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The topological entropy of cellular automata is uncomputable

Published online by Cambridge University Press:  19 September 2008

Lyman P. Hurd ,
Jarkko Kari  and
Karel Culik
Lyman P. Hurd
Affiliation:
Laboratory for Plasma Research, University of Maryland, College Park, MD 20742, USA
Jarkko Kari
Affiliation:
Department of Mathematics, University of Turku, 20500 Turku, Finland
Karel Culik
Affiliation:
Department of Computer Science, University of South Carolina, Columbia, SC 29208, USA
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Abstract

There is no algorithm which will take a description of a celluar automaton and determine whether it has zero topological entropy, or for any fixed ε > 0 compute its topological entropy to a tolerance e. Furthermore a set of aperiodic Wang tiles arising from Penrose's kite and dart tiles is used to demonstrate specific examples of cellular automata with a single periodic point but non-trivial non-wandering sets, which furthermore can be constructed to have arbitrarily high topological entropy.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 2 , June 1992 , pp. 255 - 265
Copyright
Copyright © Cambridge University Press 1992

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