nLab
programming language

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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-level 1-type/h-prop
proofgeneralized elementprogram
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 monadmodal type theory, monad (in computer science)

homotopy levels

semantics

Category theory

category theory

Concepts

Universal constructions

Theorems

Extensions

Applications

Contents

Idea

A formal language (theory) for expressing programs. Used and studied in computer science.

Examples

Types of programming languages

Specific examples

References

category: software, applications

Revised on March 20, 2013 19:00:07 by Urs Schreiber (82.169.65.155)
Edit | Back in time (6 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: lambda-calculus, fixed-point combinator, functional programming, monad (in computer science), computation, geometry of physics, list of mathematics software, continuation-passing style