Ultimate periodicity of b-recognisable sets : a quasilinear procedure

TL;DR

This paper presents a linear-time structural characterization of minimal automata recognizing ultimately periodic sets of numbers, leading to an efficient O(n log n) decision procedure for their recognition.

Contribution

It provides a new structural description of minimal automata for ultimately periodic sets and an efficient algorithm to verify this property.

Findings

Structural description of minimal automata for periodic sets

Linear-time verification of automata meeting the description

O(n log n) decision procedure for automata acceptance

Abstract

It is decidable if a set of numbers, whose representation in a base b is a regular language, is ultimately periodic. This was established by Honkala in 1986. We give here a structural description of minimal automata that accept an ultimately periodic set of numbers. We then show that it can verified in linear time if a given minimal automaton meets this description. This thus yields a O(n log(n)) procedure for deciding whether a general deterministic automaton accepts an ultimately periodic set of numbers.