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Computation in cellular automata: A selected review
 Nonstandard Computation
, 1996
"... Cellular automata (CAs) are decentralized spatially extended systems consisting of large numbers of simple identical components with local connectivity. Such systems have the potential to perform complex computations with a high degree of efficiency and robustness, as well as to model the behavior o ..."
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Cited by 44 (2 self)
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Cellular automata (CAs) are decentralized spatially extended systems consisting of large numbers of simple identical components with local connectivity. Such systems have the potential to perform complex computations with a high degree of efficiency and robustness, as well as to model the behavior of complex systems in nature. For these reasons CAs and related architectures have
Upper bound on the products of particle interactions in cellular automata
 Physica D
, 2001
"... Abstract Particlelike objects are observed to propagate and interact in many spatially extended dynamical systems. For one of the simplest classes of such systems, onedimensional cellular automata, we establish a rigorous upper bound on the number of distinct products that these interactions can ..."
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Cited by 12 (0 self)
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Abstract Particlelike objects are observed to propagate and interact in many spatially extended dynamical systems. For one of the simplest classes of such systems, onedimensional cellular automata, we establish a rigorous upper bound on the number of distinct products that these interactions can generate. The upper bound is controlled by the structural complexity of the interacting particles a quantity which is defined here and which measures the amount of spatiotemporal information that a particle stores. Along the way we establish a number of properties of domains and particles that follow from the computational mechanics analysis of cellular automata; thereby elucidating why that approach is of general utility. The upper bound is tested against several relatively complex domainparticle cellular automata and found to be tight.
The Topological Skeleton of Cellular Automaton Dynamics
, 1996
"... We have developed statistical techniques to study the structure the statetransition graphs of cellular automata with periodic boundary conditions, in the limit of large system size. We organize our results around the concept of a topological skeleton. The topological skeleton is the set of physical ..."
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Cited by 7 (1 self)
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We have developed statistical techniques to study the structure the statetransition graphs of cellular automata with periodic boundary conditions, in the limit of large system size. We organize our results around the concept of a topological skeleton. The topological skeleton is the set of physically relevant states. Covering this skeleton is a surface, typically thin and dense, which contains the bulk of the set of states. States in the skeleton have some long histories. States on the surface, by contrast, have only short histories; they are reached only near the beginning of cellular automaton evolution. We study in detail a sequence of rules which exhibit mostly skeletal to mostly surface structure.
Spectral domain boundaries cellular automata
, 2007
"... Abstract. Let AZD be the Cantor space of ZDindexed configurations in a finite alphabet A, and let σ be the ZDaction of shifts on AZD. A cellular automaton is a continuous, σcommuting selfmap Φ of AZD, and a Φinvariant subshift is a closed, (Φ, σ)invariant subset A ⊂ AZD. Suppose a ∈ AZD is Aa ..."
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Cited by 5 (5 self)
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Abstract. Let AZD be the Cantor space of ZDindexed configurations in a finite alphabet A, and let σ be the ZDaction of shifts on AZD. A cellular automaton is a continuous, σcommuting selfmap Φ of AZD, and a Φinvariant subshift is a closed, (Φ, σ)invariant subset A ⊂ AZD. Suppose a ∈ AZD is Aadmissible everywhere except for some small region we call a defect. It has been empirically observed that such defects persist under iteration of Φ, and often propagate like ‘particles ’ which coalesce or annihilate on contact. We use spectral theory to explain the persistence of some defects under Φ, and partly explain the outcomes of their collisions. An oftenobserved phenomenon in cellular automata is the emergence and persistence of homogeneous ‘domains ’ (each characterized by a particular spatial pattern), separated by defects (analogous to ‘domain boundaries ’ or ‘kinks ’ in a crystalline solid) which evolve over time, propagating and occasionally colliding. Such defects were first empirically observed by Grassberger in the ‘elementary ’ cellular automata
Algebraic invariants for crystallographic defects in cellular automata
, 2005
"... Let AZD be the Cantor space of ZDindexed configurations in a finite alphabet A, and let σ be the ZDaction of shifts on AZD. A cellular automaton is a continuous, σcommuting selfmap Φ of AZD, and a Φinvariant subshift is a closed, (Φ, σ)invariant subset A ⊂ AZD. Suppose a ∈ AZD is Aadmissible ..."
