Dialectica Logical Principles

TL;DR

This paper provides a categorical framework for G"odel's Dialectica interpretation, linking logical principles with category theory to facilitate proof analysis and extraction of computational content.

Contribution

It offers a hyperdoctrine characterization of Dialectica, connecting it precisely to its logical description and extending its categorical understanding.

Findings

Derived the soundness of implication in the categorical setting

Established a tight correspondence between logic and category theory for Dialectica

Extended the interpretation to include principles beyond intuitionistic logic

Abstract

G\"odel's Dialectica interpretation was designed to obtain a relative consistency proof for Heyting arithmetic, to be used in conjunction with the double negation interpretation to obtain the consistency of Peano arithmetic. In recent years, proof theoretic transformations (so-called proof interpretations) that are based on G\"odel's Dialectica interpretation have been used systematically to extract new content from proofs and so the interpretation has found relevant applications in several areas of mathematics and computer science. Following our previous work on G\"odel fibrations, we present a (hyper)doctrine characterisation of the Dialectica which corresponds exactly to the logical description of the interpretation. To show that we derive in the category theory the soundness of the interpretation of the implication connective, as expounded on by Spector and Troelstra. This requires extra logical principles, going beyond intuitionistic logic, Markov's Principle (MP) and the Independence of Premise (IP) principle, as well as some choice. We show how these principles are satisfied in the categorical setting, establishing a tight (internal language) correspondence between the logical system and the categorical framework. This tight correspondence should come handy not only when discussing the applications of the Dialectica already known, like its use to extract computational content from (some) classical theorems (proof mining), its use to help to model specific abstract machines, etc. but also to help devise new applications.