On Constructive Cut Admissibility in Deduction Modulo

Abstract

Abstract. Deduction modulo is a theoretical framework which allows the introduction of computational steps in deductive systems. This approach is well suited to automated theorem proving. We describe a proofsearch method based upon tableaux for Gentzen's intuitionistic LJ extended with rewrite rules on propositions and terms . We prove its completeness with respect to Kripke structures. Then we give a soundness proof with respect to cut-free LJ modulo. This yields a constructive proof of semantic cut elimination, which we use to characterize the relation between tableaux methods and cut elimination in the intuitionistic case.

Automating Theories in Intuitionistic Logic

“…ENAR [11], generalizing resolution, as well as Tamed [5], a tableau method. These methods are complete only if the sequent calculus modulo admits cut.…”

“…Proof search methods based on deduction modulo, e.g. ENAR [11] and TaMed [5] can then be used to find proofs in those theories. These methods are complete only if the sequent calculus modulo the compatible rewrite system admits cut.…”

A Semantic Completeness Proof for TaMeD

“…The workaround should be a Skolem theorem for cut-free sequent calculus modulo, that is yet to be investigated. The link between ground tableaux and a constructive cut elimination theorem is well-known in classical logic, and studied in the intuitionistic frame in [3].…”

“…We proved this for an order condition and a positivity condition. In [3,15] some more conditions are studied from the point of view of semantic completeness (of cut-free sequent calculus and intuitionistic tableaux) such as a mix of the two previous conditions or the formulation of HOL in first-order logic modulo given in [10]. Those results should be easily extendable to the study of tableau completeness, since we already have a partial valuation.…”

Resolution is Cut-Free

“…Deduction modulo also allows a unified treatment from a theoretical point of view (no more axioms and axiomatic cuts) and from a practical one : the resolution method of [10] presented here, as well as the tableau methods of [4,5], apply to any set of rewrite rules. The main novelty of deduction modulo is the ability to rewrite atomic propositions:…”