Deciding whether a relation defined in Presburger logic can be defined in weaker logics

Abstract

Abstract. We consider logics on Z and N which are weaker than Presburger arithmetic and we settle the following decision problem: given a k-ary relation on Z and N which are first order definable in Presburger arithmetic, are they definable in these weaker logics? These logics, intuitively, are obtained by adding modulo and threshold counting predicates.