MPI-I-2009-RG1-004

Saturation-based calculi such as superposition can be successfully instantiated to decision procedures for many decidable fragments of first-order logic. In case of termination without generating an empty clause, a saturated clause set implicitly represents a minimal model for all clauses, based on the underlying term ordering of the superposition calculus. In general, it is not decidable whether a ground atom, a clause or even a formula holds in this minimal model of a satisfiable saturated clause set. Based on an extension of our superposition calculus for fixed domains with syntactic disequality constraints in a non-equational setting, we describe models given by ARM (Atomic Representations of term Models) or DIG (Disjunctions of Implicit Generalizations) representations as minimal models of finite saturated clause sets. This allows us to present several new decidability results for validity in such models. These results extend in particular the known decidability results for ARM and DIG representations.