constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
basic constructions:
strong axioms
further
The term computational type theory has been used
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).
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)
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
Discussion specifically in the context of NuPRL is in
Discussion in the sense of modal type theory where computation is exhibited by a monad (in computer science) is in
See also