Response Curves for Cellular Automata in One and Two Dimensions: An Example of Rigorous Calculations

Response Curves for Cellular Automata in One and Two Dimensions: An Example of Rigorous Calculations

Henryk Fuks (Brock University, Canada) and Andrew Skelton (Brock University, Canada)
Copyright: © 2012 |Pages: 15
DOI: 10.4018/978-1-4666-1574-8.ch016

Abstract

In this paper, the authors consider the problem of computing a response curve for binary cellular automata, that is, the curve describing the dependence of the density of ones after many iterations of the rule on the initial density of ones. The authors demonstrate how this problem could be approached using rule 130 as an example. For this rule, preimage sets of finite strings exhibit recognizable patterns; therefore, it is possible to compute both cardinalities of preimages of certain finite strings and probabilities of occurrence of these strings in a configuration obtained by iterating a random initial configuration n times. Response curves can be rigorously calculated in both one- and two-dimensional versions of CA rule 130. The authors also discuss a special case of totally disordered initial configurations, that is, random configurations where the density of ones and zeros are equal to 1/2.
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Basic Definitions

Let 978-1-4666-1574-8.ch016.m06 be called a symbol set, and let 978-1-4666-1574-8.ch016.m07 be the set of all bisequences over 978-1-4666-1574-8.ch016.m08, where by a bisequence we mean a function on 978-1-4666-1574-8.ch016.m09 to 978-1-4666-1574-8.ch016.m10. Throughout the remainder of this text the configuration space 978-1-4666-1574-8.ch016.m11 will be simply denoted by 978-1-4666-1574-8.ch016.m12.

A block of length978-1-4666-1574-8.ch016.m13 is an ordered set 978-1-4666-1574-8.ch016.m14, where 978-1-4666-1574-8.ch016.m15, 978-1-4666-1574-8.ch016.m16. Let 978-1-4666-1574-8.ch016.m17 and let 978-1-4666-1574-8.ch016.m18 denote the set of all blocks of length 978-1-4666-1574-8.ch016.m19 over 978-1-4666-1574-8.ch016.m20 and 978-1-4666-1574-8.ch016.m21 be the set of all finite blocks over 978-1-4666-1574-8.ch016.m22.

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