A Theorem Prover for Equality using Lemmas

Equality handling has been traditionally one of the weaknesses of tableau-based theorem provers in general, from which those based on the Connection Tableau Calculus are not an exception. The Intermediate Lemma Extension is based on a simple extension to Logic Programming in which positive subgoals are skipped and all clauses treated as Horn clauses, allowing to derive clauses consisting of only positive literals, called lemmas. The purpose of the extension is to replace a large search space dedicated to find a deep, closed tableau by smaller search spaces that lead to a series of more shallow tableaux. Although its ultimate goal is proof finding, the lemma generation procedure shares some similarities with another automated deduction task, consequence finding. In this project we investigate the use of the Intermediate Lemma Extension for Connection Tableau Calculus and how this could help to reduce the search space usually associated with equality problems. Furthermore, we define a sound and complete extension of the calculus with a built-in equality reasoning mechanism based on RUE resolution [13], originally defined for Disconnection Tableau Calculus [28], eliminating the necessity of explicitly using the equality axioms. Additionally, we have reviewed the Local Failure Caching procedure, a sophisticated pruning technique originally proposed for the SOL Tableau Calculus for consequence finding and have analysed how languages defined for consequences, production fields, could be adapted to lemma derivation. Resulting from that further analysis we have proposed a sound and complete generalisation of the Intermediate Lemma Extension for not only positive subgoals that has turned out to achieve the best performance results.