An efficient algorithm to decide periodicity of b-recognisable sets using LSDF convention

Given an integer base b > 1, a set of integers is represented in base b by a language over {0, 1, ..., b − 1}. The set is said to be b-recognisable if its representation is a regular language. It is known that ultimately periodic sets are b-recognisable in every base b, and Cobham’s theorem implies the converse: no other set is b-recognisable in every base b. We are interested in deciding whether a b-recognisable set of integers (given as a finite automaton) is eventually periodic. Honkala showed in 1986 that this problem is decidable. Leroux used in 2005 the convention to write integers with the least significant digit first (LSDF), and designed a quadratic algorithm to solve a more general problem. We use here LSDF convention as well and give a structural description of the minimal automata that accept periodic sets of integers. We then show that it can be verified in linear time if a given minimal automaton meets this description. This yields a O(bn log(n)) procedure to decide whether a general deterministic automaton accepts an ultimately periodic set of numbers.