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Cobham's Theorem seen through Büchi's Theorem

  • Christian Michaux
  • Roger Villemaire
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 700)

Abstract

Cobham's Theorem says that for k and l multiplicatively independent (i.e. for any nonzero integers r and s we have κ r ≠ l3), a subset of ℕ which is κ- and l-recognizable is recognizable.

Here we give a new proof of this result using a combinatorial property of subsets of ℕ which are not first-order definable in Presburger Arithmetic (i.e. which are not ultimately periodic). The crucial lemma shows that an L ⊑ ℕ is first-order definable in Presburger Arithmetic iff any subset of ℕ first-order definable in<ℕ,+, L>is non-expanding (i.e. the distance between two consecutive elements is bounded).

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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Christian Michaux
    • 1
  • Roger Villemaire
    • 2
  1. 1.Faculté des SciencesUniversité de Mons-HainautMonsBelgium
  2. 2.Département de mathématiques et d'informatiqueUniversité du Québec à MontréalMontréal (Québec)Canada

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