ICALP 1993: Automata, Languages and Programming pp 325-334

# 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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