Reduction rules and universal variables for first order tableaux and DPLL

Recent experimental results have shown that the strength of resolution, the propositional DPLL procedure, the KSAT procedure for description logics, or related tableau-like implementations such as DLP, is due to reduction rules which propagate constraints and prune the search space. Yet, for first order logic such reduction rules are only known for resolution. The lack of reduction rules for first order tableau calculi (DPLL can be seen as a tableau calculus with semantic branching) is one of the causes behind the lack of efficient first order DPLL-like procedures. The difficulty is that first order splitting rules force tableau and DPLL calculi to use rigid variables which hinder reduction rules such as unit subsumption or unit propagation. This paper shows how reduction and simplification rules can be lifted to a first order tableau-like calculus using universal variables and a new rule called renaming. It also shows how to optimize the calculus for exploiting universal variables and reduction rules when semantic branching aka split DPLL-style is chosen as the inference rule with DFS-search.