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

  • Authors:

  • Anne Lacroix;Narad Rampersad;Michel Rigo;É/lise Vandomme

  • Affiliations:

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

  • Venue:
  • Fundamenta Informaticae - Words, Graphs, Automata, and Languages; Special Issue Honoring the 60th Birthday of Professor Tero Harju
  • Year:
  • 2012

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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.