TaMeD: A Tableau Method for Deduction Modulo

Abstract

Abstract. Deduction modulo is a formalism introduced to separate cleanly computations and deductions by reasoning modulo a congruence on propositions. A sequent calculus modulo has been defined by Dowek, Hardin and Kirchner as well as a resolution-based proof search method called Extended Narrowing And Resolution (ENAR), in which the congruences are handled through rewrite rules on terms and atomic propositions. We define a tableau-based proof search method, called Tableau Method for Deduction modulo (TaMeD), for theorem proving modulo. We then give a syntactic proof of the completeness of the method with respect to provability in the sequent calculus modulo. Moreover, we follow in our proofs the same steps as the ENAR method in such a way that it allows to try and compare the characteristics of both methods.

A Semantic Completeness Proof for TaMeD

“…We introduce in this section a new version of the tableaux modulo of [2]. We use constrained a form of constrained tableaux which borrows ideas from those of [5,6,7,13].…”

“…The need for a model construction forced us to give this proof-search algorithm. This is an improvement over [2]. -Section 4 eventually presents this new semantic completeness proof.…”

“…One framework to handle such rewrite rules is deduction modulo [9]. The axiom ∀x ∀y (x * y = 0 ⇐⇒ (x = 0 ∨ y = 0)) yields by orientation the rewrite rule x * y = 0 → x = 0 ∨ y = 0 , which is useful to prove ∃z(a * a = z ⇒ a = z) by adapted automated deduction methods (see [2,9]) which handle propositional rewriting by extended narrowing.…”

“…In [2], a syntactic completeness proof of a tableau method for deduction modulo (TaMeD) is given. Our semantic completeness proof sheds.…”

“…The use of general semantic methods allows us to precisely give some categories of rewrite systems for which the completeness theorem holds. It is an improvement over both [2,9] and [19]. The completeness proofs of the first two papers assume that certain properties such as cut elimination in the sequent calculus modulo are enjoyed by the considered rewrite systems, without describing any possible candidate.…”