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
logic | category theory | type theory |
---|---|---|
true | terminal object/(-2)-truncated object | h-level 0-type/unit 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
</table>
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
Computational Reality_ (pdf)
Discussion in the sense of modal type theory where computation is exhibited by a monad (in computer science) is in
See also
Last revised on October 14, 2016 at 01:22:02. See the history of this page for a list of all contributions to it.