Abstract
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. We extend our superposition calculus for fixed domains with syntactic disequality constraints in a non-equational setting. Based on this calculus, we present several new decidability results for validity in the minimal model of a satisfiable finitely saturated clause set that in particular extend the decidability results known for ARM (Atomic Representations of term Models) and DIG (Disjunctions of Implicit Generalizations) model representations.
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