Decomposition of Decidable First-Order Logics over Integers and Reals

We tackle the issue of representing infinite sets of real- valued vectors. This paper introduces an operator for combining integer and real sets. Using this operator, we decompose three well-known logics extending Presburger with reals. Our decomposition splits a logic into two parts : one integer, and one decimal (i.e. on the interval [0,1]). We also give a basis for an implementation of our representation.