nLab
intensional type theory

Context

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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty 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
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

homotopy levels

semantics

Contents

Idea

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.

Properties

Decidability

Only the intensional but not the extensional Martin-Löf type theory is decidable. (Martin-Löf, Hofmann).

Examples

  • CiC?

References

  • 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)

Revised on November 30, 2012 01:06:32 by Urs Schreiber (82.169.65.155)