Presburger arithmetic and recognizability of sets of natural numbers by automata: New proofs of Cobham's and Semenov's theorems

Abstract

Let N be the set of nonnegative integers. We show the two following facts about Presburger's arithmetic: 1. 1. Let L ⊆ N . If L is not definable in 〈 N , +〉 then there is an L′ ⊆ N definable in 〈 N,+,L〉 , such that there is no bound on the distance between two consecutive elements of L′. (Actually we give in Theorem 3.7 two explicit sets one of which can be chosen to be L′) and 2. 2. L ⊆ N n is definable in 〈 N , +〉 if and only if every subset of N which is definable in 〈 N, +, L〉 is definable in 〈 N , +〉. (Theorem 5.1) These two Theorems are of independent interest but we will get from them new proofs of Cobham's and Semenov's Theorems (Cobham's Theorem being the case n = 1 of Semenov's Theorem); Semenov's Theorem is: Let k and l be multiplicatively independent (i.e have no nondashtrivial common power). If L ⊆ N n is definable in 〈 N, +, V k〉 and in 〈 N, +,V l〉 then L is recognizable (i.e definable in 〈 N , +〉). Here V m is the function which sends a nonzero natural number to the greatest power of m dividing it.