Combining Satisfiability Procedures for Unions of Theories with a Shared Counting Operator

We present some decidability results for the universal fragment of theories modeling data structures and endowed with arithmetic constraints. More precisely, all the theories taken into account extend a theory that constrains the function symbol for the successor. A general decision procedure is obtained, by devising an appropriate calculus based on superposition. Moreover, we derive a decidability result for the combination of the considered theories for data structures and some fragments of arithmetic by applying a general combination schema for theories sharing a common subtheory. The effectiveness of the resulting algorithm is ensured by using the proposed calculus and a careful adaptation of standard methods for reasoning about arithmetic, such as Gauss elimination, Fourier-Motzkin elimination and Groebner bases computation.