Why Are Proofs Relevant in Proof-Relevant Models?

Relational models of λ-calculus can be presented as type systems, the relational interpretation of a λ-term being given by the set of its typings. Within a distributors-induced bicategorical semantics generalizing the relational one, we identify the class of ‘categorified’ graph models and show that...

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Bibliographic Details
Published in:Proceedings of ACM on programming languages Vol. 7; no. POPL; pp. 218 - 248
Main Authors: Kerinec, Axel, Manzonetto, Giulio, Olimpieri, Federico
Format: Journal Article
Language:English
Published: New York, NY, USA ACM 09.01.2023
Subjects:
Categorical semantics
Lambda calculus
Theory of computation
Intersection Types
Approximation Theorem
Lambda calculus
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ISSN:2475-1421, 2475-1421
Online Access:Get full text
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Summary:Relational models of λ-calculus can be presented as type systems, the relational interpretation of a λ-term being given by the set of its typings. Within a distributors-induced bicategorical semantics generalizing the relational one, we identify the class of ‘categorified’ graph models and show that they can be presented as type systems as well. We prove that all the models living in this class satisfy an Approximation Theorem stating that the interpretation of a program corresponds to the filtered colimit of the denotations of its approximants. As in the relational case, the quantitative nature of our models allows to prove this property via a simple induction, rather than using impredicative techniques. Unlike relational models, our 2-dimensional graph models are also proof-relevant in the sense that the interpretation of a λ-term does not contain only its typings, but the whole type derivations. The additional information carried by a type derivation permits to reconstruct an approximant having the same type in the same environment. From this, we obtain the characterization of the theory induced by the categorified graph models as a simple corollary of the Approximation Theorem: two λ-terms have isomorphic interpretations exactly when their B'ohm trees coincide.
ISSN:2475-1421
2475-1421
DOI:10.1145/3571201