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

TL;DR

This paper characterizes regular languages obeying the zero-one law using automata theory and algebraic structures, providing an effective linear-time algorithm for testing this property and exploring its logical implications.

Contribution

It offers a new automata-theoretic and algebraic characterization of zero-one languages, along with an efficient testing algorithm and insights into their logical aspects.

Findings

A regular language obeys the zero-one law iff its syntactic monoid has a zero element.

Provides a linear-time algorithm to test the zero-one law for regular languages.

Discusses the logical aspects of the zero-one law in the context of regular languages.

Abstract

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.