Cellular automata, \(\omega{} \omega\)-regular sets, and sofic systems (Q1179180)

Several mechanisms for defining sets of biinfinite words (\(\omega\omega\)- languages) namely \(\omega\omega\)-finite automata, adherences of regular languages and sofic systems are investigated and compared. It is shown that \(C\subseteq S^ Z\) is a sofic system iff \(C\) is the adherence of a regular language, iff \(C\) is a topologically closed \(\omega\omega\)- regular set. This paper also gives another complete proof of the closure of the \(\omega\omega\)-regular sets under \(\omega\omega\)-rational relations and under Boolean operations. Finally, Hurd's conjecture on bi- extensible subsets of languages is disproved.