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单词 CombinatoryLogic
释义

combinatory logic


Combinatory logicMathworldPlanetmath was invented by Moses Schönfinkel in the early 1920s, and was mostly developed by Haskell Curry. The idea was to reduce the notation of logic to the simplest terms possible. As such, combinatory logic consists only of combinators, combinationMathworldPlanetmath operationsMathworldPlanetmath, and no free variablesMathworldPlanetmath.

A combinator is simply a function with no free variables. A free variable is any variable referred to in a function that is not a parameter of that function. The operation of combination is then simply the application of a combinator to its parameters. Combination is specified by simple juxtaposition of two terms, and is left-associative. Parentheses may also be present to override associativity. For example

fgxy=(fg)xy=((fg)x)y

All combinators in combinatory logic can be derived from two basic combinators, S and K. They are defined as

Sfgx=fx(gx)
Kxy=x

Reference is sometimes made to a third basic combinator, I, which can be defined in terms of S and K.

Ix=SKKx=x

Combinatory logic where I is considered to be derived from S and K is sometimes known as pure combinatory logic.

Combinatory logic and lambda calculusMathworldPlanetmath are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath. However, lambda calculus is more concise than combinatory logic; an expression of size 𝒪(n) in lambda calculus is equivalent to an expression of size 𝒪(n2) in combinatory logic.

For example, Sfgx=fx(gx) in combinatory logic is equivalent toS=(λf(λg(λx((fx)(gx))))), and Kxy=xis equivalent to K=(λx(λyx)).

Titlecombinatory logic
Canonical nameCombinatoryLogic
Date of creation2013-03-22 12:32:29
Last modified on2013-03-22 12:32:29
OwnerLogan (6)
Last modified byLogan (6)
Numerical id5
AuthorLogan (6)
Entry typeDefinition
Classificationmsc 03B40
Related topicLambdaCalculus
Related topicSubstitutability
Definescombinator
Definescombination
Definesfree variable
Definespure combinatory logic
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