Cellular automata in operational probabilistic theories

Lecture Notes in Computer Science, 2002

The Ising-like phase transition is considered in probabilistic cellular automata (CA). The nonequilibrium CA with Toom rule are compared to standard equilibrium lattice systems to verify influence of synchronous vs asynchronous updating. It was observed by Marcq et al. [Phys.Rev.E 55(1997) 2606] that the mode of updating separates systems of coupled map lattices into two distinct universality classes. The similar partition holds in case of CA. CA with Toom rule and synchronous updating represent the weak universality class of the Ising model, while Toom CA with asynchronous updating fall into the Ising universality class.

Statistics & Probability Letters, 2005

We give a necessary and sufficient condition for the existence of an increasing coupling of N (N 2) synchronous dynamics on S Z d (PCA). Increasing means the coupling preserves stochastic ordering. We first present our main construction theorem in the case where S is totally ordered; applications to attractive PCA's are given. When S is only partially ordered, we show on two examples that a coupling of more than two synchronous dynamics may not exist. We also prove an extension of our main result for a particular class of partially ordered spaces.

Springer eBooks, 2018

Cellular Automata are discrete-time dynamical systems on a spatially extended discrete space which provide paradigmatic examples of nonlinear phenomena. Their stochastic generalizations, i.e., Probabilistic Cellular Automata, are discrete time Markov chains on lattice with finite single-cell states whose distinguishing feature is the parallel character of the updating rule. We review the some of the results obtained about the metastable behavior of Probabilistic Cellular Automata and we try to point out difficulties and peculiarities with respect to standard Statistical Mechanics Lattice models.

Journal of Statistical Physics, 1986

Cellular automata are discrete mathematical systems that generate diverse, often complicated, behavior using simple deterministic rules. Analysis of the local structure of these rules makes possible a description of the global properties of the associated automata. A class of cellular automata that generate infinitely many aperiodic temporal sequences is defined, as is the set of rules for which inverses exist. Necessary and sufficient conditions are derived characterizing the classes of "nearest-neighbor" rules for which arbitrary finite initial conditions (i) evolve to a homogeneous state; (ii) generate at least one constant temporal sequence.

Emergence, Complexity and Computation, 2018

The Emergence, Complexity and Computation (ECC) series publishes new developments, advancements and selected topics in the fields of complexity, computation and emergence. The series focuses on all aspects of reality-based computation approaches from an interdisciplinary point of view especially from applied sciences, biology, physics, or chemistry. It presents new ideas and interdisciplinary insight on the mutual intersection of subareas of computation, complexity and emergence and its impact and limits to any computing based on physical limits (thermodynamic and quantum limits, Bremermann's limit, Seth Lloyd limits…) as well as algorithmic limits (Gödel's proof and its impact on calculation, algorithmic complexity, the Chaitin's Omega number and Kolmogorov complexity, non-traditional calculations like Turing machine process and its consequences,…) and limitations arising in artificial intelligence field. The topics are (but not limited to) membrane computing, DNA computing, immune computing, quantum computing, swarm computing, analogic computing, chaos computing and computing on the edge of chaos, computational aspects of dynamics of complex systems (systems with self-organization, multiagent systems, cellular automata, artificial life,…), emergence of complex systems and its computational aspects, and agent based computation. The main aim of this series it to discuss the above mentioned topics from an interdisciplinary point of view and present new ideas coming from mutual intersection of classical as well as modern methods of computation. Within the scope of the series are monographs, lecture notes, selected contributions from specialized conferences and workshops, special contribution from international experts.

1989

In this thesis we investigate the theoretical nature of the mathematical structures termed cellular automata. Chapter 1: Reviews the origin and history of cellular automata in order to place the current work into context. Chapter 2: Develops a cellular automata framework which contains the main aspects of cellular automata structure which have appeared in the literature. We present a scheme for specifying the cellular automata rules for this general model and present six examples of cellular automata within the model. Chapter 3: Here we develop a statistical mechanical model of cellular automata behaviour. We consider the relationship between variations within the model and their relationship to dynamical systems. We obtain results on the variance of the state changes, scaling of the cellular automata lattice, the equivalence of noise, spatial mixing of the lattice states and entropy, synchronous and asynchronous cellular automata and the equivalence of the rule probability and the ...

Electronic Proceedings in Theoretical Computer Science, 2009

The goal of this paper is to show why the framework of communication complexity seems suitable for the study of cellular automata. Researchers have tackled different algorithmic problems ranging from the complexity of predicting to the decidability of different dynamical properties of cellular automata. But the difference here is that we look for communication protocols arising in the dynamics itself. Our work is guided by the following idea : if we are able to give a protocol describing a cellular automaton, then we can understand its behavior.

Cellular automata are discrete dynamical systems, of simple construction but complex and varied behaviour. Algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata. The complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities. The systems are usually irreversible, and are found to evolve through transients to attractors consisting of cycles sometimes containing a large number of configurations.

Mathematical Structures in Computer Science, 2002

For lack of composability of their morphisms, probability spaces, and hence probabilistic automata, fail to form categories; however, they t into the more general framework of precategories, which are introduced and studied here. In particular, the notion of adjunction and weak adjunction for precategories is presented and justi ed in detail. As an immediate bene t, a concept of (weak) product for precategories is obtained. Thus, universal properties can be used for characterizing well-known basic constructions in the theory of probabilistic automata: The aggregation of two automata is shown to be a weak product, whereas restriction and interconnection of automata are recognized as Cartesian lifts. Finally, we establish that the precategory of decision trees is core exive in the precategory of probabilistic automata.

International Conference on Cellular Automata for Research and Industry, 2008

We consider two formulations of a cellular automata: the first one uses a gather-update paradigm and the second one a collision-propagation paradigm. We show the equivalence of both descriptions and, using the latter paradigm, we propose a simple way to define a Cellular Automata on a graph with arbitrary topology. Finally, we exploit the duality of formulation to reconsider the problem of characterizing invertible cellular automata.