Regular Derivations in Basic Superposition-Based Calculi

We prove the completeness of the regular strategy of derivations for superposition-based calculi. The regular strategy was pioneered by Kanger in [Kan63], who proposed that all equality inferences take place before all other steps in the proof. We show that the strategy is complete with the elimination of tautologies. The implication of our result is the completeness of non-standard selection functions by which in non-relational clauses only equality literals (and all of them) are selected.

Mechanising first-order temporal resolution

First-order temporal logic is a concise and powerful notation, with many potential applications in both Computer Science and Artificial Intelligence. While the full logic is highly complex, recent work on monodic first-order temporal logics has identified important enumerable and even decidable fragments. Although a complete and correct resolution-style calculus has already been suggested for this specific fragment, this calculus involves constructions too complex to be of practical value. In this paper, we develop a machine-oriented clausal resolution method which features radically simplified proof search. We first define a normal form for monodic formulae and then introduce a novel resolution calculus that can be applied to formulae in this normal form. By careful encoding, parts of the calculus can be implemented using classical first-order resolution and can, thus, be efficiently implemented. We prove correctness and completeness results for the calculus and illustrate it on a comprehensive example. An implementation of the method is briefly discussed.