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Continuous spatial automaton
About: Continuous spatial automaton is a research topic. Over the lifetime, 1530 publications have been published within this topic receiving 33809 citations.
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TL;DR: Analysis is given of ''elementary'' cellular automata consisting of a sequence of sites with values 0 or 1 on a line, with each site evolving deterministically in discrete time steps according to p definite rules involving the values of its nearest neighbors.
Abstract: Cellular automata are used as simple mathematical models to investigate self-organization in statistical mechanics. A detailed analysis is given of "elementary" cellular automata consisting of a sequence of sites with values 0 or 1 on a line, with each site evolving deterministically in discrete time steps according to definite rules involving the values of its nearest neighbors. With simple initial configurations, the cellular automata either tend to homogeneous states, or generate self-similar patterns with fractal dimensions \ensuremath{\simeq} 1.59 or \ensuremath{\simeq} 1.69. With "random" initial configurations, the irreversible character of the cellular automaton evolution leads to several self-organization phenomena. Statistical properties of the structures generated are found to lie in two universality classes, independent of the details of the initial state or the cellular automaton rules. More complicated cellular automata are briefly considered, and connections with dynamical systems theory and the formal theory of computation are discussed.
2,860 citations
TL;DR: It is proved that the reachability problem is undecidable for timed automata augmented with a single stopwatch, and an (optimal) PSPACE reachability algorithm is given for the case of initialized rectangular automata.
903 citations
Book•
21 Jun 2019
TL;DR: Primary Papers * Statistical Mechanics of Cellular Automata * Algebraic Properties of CellularAutomaton * Universality and Complexity in Cellular Automaton * Computation Theory of Cellular automaton * Undecidability and Intractability in Theoretical Physics
Abstract: Primary Papers * Statistical Mechanics of Cellular Automata * Algebraic Properties of Cellular Automata * Universality and Complexity in Cellular Automata * Computation Theory of Cellular Automata * Undecidability and Intractability in Theoretical Physics * Two-Dimensional Cellular Automata * Origins of Randomness in Physical Systems * Thermodynamics and Hydrodynamics of Cellular Automata * Random Sequence Generation by Cellular Automata * Approaches to Complexity Engineering * Minimal Cellular Automaton Approximations to Continuum Systems * Cellular Automaton Fluids: Basic Theory Additional And Survey Papers * Cellular Automata * Computers in Science and Mathematics * Geometry of Binomial Coefficients * Twenty Problems in the Theory of Cellular Automata * Cryptography with Cellular Automata * Complex Systems Theory * Cellular Automaton Supercomputing Appendices * Tables of Cellular Automaton Properties * Scientific Bibliography of Stephen Wolfram
601 citations
TL;DR: The sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages and it is suggested that such undecidability is common in these and other dynamical systems.
Abstract: Self-organizing behaviour in cellular automata is discussed as a computational process. Formal language theory is used to extend dynamical systems theory descriptions of cellular automata. The sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages. Many examples are given. The sizes of the minimal grammars for these languages provide measures of the complexities of the sets. This complexity is usually found to be non-decreasing with time. The limit sets generated by some classes of cellular automata correspond to regular languages. For other classes of cellular automata they appear to correspond to more complicated languages. Many properties of these sets are then formally non-computable. It is suggested that such undecidability is common in these and other dynamical systems.
579 citations
TL;DR: A direct connection between a cellular automaton and integrable nonlinear wave equations is shown and a general method for constructing suchintegrable cellular automata and their $N$-soliton solutions is proposed.
Abstract: We show a direct connection between a cellular automaton and integrable nonlinear wave equations. We also present the $N$-soliton formula for the cellular automaton. Finally, we propose a general method for constructing such integrable cellular automata and their $N$-soliton solutions.
505 citations