A Computational Interpretation of Forcing in Type Theory

Coquand, Thierry; Jaber, Guilhem

doi:10.1007/978-94-007-4435-6_10

Cited by 10 publications

(15 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If t1 computes to exca then this exception might get "caught" if t2 computes to a canonical expression "before" t1. Once we add non-determinism to Nuprl, we might be able to use non-deterministic computations to compute the modulus of continuity of functions in a similar fashion as done by Coquand and Jaber [31]. This is left for future work.…”

Section: Discussionmentioning

confidence: 99%

“…This is not the first (formal) proof that a type theory satisfies Brouwer's continuity principle. Coquand and Jaber [30,31] proved the uniform continuity of a Martin-Löf-like intensional type theory using forcing [9,11,24,25,61]. Their method consists in adding a generic element f as a constant to the language that stands for a Cohen real of type 2 N , and defining the forcing conditions as approximations of f, i.e., finite sub-graphs of f. They then define a suitable computability predicate that expresses when a term is a computable term of some type up to approximations given by the forcing conditions.…”

Section: Introductionmentioning

confidence: 99%

“…Escardó and Xu [39] showed that in the case of uniform continuity Σ can equivalently be Σ or ⇃Σ. In [31], Coquand and Jaber provide a Haskell realizer that computes the uniform modulus of continuity of a functional on the Cantor space 4 . Similarly, Escardó and Xu [89] proved that the definable functionals of Gödel's system T [46] are uniformly continuous on the Cantor space (without assuming classical logic or the Fan Theorem).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A nominal exploration of intuitionism

Rahli

Bickford

2016

Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs

View full text Add to dashboard Cite

This papers extends the Nuprl proof assistant (a system representative of the class of extensional type theoriesà la Martin-Löf) with named exceptions and handlers, as well as a nominal fresh operator. Using these new features, we prove a version of Brouwer's Continuity Principle for numbers. We also provide a simpler proof of a weaker version of this principle that only uses diverging terms. We prove these two principles in Nuprl's meta-theory using our formalization of Nuprl in Coq and show how we can reflect these metatheoretical results in the Nuprl theory as derivation rules. We also show that these additions preserve Nuprl's key meta-theoretical properties, in particular consistency and the congruence of Howe's computational equivalence relation. Using continuity and the fan theorem we prove important results of Intuitionistic Mathematics: Brouwer's continuity theorem and bar induction on monotone bars.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A nominal exploration of intuitionism

Rahli

Bickford

2016

Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs

View full text Add to dashboard Cite

show abstract

“…In [34], Coquand and Jaber provide a Haskell realizer that computes the uniform modulus of continuity of a functional on the Cantor space ¶ .…”

Section: Introductionmentioning

confidence: 99%

Validating Brouwer's continuity principle for numbers using named exceptions

Rahli

Bickford

2017

Math. Struct. Comp. Sci.

View full text Add to dashboard Cite

This paper extends the Nuprl proof assistant (a system representative of the class of extensional type theories with dependent types) with named exceptions and handlers, as well as a nominal fresh operator. Using these new features, we prove a version of Brouwer's Continuity Principle for numbers. We also provide a simpler proof of a weaker version of this principle that only uses diverging terms. We prove these two principles in Nuprl's metatheory using our formalization of Nuprl in Coq and reflect these metatheoretical results in the Nuprl theory as derivation rules. We also show that these additions preserve Nuprl's key metatheoretical properties, in particular consistency and the congruence of Howe's computational equivalence relation. Using continuity and the fan theorem we prove important results of Intuitionistic Mathematics: Brouwer's continuity theorem; bar induction on monotone bars; and the negation of the law of excluded middle.

show abstract

“…Our work is also related to Coquand and Jaber's forcing model [11,12], which instead uses the semilattice of finite binary sequences under the prefix order as the underlying category of the site, modelling the idea of a generic infinite binary sequence. They iterate their construction in order to be able to model the fan functional, and our model can be regarded as accomplishing this iteration directly in a single step (personal communication with Coquand).…”

Section: Introductionmentioning

confidence: 99%

A constructive manifestation of the Kleene–Kreisel continuous functionals

Escardó

2016

Annals of Pure and Applied Logic

View full text Add to dashboard Cite

We identify yet another category equivalent to that of Kleene-Kreisel continuous functionals. Reasoning constructively and predicatively, all functions from the Cantor space to the natural numbers are uniformly continuous in this category. We do not need to assume Brouwerian continuity axioms to prove this, but, if we do, then we can show that the full type hierarchy is equivalent to our manifestation of the continuous functionals. We construct this manifestation within a category of concrete sheaves, called C-spaces, which form a locally cartesian closed category, and hence can be used to model system T and dependent types. We show that this category has a fan functional and validates the uniform-continuity principle in these theories. Our development is within informal constructive mathematics, along the lines of Bishop mathematics. However, in order to extract concrete computational content from our constructions, we formalized it in intensional Martin-Löf type theory, in Agda notation, and we discuss the main technical aspects of this at the end of the paper.

show abstract

A Computational Interpretation of Forcing in Type Theory

Cited by 10 publications

References 12 publications

A nominal exploration of intuitionism

A nominal exploration of intuitionism

Validating Brouwer's continuity principle for numbers using named exceptions

A constructive manifestation of the Kleene–Kreisel continuous functionals

Contact Info

Product

Resources

About