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

TL;DR

This paper investigates the long-term behavior of cellular automata under probability measures, characterizing persistent words and analyzing the computational complexity of their associated languages, revealing non-recursive and non-enumerable properties.

Contribution

It provides a characterization of persistent languages for certain cellular automata and demonstrates the non-recursive and non-enumerable nature of their limit sets.

Findings

Persistent languages can be non-recursive.

Set of quasi-nilpotent automata is neither recursively nor co-recursively enumerable.

Characterization of persistent words for automata with Bernoulli 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.