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

## TL;DR

This paper presents a quasi-linear algorithm for deciding whether a $b$-recognisable set of integers, represented by a finite automaton in MSDF convention, is eventually periodic, extending previous work limited to LSF representation.

## Contribution

The paper introduces an efficient algorithm for determining periodicity of $b$-recognisable sets in MSDF notation, which was previously only addressed in LSF notation.

## Key findings

- Algorithm runs in quasi-linear time.
- Extends decidability results to MSDF digit order.
- Provides practical decision procedure for periodicity.

## Abstract

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 eventually 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 that this problem decidable in 1986 and recent developments give efficient decision algorithms. However, they only work when the integers are written with the least significant digit first. In this work, we consider the natural order of digits (Most Significant Digit First) and give a quasi-linear algorithm to solve the problem in this case.