A Semantic Completeness Proof for TaMeD

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 first-order 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.

[1]  Olivier Hermant.  Méthodes sémantiques en déduction modulo , 2005 .

[2]  Melvin Fitting,et al.  First-Order Logic and Automated Theorem Proving , 1990, Graduate Texts in Computer Science.

[3]  Domenico Cantone,et al.  A Sound Framework for δ-Rule Variants in Free-Variable Semantic Tableaux , 2007, Journal of Automated Reasoning.

[4]  Olivier Hermant,et al.  Semantic Cut Elimination in the Intuitionistic Sequent Calculus , 2005, TLCA.

[5]  Eric Deplagne.  Sequent Calculus Viewed Modulo , 2000 .

[6]  Peter H. Schmitt,et al.  The liberalized δ-rule in free variable semantic tableaux , 2004, Journal of Automated Reasoning.

[7]  Martin Giese,et al.  Incremental Closure of Free Variable Tableaux , 2001, IJCAR.

[8]  Claude Kirchner,et al.  Theorem Proving Modulo , 2003, Journal of Automated Reasoning.

[9]  Jean-Louis Krivine.  Une preuve formelle et intuitionniste du théorème de complétude de la logique classique , 1996, Bull. Symb. Log..

[10]  Gilles Dowek,et al.  Proof normalization modulo , 2003, Journal of Symbolic Logic.

[11]  M. Fitting.  First-order logic and automated theorem proving (2nd ed.) , 1996 .

[12]  Andrei Voronkov,et al.  What You Always Wanted to Know about Rigid E-Unification , 1996, Journal of Automated Reasoning.

[13]  Bernhard Beckert,et al.  The Even More Liberalized delta-Rule in Free Variable Semantic Tableaux , 1993, Kurt Gödel Colloquium.

[14]  Andrei Voronkov,et al.  Equality Reasoning in Sequent-Based Calculi , 2001, Handbook of Automated Reasoning.

[15]  Richard Bonichon,et al.  On Constructive Cut Admissibility in Deduction Modulo , 2006, TYPES.

[16]  Richard Bonichon,et al.  TaMeD: A Tableau Method for Deduction Modulo , 2004, IJCAR.

[17]  Andrei Voronkov,et al.  The Undecidability of Simultaneous Rigid E-Unification , 1996, Theor. Comput. Sci..

[18]  Jürgen Stuber.  A Model-Based Completeness Proof of Extended Narrowing and Resolution , 2001, IJCAR.

[19]  Gilles Dowek,et al.  Arithmetic as a Theory Modulo , 2005, RTA.