Zero-One Law for Regular Languages and Semigroups with Zero

TL;DR

This paper characterizes regular languages with the zero-one law, showing they are closed under Boolean operations, and provides algebraic and automata criteria, along with efficient algorithms for testing this property.

Contribution

It establishes a complete algebraic and automata-based characterization of zero-one law regular languages and introduces an efficient testing algorithm.

Findings

Zero-one languages are closed under Boolean operations and quotients.

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

An O(n log n) algorithm tests zero-one law for regular languages.

Abstract

A regular language has the zero-one law if its asymptotic density converges to either zero or one. We prove that the class of all zero-one languages is closed under Boolean operations and quotients. Moreover, we prove that a regular language has the zero-one law if and only if its syntactic monoid has a zero element. Our proof gives both algebraic and automata characterisation of the zero-one law for regular languages, and it leads the following two corollaries: (i) There is an O(n log n) algorithm for testing whether a given regular language has the zero-one law. (ii) The Boolean closure of existential first-order logic over finite words has the zero-one law.