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Cited by 4 (4 self)
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Let AZD be the Cantor space of ZDindexed configurations in a finite alphabet A, and let σ be the ZDaction of shifts on AZD. A cellular automaton is a continuous, σcommuting selfmap Φ of AZD, and a Φinvariant subshift is a closed, (Φ, σ)invariant subset A ⊂ AZD. Suppose a ∈ AZD is Aadmissible everywhere except for some small region we call a defect. It has been empirically observed that such defects persist under iteration of Φ, and often propagate like ‘particles ’ which coalesce or annihilate on contact. We construct algebraic invariants for these defects, which explain their persistence under Φ, and partly explain the outcomes of their collisions. Some invariants are based on the cocycles of multidimensional subshifts; others arise from the higherdimensional (co)homology/homotopy groups for subshifts, obtained by generalizing the ConwayLagarias tiling groups and the GellerPropp fundamental group.
Cellular Automata for Contour Dynamics
 Physica D
, 1995
"... The dynamics of deterministic partially permutive cellular automata in two dimensions is considered. In particular we lay out the design principles and analyze the dynamics of generalized votertype rules. They provide a two parameter model class of cellular automata which exhibit annealing, diffusi ..."
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Cited by 2 (0 self)
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The dynamics of deterministic partially permutive cellular automata in two dimensions is considered. In particular we lay out the design principles and analyze the dynamics of generalized votertype rules. They provide a two parameter model class of cellular automata which exhibit annealing, diffusive and critical domain behavior. We analyze the dynamics rigorously in certain cases and also show that all types of behavior are very close to their ideal counterparts observed in probabilistic models based on strong independence assumptions. This is further evidence that the statistical mechanics extends to cover also classes of purely deterministic dynamics. Keywords: Cellular automaton, partial permutivity, votermodel, annealing and critical dynamics. Running title: Cellular automata for contour dynamics Research partially supported by the Academy of Finland and The Santa Fe Institute Introduction For already some time it has been known that deterministic cellular automata (c.a.) ...
Is It Alive Or is It a Cellular Automaton?
"... this article we will make a brief excursion to this territory. 1. What are they? ..."
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this article we will make a brief excursion to this territory. 1. What are they?
Voter Dynamics In Deterministic Cellular Automata
, 1995
"... A deterministic counterpart is introduced to the voter model studied in probabilistic particle systems. Here we investigate the ingredients of the rule as well as its annealing domain dynamics. It is a representative from a larger class of cellular automata with the common property that they all sho ..."
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A deterministic counterpart is introduced to the voter model studied in probabilistic particle systems. Here we investigate the ingredients of the rule as well as its annealing domain dynamics. It is a representative from a larger class of cellular automata with the common property that they all show behavior previously associated to lattice models with strong independence assumptions. The results indicate that purely deterministic schemes are capable of producing behavior characteristic to disordered systems of statistical mechanics. INTRODUCTION We will introduce design principles by which one can build up cellular automata (c.a.) rules with dynamical properties remarkably close to the probabilistic voter models (see e.g. Durrett). The motivation for this comes from the currently unsatisfactory level of understanding of the dynamical capabilities of c.a. In particular their statistical mechanics has been described in a number of simulation studies but the fundamental theory is miss...
Defect Particle Kinematics . . .
, 2007
"... Let A Z be the Cantor space of biinfinite sequences in a finite alphabet A, and let σ be the shift map on A Z. A cellular automaton is a continuous, σcommuting selfmap Φ of A Z, and a Φinvariant subshift is a closed, (Φ,σ)invariant subset S ⊂ A Z. Suppose a ∈ A Z is Sadmissible everywhere exce ..."
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Let A Z be the Cantor space of biinfinite sequences in a finite alphabet A, and let σ be the shift map on A Z. A cellular automaton is a continuous, σcommuting selfmap Φ of A Z, and a Φinvariant subshift is a closed, (Φ,σ)invariant subset S ⊂ A Z. Suppose a ∈ A Z is Sadmissible everywhere except for some small region we call a defect. It has been empirically observed that such defects persist under iteration of Φ, and often propagate like ‘particles’. We characterize the motion of these particles, and show that it falls into several regimes, ranging from simple deterministic motion, to generalized random walks, to complex motion emulating Turing machines or pushdown automata. One consequence is that some questions about defect behaviour are formally undecidable.