Deciding Effectively Propositional Logic with Equality

Effectively Propositional Logic (EPR), also known as the Bernays-Schoenfinkel class, allows encoding problems that are propositional in nature, but EPR encodings can be exponentially more succinct than purely propositional logic encodings. We recently developed a DPLL-based decision procedure that builds on top of efficient SAT solving techniques to handle the propositional case efficiently while maintaining the succinctness offered by the EPR representation. To achieve the effect, it uses sets of substitutions encoded as binary decision diagrams. It is possible to reduce EPR formulas with equality to pure EPR, but the reduction requires adding axioms for equality and congruence. This approach potentially increases the search space and could defeat the efficiency we are aiming to achieve. We here provide a calculus and decision procedure that handles equality natively. The procedure builds in equality propagation, and allows reducing dependencies on equalities during conflict resolution.