Skip to Main content Skip to Navigation
Preprints, Working Papers, ...

A completeness theorem for strong normalization in minimal deduction modulo

Denis Cousineau 1, 2
2 TYPICAL - Types, Logic and computing
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France
Abstract : 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.
Document type :
Preprints, Working Papers, ...
Complete list of metadata

Cited literature [16 references]  Display  Hide  Download

https://hal.inria.fr/inria-00370379
Contributor : Denis Cousineau <>
Submitted on : Sunday, May 10, 2009 - 3:58:41 PM
Last modification on : Thursday, March 5, 2020 - 6:26:31 PM
Long-term archiving on: : Saturday, November 26, 2016 - 8:39:30 AM

File

CompletenessMDM.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : inria-00370379, version 2

Collections

Citation

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

Share

Metrics

Record views

341

Files downloads

190