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151
Revisiting the edge of chaos: Evolving cellular automata to perform computations
 Complex Systems
, 1993
"... We present results from an experiment similar to one performed by Packard [24], in which a genetic algorithm is used to evolve cellular automata (CA) to perform a particular computational task. Packard examined the frequency of evolved CA rules as a function of Langton’s λ parameter [17], and interp ..."
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Cited by 99 (10 self)
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We present results from an experiment similar to one performed by Packard [24], in which a genetic algorithm is used to evolve cellular automata (CA) to perform a particular computational task. Packard examined the frequency of evolved CA rules as a function of Langton’s λ parameter [17], and interpreted the results of his experiment as giving evidence for the following two hypotheses: (1) CA rules able to perform complex computations are most likely to be found near “critical ” λ values, which have been claimed to correlate with a phase transition between ordered and chaotic behavioral regimes for CA; (2) When CA rules are evolved to perform a complex computation, evolution will tend to select rules with λ values close to the critical values. Our experiment produced very different results, and we suggest that the interpretation of the original results is not correct. We also review and discuss issues related to λ, dynamicalbehavior classes, and computation in CA. The main constructive results of our study are identifying the emergence and competition of computational strategies and analyzing the central role of symmetries in an evolutionary system. In particular, we demonstrate how symmetry breaking can impede the evolution toward higher computational capability.
Evolutionary games on graphs
, 2007
"... Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to ..."
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Cited by 54 (0 self)
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Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in nonequilibrium statistical physics. This review gives a tutorialtype overview of the field for physicists. The first four sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fifth section surveys the topological complications implied by nonmeanfieldtype social network structures in general. The next three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner’s Dilemma, the Rock–Scissors–Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.
Turbulent Pattern Bases for Cellular Automata
 Physica D
, 1993
"... Unpredictable patterns generated by cellular automata (CA) can be decomposed with respect to a turbulent, positive entropy rate pattern basis. The resulting filtered patterns uncover significant structural organization in a CA's dynamics and information processing capabilities. We illustrate the dec ..."
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Cited by 46 (14 self)
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Unpredictable patterns generated by cellular automata (CA) can be decomposed with respect to a turbulent, positive entropy rate pattern basis. The resulting filtered patterns uncover significant structural organization in a CA's dynamics and information processing capabilities. We illustrate the decomposition technique by analyzing a binary, range2 cellular automaton having two invariant chaotic domains of different complexities and entropies. Once identified, the domains are seen to organize the CA's state space and to dominate its evolution. Starting from the domains' structures, we show how to construct a finitestate transducer that performs nonlinear spatial filtering such that the resulting spacetime patterns reveal the domains and the intervening walls and dislocations. To show the statistical consequences of domain detection, we compare the entropy and complexity densities of each domain with the globally averaged quantities. A more graphical comparison uses difference patter...
Classifying Cellular Automata Automatically; Finding gliders, filtering, and relating spacetime patterns, attractor basins, and the Z parameter
 Complexity
, 1998
"... CA rules can be classied automatically for a spectrum of ordered, complex and chaotic dynamics, by a measure of the variance of inputentropy over time. Rules that support interacting gliders and related complex dynamics can be identied, giving an unlimited source for further study. The distribution ..."
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Cited by 45 (3 self)
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CA rules can be classied automatically for a spectrum of ordered, complex and chaotic dynamics, by a measure of the variance of inputentropy over time. Rules that support interacting gliders and related complex dynamics can be identied, giving an unlimited source for further study. The distribution of rule classes in rulespace can be shown. A byproduct of the method allows the automatic \ltering" of CA spacetime patterns to show up gliders and related emergent congurations more clearly. The classication seems to correspond to our subjective judgment of spacetime dynamics. There are also approximate correlations with global measures on convergence in attractor basins, characterized by the distribution of indegree sizes in their branching structure, and to the rule parameter, Z. Based on computer experiments using the software Discrete Dynamics Lab (DDLab)[22], this paper explains the methods and presents results for 1d CA. 1 Introduction Cellular automata (CA) are a much stud...
Computational Mechanics of Cellular Automata: An Example
, 1995
"... We illustrate and extend the techniques of computational mechanics in explicating the structures that emerge in the spacetime behavior of elementary onedimensional cellular automaton rule 54. The CA's dominant regular domain is identified and a domain filter is constructed to locate and classify d ..."
