Syntactic Complexity of Ultimately Periodic Sets of Integers and Application to a Decision Procedure

Issue title: Words, Graphs, Automata, and Languages; Special Issue Honoring the 60th Birthday of Professor Tero Harju

Article type: Research Article

Authors: Lacroix, Anne | Rampersad, Narad | Rigo, Michel | Vandomme, Élise

Affiliations: Institute of Mathematics, University of Liège, Grande Traverse 12 (B 37), B-4000 Liège, Belgium, {a.lacroix,e.vandomme,M.Rigo}@ulg.ac.be; narad.rampersad@gmail.com

Note: [] Address for correspondence: Institute of Mathematics, University of Liège, Grande Traverse 12 (B 37), B-4000 Liège, Belgium

Abstract: We compute the cardinality of the syntactic monoid of the language 0* repb(m$\mathbb{N}$) made of base b expansions of the multiples of the integer m. We also give lower bounds for the syntactic complexity of any (ultimately) periodic set of integers written in base b. We apply our results to a well studied problem: decide whether or not a b-recognizable set of integers is ultimately periodic.

DOI: 10.3233/FI-2012-677

Journal: Fundamenta Informaticae, vol. 116, no. 1-4, pp. 175-187, 2012