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On the Complexity of Limit Sets of Cellular Automata Associated with Probability Measures

Abstract : We study the notion of limit sets of cellular automata associated with probability measures (mu-limit sets). This notion was introduced by P. Kurka and A. Maass. It is a refinement of the classical notion of omega-limit sets dealing with the typical long term behavior of cellular automata. It focuses on the words whose probability of appearance does not tend to 0 as time tends to infinity (the persistent words). In this paper, we give a characterisation of the persistent language for non sensible cellular automata associated with Bernouilli measures. We also study the computational complexity of these languages. We show that the persistent language can be non-recursive. But our main result is that the set of quasi-nilpotent cellular automata (those with a single configuration in their mu-limit set) is neither recursively enumerable nor co-recursively enumerable.
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https://hal.archives-ouvertes.fr/hal-00022186
Contributor : Guillaume Theyssier Connect in order to contact the contributor
Submitted on : Monday, October 2, 2006 - 10:42:30 AM
Last modification on : Thursday, October 7, 2021 - 3:04:32 AM
Long-term archiving on: : Monday, September 20, 2010 - 5:05:12 PM

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Laurent Boyer, Victor Poupet, Guillaume Theyssier. On the Complexity of Limit Sets of Cellular Automata Associated with Probability Measures. Slovak Society for Computer Science ; Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Aug 2006, Stará Lesná, pp.190-201. ⟨hal-00022186v2⟩

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