An Efficient Algorithm to Decide Periodicity of b-Recognisable Sets Using MSDF Convention

Authors Bernard Boigelot, Isabelle Mainz, Victor Marsault, Michel Rigo

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Bernard Boigelot
Isabelle Mainz
Victor Marsault
Michel Rigo

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Bernard Boigelot, Isabelle Mainz, Victor Marsault, and Michel Rigo. An Efficient Algorithm to Decide Periodicity of b-Recognisable Sets Using MSDF Convention. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 118:1-118:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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 is 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.
  • integer-base systems
  • automata
  • recognisable sets
  • periodic sets


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