LAMBDA CALCULUS - cs.umd.edu · Review Ocaml Lambda Calculus fun x e //anonymous function applies...

LAMBDA CALCULUS

Review

Ocaml Lambda Calculus

fun x e //anonymous function applies parameter x

to expression e

λx.e

Review

Ocaml Lambda Calculus

fun x e //anonymous function applies parameter x

on expression e

let x = e1 in e2

λx.e

(λx.e2)e1 //evaluate e2 with x replaced by e1

Beta Reduction

Given: (λx.e2)e1 //evaluate e2 with x replaced by e1

(λx.x)z

Beta Reduction


(λx.x)z z

(λx.y)z

Beta Reduction


(λx.x)z z

(λx.y)z y

(λx.x y)z

Beta Reduction


(λx.x)z z

(λx.y)z y

(λx.x y)z z y

(λx.λz.x z)y

Beta Reduction


(λx.x)z z

(λx.y)z y

(λx.x y)z z y

(λx.λz.x z)y (λx.(λz.(x z)))y

e2 e1

Beta Reduction


(λx.x)z z

(λx.y)z y

(λx.x y)z z y

(λx.λz.x z)y (λx.(λz.(x z)))y λz.y z

e2 e1

Beta Reduction – More Practice


(λx.xy)(λz.z)

(λx.λy.xy)z

(λx.λy.xy)(λz.zz)x

Beta Reduction – More Practice


(λx.xy)(λz.z) (λz.z)y y

(λx.λy.xy)z λy.zy

(λx.λy.xy)(λz.zz)x (λy.(λz.zz)y)x

(λz.zz)x xx

Static Scoping

(λx.x(λx.x))z

Static Scoping & Alpha Conversion

(λx.x(λx.x))z (λx.x(λy.y))z



z(λy.y)



z(λy.y)

i.e. (λx.λy.xy)y



z(λy.y)

i.e. (λx.λy.xy)y (λx.λz.xz)y λz.yz

Lambda Calculus Encodings

Given: not = λx.((x false)true)

true= λx.λy.x

false= λx.λy.y

Prove: not(not true) = true



true= λx.λy.x

false= λx.λy.y


λx.((x false)true)(not true)



true= λx.λy.x

false= λx.λy.y



((not true) false) true



true= λx.λy.x

false= λx.λy.y




((λx.((x false) true) true)

false) true



true= λx.λy.x

false= λx.λy.y





false) true

(((true false) true) false)

true



true= λx.λy.x

false= λx.λy.y





false) true


true

((((λx.λy.x) false) true)

false) true



true= λx.λy.x

false= λx.λy.y





false) true


true


false) true

(((λy.false) true) false)

true



true= λx.λy.x

false= λx.λy.y





false) true


true


false) true


true

((false) false) true



true= λx.λy.x

false= λx.λy.y





false) true


true


false) true


true


((λx.λy.y) false) true



true= λx.λy.x

false= λx.λy.y





false) true


true


false) true


true



(λy.y) true



true= λx.λy.x

false= λx.λy.y





false) true


true


false) true


true



(λy.y) true = true


Given: or = λx.λy.((x true) y)

true = λx.λy.x

false = λx.λy.y

Prove: or false true = true


Given: or = λx.λy.((x true) y)

true = λx.λy.x

false = λx.λy.y

Prove: or false true = true

λx.λy.((x true) y) false true

λy.((false true) y) true

(false true) true

((λx.λy.y) true) true

(λy.y) true

true


Given: if a then b else c = abc

true = λx.λy.x

false = λx.λy.y

Prove: if false then x else y = y


Given: if a then b else c = abc

true = λx.λy.x

false = λx.λy.y

Prove: if false then x else y = y

false x y

(λx.λy.y) x y

(λy.y) y

y

LAMBDA CALCULUS - cs.umd.edu · Review Ocaml Lambda Calculus fun x e //anonymous function applies...

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