Finite Instantiations for Integer Difference Logic

The last few years have seen the advent of a new breed of decision procedures for various fragments of first-order logic based on propositional abstraction. A lazy satisfiability checker for a given fragment of first-order logic invokes a theory-specific decision procedure (a theory solver) on (partial) satisfying assignments for the abstraction. If the assignment is found to be consistent in the given theory, then a model for the original formula has been found. Otherwise, a refinement of the propositional abstraction is extracted from the proof of inconsistency and the search is resumed. We describe a theory solver for integer difference logic that is effective when the formula to be decided contains equality and disequality (negated equality) constraints so that the decision problem partakes of the nature of the pigeonhole problem. We propose a reduction of the problem to propositional satisfiability by computing bounds on a sufficient subset of solutions, and present experimental evidence for the efficiency of this approach