Linking Focusing and Resolution with Selection

Guillaume Burel (1, 2, 3)

1 ENSIIE - Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise
2 METHODES-SAMOVAR - Méthodes et modèles pour les réseaux
3 DEDUCTEAM - Deduction modulo, interopérabilité et démonstration automatique

Focusing and selection are techniques that shrink the proof search space for respectively sequent calculi and resolution. To bring out a link between them, we generalize them both: we introduce a sequent calculus where each occurrence of an atom can have a positive or a negative polarity; and a resolution method where each literal, whatever its sign, can be selected. We prove the equivalence between cut-free proofs in this sequent calculus and derivations of the empty clause in that resolution method. Such a generalization is naturally not semi-complete in general; we present three complete instances: first, our framework allows us to show that usual focusing corresponds to hyperresolution and semantic resolution; the second instance is deduction modulo theory; and a new setting extends deduction modulo theory with rewriting rules having several left-hand sides, therefore restricting even more the proof search space.

Mots clés: proof theory; sequent calculus; refinements of resolution; deduction modulo theory; polarization

Domaines: Logique en informatique [cs.LO]

Guillaume Burel. Linking Focusing and Resolution with Selection. 2017. ⟨hal-01670476v1⟩