Homology, Homotopy and Applications

On the geometry of intuitionistic S4 proofs

Jean Goubault-Larrecq and Éric Goubault

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Abstract

The Curry-Howard correspondence between formulas and types, proofs and programs, proof simplification and program execution, also holds for intuitionistic modal logic S4. It turns out that the S4 modalities translate as a monoidal comonad on the space of proofs, giving rise to a canonical augmented simplicial structure. We study the geometry of these augmented simplicial sets, showing that each type gives rise to an augmented simplicial set which is a disjoint sum of nerves of finite lattices of points, plus isolated $(-1)$-dimensional subcomplexes. As an application, we give semantics of modal proofs (a.k.a., programs) in categories of augmented simplicial sets and of topological spaces, and prove a completeness result in the style of Friedman: if any two proofs have the same denotations in each augmented simplicial model, then they are convertible. This result rests both on the fine geometric structure of the constructed spaces of proofs and on properties of subscone categories--the categorical generalization of the notion of logical relations used in lambda-calculus.

Article information

Source
Homology Homotopy Appl., Volume 5, Number 2 (2003), 137-209.

Dates
First available in Project Euclid: 28 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.hha/1088453324

Mathematical Reviews number (MathSciNet)
MR1994943

Zentralblatt MATH identifier
1022.03010

Citation

Goubault-Larrecq, Jean; Goubault, Éric. On the geometry of intuitionistic S4 proofs. Homology Homotopy Appl. 5 (2003), no. 2, 137--209. https://projecteuclid.org/euclid.hha/1088453324


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