Cobham’s theorem seen through Büchi’s theorem

Abstract

Cobham’s Theorem says that for k and l multiplicatively independent (i.e. for any nonzero integers r and s we have kr ≠ ls), a subset of IN which is k-and/-recognizable is recognizable. Here we give a new proof of this result using a combinatorial property of subsets of IN which are not first-order definable in Presburger Arithmetic (i.e. which are not ultimately periodic). The crucial lemma shows that an L ⊆ IN is first-order definable in Presburger Arithmetic iff any subset of IN first-order definable in < IN, +, L > is non-expanding (i.e. the distance between two consecutive elements is bounded).