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diff --git a/lambda-calculus.html.markdown b/lambda-calculus.html.markdown index 6103c015..53a7a7cd 100644 --- a/lambda-calculus.html.markdown +++ b/lambda-calculus.html.markdown @@ -3,6 +3,7 @@ category: Algorithms & Data Structures name: Lambda Calculus contributors: - ["Max Sun", "http://github.com/maxsun"] + - ["Yan Hui Hang", "http://github.com/yanhh0"] --- # Lambda Calculus @@ -54,7 +55,7 @@ Although lambda calculus traditionally supports only single parameter functions, we can create multi-parameter functions using a technique called [currying](https://en.wikipedia.org/wiki/Currying). -- `(λx.λy.λz.xyz)` is equivalent to `f(x, y, z) = x(y(z))` +- `(λx.λy.λz.xyz)` is equivalent to `f(x, y, z) = ((x y) z)` Sometimes `λxy.<body>` is used interchangeably with: `λx.λy.<body>` @@ -84,9 +85,9 @@ Using `IF`, we can define the basic boolean logic operators: `a OR b` is equivalent to: `λab.IF a T b` -`a NOT b` is equivalent to: `λa.IF a F T` +`NOT a` is equivalent to: `λa.IF a F T` -*Note: `IF a b c` is essentially saying: `IF(a(b(c)))`* +*Note: `IF a b c` is essentially saying: `IF((a b) c)`* ## Numbers: @@ -110,12 +111,109 @@ we use the successor function `S(n) = n + 1` which is: Using successor, we can define add: -`ADD = λab.(a S)n` +`ADD = λab.(a S)b` **Challenge:** try defining your own multiplication function! +## Get even smaller: SKI, SK and Iota + +### SKI Combinator Calculus + +Let S, K, I be the following functions: + +`I x = x` + +`K x y = x` + +`S x y z = x z (y z)` + +We can convert an expression in the lambda calculus to an expression +in the SKI combinator calculus: + +1. `λx.x = I` +2. `λx.c = Kc` +3. `λx.(y z) = S (λx.y) (λx.z)` + +Take the church number 2 for example: + +`2 = λf.λx.f(f x)` + +For the inner part `λx.f(f x)`: + +``` + λx.f(f x) += S (λx.f) (λx.(f x)) (case 3) += S (K f) (S (λx.f) (λx.x)) (case 2, 3) += S (K f) (S (K f) I) (case 2, 1) +``` + +So: + +``` + 2 += λf.λx.f(f x) += λf.(S (K f) (S (K f) I)) += λf.((S (K f)) (S (K f) I)) += S (λf.(S (K f))) (λf.(S (K f) I)) (case 3) +``` + +For the first argument `λf.(S (K f))`: + +``` + λf.(S (K f)) += S (λf.S) (λf.(K f)) (case 3) += S (K S) (S (λf.K) (λf.f)) (case 2, 3) += S (K S) (S (K K) I) (case 2, 3) +``` + +For the second argument `λf.(S (K f) I)`: + +``` + λf.(S (K f) I) += λf.((S (K f)) I) += S (λf.(S (K f))) (λf.I) (case 3) += S (S (λf.S) (λf.(K f))) (K I) (case 2, 3) += S (S (K S) (S (λf.K) (λf.f))) (K I) (case 1, 3) += S (S (K S) (S (K K) I)) (K I) (case 1, 2) +``` + +Merging them up: + +``` + 2 += S (λf.(S (K f))) (λf.(S (K f) I)) += S (S (K S) (S (K K) I)) (S (S (K S) (S (K K) I)) (K I)) +``` + +Expanding this, we would end up with the same expression for the +church number 2 again. + +### SK Combinator Calculus + +The SKI combinator calculus can still be reduced further. We can +remove the I combinator by noting that `I = SKK`. We can substitute +all `I`'s with `SKK`. + +### Iota Combinator + +The SK combinator calculus is still not minimal. Defining: + +``` +ι = λf.((f S) K) +``` + +We have: + +``` +I = ιι +K = ι(ιI) = ι(ι(ιι)) +S = ι(K) = ι(ι(ι(ιι))) +``` + ## For more advanced reading: 1. [A Tutorial Introduction to the Lambda Calculus](http://www.inf.fu-berlin.de/lehre/WS03/alpi/lambda.pdf) 2. [Cornell CS 312 Recitation 26: The Lambda Calculus](http://www.cs.cornell.edu/courses/cs3110/2008fa/recitations/rec26.html) -3. [Wikipedia - Lambda Calculus](https://en.wikipedia.org/wiki/Lambda_calculus)
\ No newline at end of file +3. [Wikipedia - Lambda Calculus](https://en.wikipedia.org/wiki/Lambda_calculus) +4. [Wikipedia - SKI combinator calculus](https://en.wikipedia.org/wiki/SKI_combinator_calculus) +5. [Wikipedia - Iota and Jot](https://en.wikipedia.org/wiki/Iota_and_Jot) |