Simpliication with Renaming: a General Proof Technique for Tableau and Sequent-based Provers

Tableau and sequent calculi are the basis for most popular interactive theorem provers for hardware and software veri cation. Yet, when it comes to decision procedures or automatic proof search, tableaux are orders of magnitude slower than Davis-Putnam, SAT based procedures or other techniques based on resolution. To meet this challenge, this paper proposes a theoretical innovation: the rule of simpli cation, which plays for tableaux the same role of subsumption for resolution and unit for Davis-Putnam. This technique gives an unifying view of a number of tableaux-like calculi such as DPLL, KE, HARP, hyper-tableaux etc. For instance, the stand-alone nature of the rst-order Davis-Putnam-Longeman-Loveland procedure can be explained away as a case of Smullyan tableau with propositional simpli cation. Beside its computational e ectiveness, the simplicity and generality of simpli cation make possible its extension in a uniform way. We de ne it for propositional and rst order logic and a wide range of modal logics. For a fulledged rst order simpli cation we combine it with another technique, renaming, which subsumes the use of free universal variables in sequent and tableau calculi. New experimental results are given for random SAT and the IFIP benchmarks for hardware veri cation. CONTENTS i