Typed realizability for first-order classical analysis

@article{Blot2015TypedRF,
  title={Typed realizability for first-order classical analysis},
  author={Valentin Blot},
  journal={Log. Methods Comput. Sci.},
  year={2015},
  volume={11}
}
  • Valentin Blot
  • Published 16 December 2015
  • Computer Science
  • Log. Methods Comput. Sci.
We describe a realizability framework for classical first-order logic in which realizers live in (a model of) typed {\lambda}{\mu}-calculus. This allows a direct interpretation of classical proofs, avoiding the usual negative translation to intuitionistic logic. We prove that the usual terms of G\"odel's system T realize the axioms of Peano arithmetic, and that under some assumptions on the computational model, the bar recursion operator realizes the axiom of dependent choice. We also perform a… 

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References

SHOWING 1-10 OF 51 REFERENCES

Game semantics and realizability for classical logic

This thesis investigates two realizability models for classical logic built on HO game semantics. The main motivation is to have a direct computational interpretation of classical logic, arithmetic

A Curry-Howard foundation for functional computation with control

The goal is that ¿µv and µPCFv respectively should be to functional computation with first-class access to the flow of control what ¿-calculus and PCF respectively are to pure functional programming.

A proof-theoretic foundation of abortive continuations

Abstract We give an analysis of various classical axioms and characterize a notion of minimal classical logic that enforces Peirce’s law without enforcing Ex Falso Quodlibet. We show that a “natural”

Lambda-Mu-Calculus: An Algorithmic Interpretation of Classical Natural Deduction

This paper presents a way of extending the paradigm "proofs as programs" to classical proofs, which can be seen as a simple extension of intuitionistic natural deduction, whose algorithmic interpretation is very well known.

A formulae-as-type notion of control

It is proved that all evaluations of typed terms in Idealized Scheme are finite, and the existence of computationally interesting “classical programs” is illustrated by the definition of conjunctively, disjunctive, and existential types using standard classical definitions.

Typed lambda-calculus in classical Zermelo-Frænkel set theory

A type system of simple types, which uses the intuitionistic propositional calculus, with the only connective →, and a system closely related to the latter, called the λc-calculus, which has the normalization property.

Existential witness extraction in classical realizability and via a negative translation

It is shown that in the Sigma^0_1-case, Krivine's witness extraction method reduces to Friedman's through a well-suited negative translation to intuitionistic second-order arithmetic.

The Peirce translation

A semantic view of classical proofs: type-theoretic, categorical, and denotational characterizations

  • C. Ong
  • Philosophy
    Proceedings 11th Annual IEEE Symposium on Logic in Computer Science
  • 1996
This extended abstract outlines a semantic theory of classical proofs based on a variant of Parigot's /spl lambda//spl mu/-calculus, but presented here as a type theory.
...