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Dynamics, Computation, and the \Edge of Chaos": a Re-Examination

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Dynamics, Computation, and the \Edge of Chaos Dynamics Computation and the Edge of Chaos A ReExamination Melanie Mitchell James PCrutcheld and Peter T Hrab er Santa Fe Institute Working Pap er To app ear in G Cowan D Pines and D Melzner editors Integrative Themes Santa Fe Institute Stuides in the Sciences of Complexity Pro ceedings Volume Reading MA AddisonWesley Abstract In this pap er we review previous work and present new work concerning the relationship b e tween dynamical systems theory and computation In particular we review work by Langton and Packard on the relationship b etween dynamical b ehavior and computational capability in cellular automata CAs We present results from an exp eriment similar to the one describ ed byPackard whichwas cited as evidence for the hyp othesis that rules capable of p erforming complex computations are most likely to b e found at a phase tran sition b etween ordered and chaotic b ehavioral regimes for CAs the edge of chaos Our exp eriment pro duced very dierent results from the original exp eriment and we suggest that the interpretation of the original results is not correct Weconcludeby discussing general issues related to dynamics computation and the edge of chaos in cellular automata Intro duction A central goal of the sciences of complex systems is to understand the laws and mechanisms by which complicated coherent global b ehavior can emerge from the collective activities of relatively simple lo cally interacting comp onents Given the diversity of systems falling into this broad class the discovery of any commonalities or universal laws underlying such sys tems will require very general theoretical frameworks Twosuch frameworks are dynamical systems theory and the theory of computation These have indep endently provided p ower ful to ols for understanding and describing common prop erties of a wide range of complex systems Dynamical systems theory has develop ed as one of the main alternatives to analytic closedform exact solutions of complex systems Typically a system is considered to b e solved when one can write down a nite set of nite expressions that can b e used to predict the state of the system at time tgiven the state of the system at some initial time t Using existing mathematical metho ds such solutions are generally not p ossible for most complex systems of interest The central contribution of dynamical systems theory to Santa Fe Institute Old Pecos Trail Suite A Santa Fe New Mexico USA Email mmsantafeedu pthsantafeedu Physics Department University of California Berkeley CA USA Email chaosgo jirab erkeleyedu mo dern science is that exact solutions are not necessary for understanding and analyzing a nonlinear pro cess Instead of deriving exact single solutions or opting for coarse statistical descriptions the emphasis of dynamical systems theory is on describing the geometrical and top ological structure of ensembles of solutions In other words dynamical systems theory gives a geometric view of a pro cesss structural elements such as attractors basins and separatrices It is thus distinguished from a purely probabilistic approach such as statistical mechanics in which geometric structures are not considered Dynamical systems theory also addresses the question of what structures are generic that is what b ehavior typ es are typical across the sp ectrum of complex systems In contrast to fo cusing on how geometric structures are constrained in a state space computation theory fo cuses on how basic informationpro cessing elementsstorage logical gates stacks queues pro duction rules and the likecan b e combined to eect a given informationpro cessing task As such computation theory is a theory of organization and the functionality supp orted by organization When adapted to analyze complex systems it provides a framework for describing b ehaviors as computations of varying structure For example if the global mapping from initial to nal states is considered as a computation then the question is what function is b eing computed by the global dynamics Another range of examples concern limitations imp osed by the equations of motion on information pro cessing can a given complex system b e designed to emulate a universal Turing machine In contrast to this sort of engineering question one is also interested in the intrinsic computational capability of a given complex system that is what informationpro cessing structures are intrinsic in its b ehavior Dynamical systems theory and computation theory have historically b een applied in dep endently but there have b een some eorts to understand the relationship b etween the twothat is the relationship b etween a systems ability for information pro cessing and other measures of the systems dynamical b ehavior Relationships Between Dynamical Systems Theory and Computation Theory Computation theory develop ed from the attempt to understand informationpro cessing as p ects of systems A collo quial denition of information pro cessing might b e the trans formation of a given input to a desired output However in order to apply the notion of information pro cessing to complex systems and to relate it to dynamical systems theory the notion must b e enriched to include the production of information as well as its storage transmission and logical manipulation In addition the engineeringbased notion of desired output is not necessarily appropriate in this context the fo cus here is often on the intrin sic informationpro cessing capabilities of a dynamical system not sub ject to a particular computational goal Beginning with Kolmogorovs and Sinais adaptation of Shannons communication theory to mechanics in the late s there has b een a continuing eort to relate a nonlinear systems informationpro cessing capability and its temp oral b ehavior One result is that a deterministic chaotic system can b e viewed as a generator of information Another is that the complexity of predicting a chaotic systems b ehavior grows exp onentially with time The complexity metric here called the KolmogorovChaitin complexity uses a universal Turing machine as the deterministic prediction machine The relationship b etween the diculty of prediction and dynamical randomness is simply summarized by the statement that the growth rate of the descriptive complexity is equal to the information pro duction rate These results give a view of deterministic chaos that emphasizes the pro duction of randomness and the resulting unpredictability They are probably the earliest connections between dynamics and computation The question of what structures underlie information pro duction in dynamical systems has received attention only more recently The rst and crudest prop erty considered is the amount of memory a system employs in pro ducing apparent randomness The idea is that an ideal random pro cess uses no memory to pro duce its informationit simply ips a coin as needed Similarly a simple p erio dic pro cess requires memory only in prop ortion to the length of the pattern it rep eats Within the memorycapacity view of dynamics b oth these typ es of pro cesses are simplemore precisely they are simple to describ e statistically Between these extremes though lie the highly structured complex pro cesses that use b oth randomness and pattern storage to pro duce their b ehavior Such pro cesses are more complex to describ e statistically than are ideal random or simple p erio dic pro cesses The tradeo between structure and randomness is common to much of science The notion of statistical complexitywas intro duced to measure this tradeo Computation theory is concerned with more than information and its pro duction and storage These elements are taken as given and instead the fo cus is on how their combina tions yield more or less computational p ower Understandably there is a central dichotomy between machines with nite and innite memory On a ner scale distinctions can b e drawn among the ways in which innite memory is organizedeg as a stack a queue or a parallel array Given such considerations the question of the intrinsic computational struc ture in a dynamical system b ecomes substantially more demanding than the initial emphasis on gross measures of information storage and pro duction Several connections in this vein have b een made recently In the realm of continuousstate dynamical systems Crutcheld and Young lo oked at the relationship b etween the dynamics and computational structure of discrete time series generated by the logistic map at dierent parameter settings They found that at the onset of chaos there is an abrupt jump in computational class of the time series as measured by the formal language class required to describ e the time series In concert with Feigenbaums renormalization group analysis of the onset of chaos this result demonstrated that a dynamical systems computational capabilityin terms of the richness of b ehavior it pro ducesis qualitatively increased at a phase transition Rather than considering intrinsic computational structure a numb er of engineering suggestions have b een made that there exist physically plausible dynamical systems imple menting Turing machines These studies provided explicit constructions for several typ es of dynamical systems At this p oint it is unclear whether the resulting computational systems are genericie likely to b e constructible in other dynamical systemsor whether they are robust and reliable in information pro cessing In any case it is clear that much work has b een done to address a range of issues that relate continuousstate
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