Deciding whether a relation defined in Presburger logic can be defined in weaker logics
We consider logics on and which are weaker than Presburger arithmetic and we settle the following decision problem: given a k-ary relation on and which are first order definable in Presburger arithmetic, are they definable in these weaker logics? These logics, intuitively, are obtained by considering modulo and threshold counting predicates for differences of two variables.