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>
Intensional type theory is the flavor of type theory in which identity types are not necessarily propositions (that is, (-1)-truncated). Martin-Löf‘s original definition of identity types, and the equivalent formulation as an inductive type, are by default intensional; one has to impose extra axioms or rules in order to get extensional type theory (in which identity types are propositions).
In particular, homotopy type theory is intensional, because identity types represent path objects.
Note that some type theorists use “intensional type theory” to refer to type theory which fails to satisfy function extensionality. This is in general an orthogonal requirement to how we are using the term here.
Only the intensional but not the extensional Martin-Löf type theory is decidable. (Martin-Löf, Hofmann).
Per Martin-Löf, An intuitionistic theory of types: predicative part, Logic Colloquium ‘73
(Amsterdam) (H. E. Rose and J. C. Shepherdson, eds.), North-Holland, 1975, pp. 73-118.
Martin Hofmann, Extensional concepts in intensional type theory, Ph.D. thesis, University of
Edinburgh, (1995) (web)
Last revised on November 30, 2012 at 01:06:32. See the history of this page for a list of all contributions to it.