# nLab computational type theory

Contents

### Context

#### Constructivism, Realizability, Computability

intuitionistic mathematics

foundations

# Contents

## Idea

The term computational type theory has been used

1. generally for intuitionistic type theory in view of its computational content via the propositions-as-types and proofs-as-programs interpretation (e.g. Constable 02, Scholarpedia).

2. more specifically for intuitionistic type theory with inductive types and here specifically for the dialect of the language which is implemented in the NuPRL software (Constable et al. 86, NuPRL 05);

Constable, p. 6: $[$computational type theory$]$ considerably extended Per Martin-Löf’s Intuitionistic Type Theory (ITT) adding set types, quotient types, recursive types, partial object types (bar types)

3. for modal type theory, specifically for type theory equipped with a monad (in computer science) that preserves finite products, which exhibits a kind of computation (Benton-Bierman-de Paiva 93, Fairtlough-Mendler 02).

The internal logic of computational type theory in this sense is also called propositional lax logic (Fairtlough-Mendler 97, Crolard) or computational logic.

## References

Discussion of intuitionistic/constructive type theory (with inductive types) as such referred to as computational type theory is in

• Robert Constable, Naïve Computational Type Theory, Proof and System-Reliability NATO Science Series Volume 62, 2002, pp 213-259

Discussion specifically in the context of NuPRL is in

• Robert Constable, Stuart F. Allen, H. M. Bromley, W. R. Cleaveland, J. F. Cremer, R. W. Harper, Douglas J. Howe, T. B. Knoblock, N. P. Mendler, P. Panangaden, James T. Sasaki, and Scott F. Smith. Implementing Mathematics with the Nuprl Proof Development System. Prentice-Hall, NJ, 1986.
• Innovations in Computational Type Theory using Nuprl (pdf)

Discussion in the sense of modal type theory where computation is exhibited by a monad (in computer science) is in

• Matt Fairtlough, Michael Mendler, Propositional Lax Logic, Volume 137, Issue 1, 25 August 1997, Pages 1–33 (pdf)
• P. Benton, G. Bierman, Valeria de Paiva, Computational types from a logical perspective, J. Functional Programming 1 (1) January 1993 (web, pdf)
• Matt Fairtlough, Michael Mendler, On the Logical Content of Computational Type Theory: A Solution to Curry’s Problem, Types for Proofs and Programs, Lecture Notes in Computer Science Volume 2277, 2002, pp 63-78 (pdf)
• Tristan Crolard, Monadic reflection in lax logic pdf