Controlled integration of the cut rule into connection tableau calculi

Abstract

In this paper techniques are developed and compared that increase the inferential power of tableau systems for classical first-order logic. The mechanisms are formulated in the framework of connection tableaux, which is an amalgamation of the connection method and the tableau calculus, and a generalization of model elimination. Since connection tableau calculi are among the weakest proof systems with respect to proof compactness, and the (backward) cut rule is not suitable for the first-order case, we study alternative methods for shortening proofs. The techniques we investigate are the folding-up and the folding-down operations. Folding up represents an efficient way of supporting the basic calculus, which is top-down oriented, with lemmata derived in a bottom-up manner. It is shown that both techniques can also be viewed as controlled integrations of the cut rule. To remedy the additional redundancy imported into tableau proof procedures by the new inference rules, we develop and apply an extension of the regularity condition on tableaux and the mechanism of anti-lemmata which realizes a subsumption concept on tableaux. Using the framework of the theorem prover SETHEO, we have implemented three new proof procedures that overcome the deductive weakness of cut-free tableau systems. Experimental results demonstrate the superiority of the systems with folding up over the cut-free variant and the one with folding down.