nLab
data type

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Contents

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
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
coinductionlimitcoinductive 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, (idempotent) 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

The concept of type in computer science

References

  • Jim Morris, Types are not sets, Proceedings of the 1st annual ACM SIGACT-SIGPLAN symposium on Principles of programming languages (POPL), 1973. (doi)

  • John C. Reynolds, Towards a Theory of Type Structure, Colloquium on Programming, Paris, 9-11 April 1974. (pdf)

  • Barbara Liskov and Stephen Zilles, Programming with Abstract Data Types, Proceedings of the ACM SIGPLAN symposium on Very high level languages, pp. 50-59, 1974. (doi) (citeseer)

  • Arthur Sale, Primitive data types, The Australian Computer Journal, Vol. 9, No. 2, July 1977 (pdf)

Last revised on April 29, 2020 at 05:13:11. See the history of this page for a list of all contributions to it.

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