nLab
computation

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Contents

Context

Constructivism, Realizability, Computability

constructive mathematics, realizability, computability

intuitionistic mathematics

propositions as types, proofs as programs, computational trinitarianism

Constructive mathematics

Realizability

Computability

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/path 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

Deduction and Induction

Foundations

foundations

The basis of it all

Foundational axioms

foundational axiom

Removing axioms

Contents

Idea

A program. A construction of a term of some type. The topic of computability theory.

computability

type I computabilitytype II computability
typical domainnatural numbers ℕ\mathbb{N}Baire space of infinite sequences 𝔹=ℕ ℕ\mathbb{B} = \mathbb{N}^{\mathbb{N}}
computable functionspartial recursive functioncomputable function (analysis)
type of computable mathematicsrecursive mathematicscomputable analysis, Type Two Theory of Effectivity
type of realizabilitynumber realizabilityfunction realizability
partial combinatory algebraKleene's first partial combinatory algebraKleene's second partial combinatory algebra

References

Last revised on March 2, 2014 at 03:50:45. See the history of this page for a list of all contributions to it.

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