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>
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
Under the identifications
the following notions are equivalent:
A proof of a proposition. (In logic.)
A program with output some type. (In type theory and computer science.)
A generalized element of an object. (In category theory.)
This is referred to as “computational trinitarianism” in (Harper), where also an exposition is given.
The central dogma of computational trinitarianism holds that Logic, Languages, and Categories are but three manifestations of one divine notion of computation. There is no preferred route to enlightenment: each aspect provides insights that comprise the experience of computation in our lives.
Computational trinitarianism entails that any concept arising in one aspect should have meaning from the perspective of the other two. If you arrive at an insight that has importance for logic, languages, and categories, then you may feel sure that you have elucidated an essential concept of computation–you have made an enduring scientific discovery. (Harper)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
In the introduction of
the insight is recalled to have surfaced in the 1970s, with an early appearance in print being the monograph
Logic_, Cambridge Studies in Advanced Mathematics Vol. 7. Cambridge University Press, 1986.
See also at History of categorical semantics of linear type theory for more on this.
A exposition of the relation between the three concepts is in
Robert Harper, The Holy Trinity (2011) (web)
Dan Frumin, Computational trinitarianism, Feb 2014 (prezi slides)
An exposition with emphasis on linear logic/quantum logic and the relation to physics is in
A discussion in the context of homotopy type theory is in
For further references see at programs as proofs, propositions as types, and relation between category theory and type theory.
Last revised on February 13, 2018 at 05:16:06. See the history of this page for a list of all contributions to it.