Normalization by Completeness with Heyting Algebras

Gaëtan Gilbert (1, 2) , Olivier Hermant (3, 4, 1, 5)

1 DEDUCTEAM - Deduction modulo, interopérabilité et démonstration automatique
2 ENS de Lyon - École normale supérieure de Lyon
3 Department of Mathematics and Computer Science
4 Mines Paris - PSL (École nationale supérieure des mines de Paris)
5 CRI - Centre de Recherche en Informatique

Usual normalization by evaluation techniques have a strong relationship with completeness with respect to Kripke structures. But Kripke structures is not the only semantics that ts intuitionistic logic: Heyting algebras are a more algebraic alternative. In this paper, we focus on this less investigated area: how completeness with respect to Heyting algebras generate a normalization algorithm for a natural deduction calculus, in the propositional fragment. Our main contributions is that we prove in a direct way completeness of natural deduction with respect to Heyting algebras, that the underlying algorithm natively deals with disjunction, that we formalized those proofs in Coq, and give an extracted algorithm.

Domaines: Logique en informatique [cs.LO]

Gaëtan Gilbert, Olivier Hermant. Normalization by Completeness with Heyting Algebras. LPAR 20 : 20th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, Nov 2015, Suva, Fiji. ⟨hal-01204599⟩