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Cited by 39 (4 self)
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We illustrate and extend the techniques of computational mechanics in explicating the structures that emerge in the spacetime behavior of elementary onedimensional cellular automaton rule 54. The CA's dominant regular domain is identified and a domain filter is constructed to locate and classify defects in the domain. The primary particles are identified and a range of interparticle interactions is studied. The deterministic equation of motion of the filtered spacetime behavior is derived. Filters of increasing sophistication are constructed for the efficient gathering of particle statistics and for the identification of higherlevel defects, particle interactions, and secondary domains. We define the emergence time at which the spacetime behavior condenses into configurations consisting only of domains, particles, and particle interactions. Taken together, these techniques serve as the basis for the investigation of pattern evolution and selforganization in this representative sys...
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 36 (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
Transition Phenomena in Cellular Automata Rule Space
 Physica D
, 1990
"... We define several qualitative classes of cellular automata (CA) behavior, based on various statistical measures, and describe how the space of all cellular automata is organized. As a cellular automaton... ..."
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Cited by 30 (7 self)
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We define several qualitative classes of cellular automata (CA) behavior, based on various statistical measures, and describe how the space of all cellular automata is organized. As a cellular automaton...
The Structure of the Elementary Cellular Automata Rule Space
 Complex Systems
, 1990
"... The structure of the elementary cellular automata rule space is investigated. The probabilities for a rule to be connected to other rules in the same class #intraclass#, as well as rules in di#erent classes #interclass#, are determined. The intraclass connection probabilities vary from around ..."
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Cited by 28 (6 self)
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The structure of the elementary cellular automata rule space is investigated. The probabilities for a rule to be connected to other rules in the same class #intraclass#, as well as rules in di#erent classes #interclass#, are determined. The intraclass connection probabilities vary from around 0.3 to 0.5, an indication of the strong tendency for rules with the similar behavior to be next to each other. Rules are also grouped according to the mean #eld descriptions. The mean#eld clusters are classi#ed into three classes #nonlinear, linear, and inversely linear# according to the #hot bits" in the rule table. It is shown that such classi#cation provides another easy way to describe the rule space.
Dynamics, computation, and the “edge of chaos”: A reexamination
 Complexity:Metaphors, Models, and Reality
, 1994
"... In this paper we review previous work and present new work concerning the relationship between dynamical systems theory and computation. In particular, we review work by Langton [21] and Packard [29] on the relationship between dynamical behavior and computational capability in cellular automata (CA ..."
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Cited by 25 (2 self)
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In this paper we review previous work and present new work concerning the relationship between dynamical systems theory and computation. In particular, we review work by Langton [21] and Packard [29] on the relationship between dynamical behavior and computational capability in cellular automata (CAs). We present results from an experiment similar to the one described by Packard [29], which was cited as evidence for the hypothesis that rules capable of performing complex computations are most likely to be found at a phase transition between ordered and chaotic behavioral regimes for CAs (the “edge of chaos”). Our experiment produced very different results from the original experiment, and we suggest that the interpretation of the original results is not correct. We conclude by discussing general issues related to dynamics, computation, and the “edge of chaos ” in cellular automata. 1
Coevolving NonUniform Cellular Automata to Perform Computations
, 1996
"... A major impediment of cellular automata (CA) stems from the difficulty of utilizing their complex behavior to perform useful computations. Recent studies by [ Packard, 1988, Mitchell et al., 1994b ] have shown that CAs can be evolved to perform a computational task. In this paper nonuniform CAs are ..."
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Cited by 25 (5 self)
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A major impediment of cellular automata (CA) stems from the difficulty of utilizing their complex behavior to perform useful computations. Recent studies by [ Packard, 1988, Mitchell et al., 1994b ] have shown that CAs can be evolved to perform a computational task. In this paper nonuniform CAs are studied, where each cell may contain a different rule, in contrast to the original, uniform model. We describe experiments in which nonuniform CAs are evolved to perform the computational task using a local, coevolutionary algorithm. For radius r = 3 we attain peak performance values of 0:92 comparable to those obtained for uniform CAs (0:93 \Gamma 0:95). This is notable considering the huge search spaces involved, much larger than the uniform case. Smaller radius CAs (previously unstudied in this context) attain performance values of 0:93 \Gamma 0:94. For r = 1 this is considerably higher than the maximal possible uniform CA performance of 0:83, suggesting that nonuniformity reduces con...