nLab 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

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) 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 basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms



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)
  • 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 July 29, 2023 at 15:42:29. See the history of this page for a list of all contributions to it.