Axiom Directed Focusing

Abstract

Superdeduction and deduction modulo are two methods designed to ease the use of first-order theories in predicate logic. Superdeduction modulo combines both in a single framework. Although soundness is ensured, using superdeduction modulo to extend deduction with awkward theories can jeopardize cut-elimination or completeness w.r.t predicate logic. In this paper our aim is to design criteria for theories which will ensure these properties. We revisit the superdeduction paradigm by comparing it with the focusing approach. In particular we prove a focalization theorem for cut-free superdeduction modulo: we show that permutations of inference rules can transform any cut-free proof in deduction modulo into a cut-free proof in superdeduction modulo and conversely, provided that some hypotheses on the synchrony of reasoning axioms are verified. It implies that cut-elimination for deduction modulo and for superdeduction modulo are equivalent. Since several criteria have already been proposed for theories that do not break cut-elimination of the corresponding deduction modulo system, these criteria also imply cut-elimination of the superdeduction modulo system, provided our synchrony hypotheses hold. © 2009 Springer Berlin Heidelberg.