St\ue5lmarck\u27s Method for Automated Theorem Proving in First Order Logic

We present an extension of St\ue5lmarck\u27s method to classical first order predicate logic. St\ue5lmarck\u27s method is a satisfiability checking method for propositional logic, and it resembles tableaux and KE. Its most distinctive feature is the dilemma rule, which is an extended branching rule, that allows branches to be recombined. This is done by keeping the intersection of all derived consequences in the two branches after the merge. St\ue5lmarck\u27s method for propositional logic is known to be sound and complete, and has turned out to be efficient in industrial applications.The extension of St\ue5lmarck\u27s method uses a tableaux-like calculus with both universal and rigid variables. We define a notion of logical intersections, in which the pairwise unifiable formulas from two sets of formulas are found. We also describe how rigid variables can be transformed into universal variables, when the two branches of a dilemma rule are merged. We define the proof system rigorously, and prove that it is sound. We also prove the specialized derivations theorem, which states that instantiating a rigid variable in all steps of a proof preserves proof correctness. We then outline a proof procedure, and discuss some of its features. A prototype implementing parts of the proof procedure has been made