A completeness theorem for strong normalization in minimal deduction modulo

Denis Cousineau (1, 2)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 TYPICAL - Types, Logic and computing

Deduction modulo is an extension of first-order predicate logic where axioms are replaced by rewrite rules and where many theories, such as arithmetic, simple type theory and some variants of set theory, can be expressed. An important question in deduction modulo is to find a condition of the theories that have the strong normalization property. Dowek and Werner have given a semantic sufficient condition for a theory to have the strong normalization property: they have proved a ”soundness” theorem of the form: if a theory has a model (of a particular form) then it has the strong normalization property. In this paper, we refine their notion of model in a way allowing not only to prove soundness, but also completeness: if a theory has the strong normalization property, then it has a model of this form. The key idea of our model construction is a refinement of Girard's notion of reducibility candidates. By providing a sound and complete semantics for theories having the strong normalization property, this paper contributes to explore the idea that strong normalization is not only a proof-theoretic notion, but also a model-theoretic one.

Mots clés: Deduction modulo; rewriting; strong normalization; completeness

Domaines: Logique en informatique [cs.LO]

Denis Cousineau. A completeness theorem for strong normalization in minimal deduction modulo. 2009. ⟨inria-00370379v2⟩