This document contains a lecture on functions, λ-calculus, and Peano arithmetic. It defines functions, injections, surjections, bijections, function composition, and identity functions. It introduces λ-calculus, including β-reduction and shows how Church numerals can represent numbers. It then covers Peano arithmetic, defining numbers, addition, multiplication, and exponentiation recursively using λ-terms.

1. Truth, Deduction,
Computation
Lecture F
Untyped λ... and other things
Vlad Patryshev
SCU
2013

2. Functions
f: X → Y; X - domain, X - codomain.
∀ x:X ∃ y:Y (y=f(x))
y
y
x
A Function
x
Not a Function

3. Functions. Definitions
Monomorphism, aka injective function, aka one-to-one function
∀x ∀y (f(x)=f(y) → x=y)
f: X ↣ Y

4. Functions. Definitions
Epimorphism, aka surjective function, aka onto function
∀y ∃x (f(x)=y)
f: X ↠ Y

5. Functions. Definitions
Bijection, aka bijective function, aka one-to-one correspondence
f is injection and surjection
f: A

6. Functions. Composition
● f: X → Y
● g: Y → Z
● h=g∘f
(g∘f)(x) = g(f(x))
h
→

7. Functions. Identity
● id: X → X
● id∘f=f=f∘id
IdX
→
X
id(x)=x

8. Functions. Make a monoid
f,g: X → X
● binary operation: f∘g
● neutral element: id
● associativity: (f∘g)∘h = f∘(g∘h)

9. Currying
Having a function
f: (x1,x2,...xn) → y
is equivalent to having a function
f c: x 1 → x 2 → … → x n → y

10. Functions in Programming languages
Language
Code Sample
Java
interface Function<X,Y> {
public Y apply(x: X)
}
Function<Int,String> f =
new Function<Int,String>() {
public String apply(n: Int) {
return "n=" + n;
}
Scala
val f:(Int => String) = (n:Int) => "n=" + n
Javascript
f = function(n) { return "n=" + n }

11. Peano Arithmetic
1.
2.
3.
4.
5.
6.
7.
8.
objects are called natural numbers
equality is defined, with usual properties
0 is a natural number
function s: s(x) is natural number
∀x (s(x) ≠ 0)
∀x∀y (s(x) = s(y) → x = y)
P(0), ∀x (P(x)→P(s(x))) ⊢ ∀x P(x)
define 1 as s(0)

12. Peano Arithmetic Illustrated
Giuseppe Peano
Domino model of Peano Arithmetic

13. Peano Arithmetic: define +
● x+0 = x
● x+s(y) = s(x+y)
● have a monoid

14. Peano Arithmetic: define ×
● x×0 = 0
● x×s(y) = x + (x×y)
● have a monoid

15. Peano Arithmetic: define ↑ etc
● x↑0 = 1
● x↑s(y) = x × (x↑y)
● (strangely, no associativity)
● x⇑↑0 = 1
● x⇑s(y) = x ↑ (x⇑y)
(Some people argue lately that this makes it
inconsistent)

16. