rafael de f. ferreira (@rafaeldff)

lambda calculus

David Hilbert

2

Wir mssen wissen. Wir werden wissen.

We must know.We will know.

3

Alonzo Church

Syntax

4

t ::= x x. t t t

terms: variable

abstraction application

Example expressions

5

x x y x. x x. y

(x. x) y (x. x x) z (x. x x) (x. x x) (f. (x. f (y. x x y)) (x. f (y. x x y)))

Example expressions

6

(x. x) yvariable

abstraction

variable

application

Substitution

7

[yz](x. (x. y x)) y=

(x. (x. z x)) z

Semantics

8

(x. t1) t2 [xt2]t1 beta reduction

redex

Semantics - examples

9

(x. x) y y

Semantics - examples

10

(x. x) (y. y y) (y. y y)

Semantics - examples

11

(x.(z.x z)) y (z.y z)

12

Church Booleans

booleans

13

(t. (f. t))

(t. (f. f))

(b. (t. (f. b t f))

true

false

if

booleans - example

14

if false (x. x) (z. z)

(b. (t. (f. b t f)) false (x. x) (z. z)

(t. (f. false t f)) (x. x) (z. z)

(f. false (x. x) f) (z. z)

false (x. x) (z. z)

=

booleans - example

15

false (x. x) (z. z)

(t. (f. f)) (x. x) (z. z)

(f. f) (z. z) (z. z)

=

booleans - example

16

if false (x. x) (z. z) * (z. z)

17

Church Numerals

numerals

18

s. z. z

s. z. s z

s. z. s (s z)

s. z. s (s (s z))

c0

c1

c2

c3

numerals

19

succ = n. s. z. s (n s z)

numerals

20

succ c1

(n. s. z. s (n s z)) c1

s. z. s ((s. z. s z) s z) s. z. s ((z. s z) z) s. z. s (s z) = c2

=

=

numerals

21

plus = m. n. s. z. m s (n s z)

booleans - example

22

plus c0 c1 * c1 plus c1 c1 * c2 plus c2 c2 * c4

23

Recursion

recursion

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= (x. x x) (x. x x)

recursion

25

= (x. x x) (x. x x) (x. x x) (x. x x) (x. x x) (x. x x) (x. x x) (x. x x)

recursion

26

Y = f. (x. f (y. x x y)) (x. f (y. x x y))

recursion

27

Y = f. (x. f (y. x x y)) (x. f (y. x x y))

Y f = f (Y f) for all f

recursion

28

g = fct. n if (eq n c0) c1 (times n (fct (pred n)))

factorial = Y g

Y f = f (Y f) for all f

Propositions as TypesPhilip Wadler

Introduction to Lambda Calculus

Barendregt, Barendsen