develop cellular automata which are capable of reliable computation even when some random errors occur [24,14,13]; on the other hand, because synchronous dynamics is not a reasonable assumption to simulate real life systems. Among cellular automata a specific class was largely studied in synchronous dynamics : the elementary cellular automata (ECA). These are the simplest cellular automata. Nevertheless they exhibit complex behaviors and even Turing universality. Several studies [20,7,8,5] have focused on this class under i¾?-asynchronous dynamics where each cell has a probability i¾?to be updated independently. It has been shown that some of these cellular automata exhibit interesting behavior such as phase transition when the asynchronicity rate i¾?varies. Due to their richness of behavior, probabilistic cellular automata are also very hard to study. Almost nothing is known of their behavior [20]. Understanding these simple rules is a key step to analyze more complex systems. We present here a coupling between oriented percolation and ECA 178 and confirms observations made in [5] that percolation may arise in cellular automata. As a consequence this coupling shows that there is a positive probability that the ECA 178 does not reach a stable configuration as soon as the initial configuration is not a stable configuration and i¾?> 0.996. Experimentally, this result seems to stay true as soon as i¾?> i¾? c ≈ 0.5.

Computational Complexity of Real Functions

GROWTH OF SURFACES GENERATED BY A PROBABILISTIC CELLULAR AUTOMATION