This document provides an introduction to lambda calculus including syntax, semantics, examples, Church booleans, Church numerals, and recursion. It defines the syntax of lambda calculus terms including variables, abstraction, and application. It explains beta reduction semantics and provides examples of reductions. It introduces Church encodings for booleans and natural numbers. It also defines the Y-combinator for expressing recursion using fixed-point combinators.

1. rafael de f. ferreira (@rafaeldff)
lambda calculus

2. David Hilbert
2
“Wir müssen wissen.
Wir werden wissen.”
We must know.
We will know.

3. 3
Alonzo Church

4. Syntax
4
t ::=
x
λx. t
t t
terms:
variable
abstraction
application

5. Example expressions
5
x
x y
λx. x
λx. y
(λx. x) y
(λx. x x) z
(λx. x x) (λx. x x)
(λf. (λx. f (λy. x x y))
(λx. f (λy. x x y)))

6. Example expressions
6
(λx. x) y
variable
abstraction
variable
application

7. Substitution
7
[y↦z](λx. (λx. y x)) y
=
(λx. (λx. z x)) z

8. Semantics
8
(λx. t1) t2 ⟶ [x↦t2]t1
beta reduction
redex

9. Semantics - examples
9
(λx. x) y ⟶ y

10. Semantics - examples
10
(λx. x) (λy. y y) ⟶ (λy. y y)

11. Semantics - examples
11
(λx.(λz.x z)) y ⟶ (λz.y z)

12. 12
Church Booleans

13. booleans
13
(λt. (λf. t))
(λt. (λf. f))
(λb. (λt. (λf. b t f))
true
false
if

14. booleans - example
14
if false (λx. x) (λz. z)
(λb. (λt. (λf. b t f)) false (λx. x) (λz. z)
(λt. (λf. false t f)) (λx. x) (λz. z)
(λf. false (λx. x) f) (λz. z)
false (λx. x) (λz. z)
=
⟶
⟶
⟶

15. booleans - example
15
false (λx. x) (λz. z)
(λt. (λf. f)) (λx. x) (λz. z)
(λf. f) (λz. z)
(λz. z)
=
⟶
⟶
⟶

16. booleans - example
16
if false (λx. x) (λz. z) ⟶* (λz. z)

17. 17
Church Numerals

18. numerals
18
λs. λz. z
λs. λz. s z
λs. λz. s (s z)
λs. λz. s (s (s z))
…
c0
c1
c2
c3
…

19. numerals
19
succ = λn. λs. λz. s (n s z)

20. numerals
20
succ c1
(λn. λs. λz. s (n s z)) c1
λs. λz. s ((λs. λz. s z) s z)
λs. λz. s ((λz. s z) z)
λs. λz. s (s z) = c2
…
=
=
⟶
⟶
⟶

21. numerals
21
plus = λm. λn. λs. λz. m s (n s z)

22. booleans - example
22
plus c0 c1 ⟶* c1
plus c1 c1 ⟶* c2
plus c2 c2 ⟶* c4

23. 23
Recursion

24. recursion
24
ω = (λx. x x) (λx. x x)

25. recursion
25
ω = (λx. x x) (λx. x x)
(λx. x x) (λx. x x)
(λx. x x) (λx. x x)
(λx. x x) (λx. x x)
…
⟶
⟶
⟶
⟶
⟶

26. recursion
26
Y = λf. (λx. f (λy. x x y)) (λx. f (λy. x x y))

27. recursion
27
Y = λf. (λx. f (λy. x x y)) (λx. f (λy. x x y))
Y f = f (Y f) for all f

28. recursion
28
g = λfct. λn if (eq n c0) c1 (times n (fct (pred n)))
factorial = Y g
Y f = f (Y f) for all f

29. Propositions as Types
PhilipWadler
Introduction to
Lambda Calculus
Barendregt, Barendsen