computational type theory


Constructivism, Realizability, Computability

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type

falseinitial objectempty type

proposition(-1)-truncated objecth-proposition, mere proposition

proofgeneralized elementprogram

cut rulecomposition of classifying morphisms / pullback of display mapssubstitution

cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction

introduction rule for implicationunit for hom-tensor adjunctioneta conversion

logical conjunctionproductproduct type

disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)

implicationinternal homfunction type

negationinternal hom into initial objectfunction type into empty type

universal quantificationdependent productdependent product type

existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)

equivalencepath space objectidentity type

equivalence classquotientquotient type

inductioncolimitinductive type, W-type, M-type

higher inductionhigher colimithigher inductive type

completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set

setinternal 0-groupoidBishop set/setoid

universeobject classifiertype of types

modalityclosure operator, (idemponent) monadmodal type theory, monad (in computer science)

linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation

proof netstring diagramquantum circuit

(absence of) contraction rule(absence of) diagonalno-cloning theorem

synthetic mathematicsdomain specific embedded programming language


homotopy levels


Deduction and Induction




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.


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

See also

  • Fairouz Kamareddine, Twan Laan and Robert Constable (2012) Russell’s Orders in Kripke’s Theory of Truth and Computational Type Theory. In Dov. M Gabbay, Akihiro Kanamori and John Woods, (editors) Sets and Extensions in the Twentieth Century, 6, HHL, : San Diego: North Holland, 2012, pp. 801-845.

Last revised on October 14, 2016 at 01:22:02. See the history of this page for a list of all contributions to it.