An Automata Theoretic Approach to the Zero-One Law for Regular Languages: Algorithmic and Logical Aspects

Ryoma Sin'ya
(Tokyo Institute of Technology)

A zero-one language L is a regular language whose asymptotic probability converges to either zero or one. In this case, we say that L obeys the zero-one law. We prove that a regular language obeys the zero-one law if and only if its syntactic monoid has a zero element, by means of Eilenberg's variety theoretic approach. Our proof gives an effective automata characterisation of the zero-one law for regular languages, and it leads to a linear time algorithm for testing whether a given regular language is zero-one. In addition, we discuss the logical aspects of the zero-one law for regular languages.

In Javier Esparza and Enrico Tronci: Proceedings Sixth International Symposium on Games, Automata, Logics and Formal Verification (GandALF 2015), Genoa, Italy, 21-22nd September 2015, Electronic Proceedings in Theoretical Computer Science 193, pp. 172–185.
Published: 23rd September 2015.

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