A Semantic Completeness Proof for TaMeD

Abstract

Abstract. Deduction modulo is a theoretical framework designed to introduce computational steps in deductive systems. This approach is well suited to automated theorem proving and a tableau method for firstorder classical deduction modulo has been developed. We reformulate this method and give an (almost constructive) semantic completeness proof. This new proof allows us to extend the completeness theorem to several classes of rewrite systems used for computations in deduction modulo. We are then able to build a counter-model when a proof fails for these systems.

Automated Deduction in the B Set Theory using Typed Proof Search and Deduction Modulo

“…In fact, elaborating a theory modulo does not only consist in turning axioms into rewrite rules, but we must be careful to keep cut-free completeness, especially when this theory is the heart of a proof search method. The problem of cut elimination is known to be very difficult in deduction modulo [8], and we therefore leave these questions for future work.…”

First-Order Automated Reasoning with Theories: When Deduction Modulo Theory Meets Practice

“…-The Lindenbaum construction is too weak to show cut-free completeness: a cut is required to show transitivity of the order relation. To solve this, we can either refine the Gödel-Henkin approach, which results in tableau-like completeness proofs [31], or define the algebra differently, along the lines of Maehara [80], Okada [88], and Lipton and DeMarco [79]. Sketching such proofs, which can be found in [31,32] and depend on the rewrite system, is beyond the scope of this paper.…”

Resolution is Cut-Free

“…A natural deduction modulo [15], a sequent calculus modulo [10], or even a tableau method modulo [4,5,7], may be defined in a natural way, in intuitionistic or classical settings. In Section 2 below we present the version of classical sequent calculus we will use.…”

“…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:…”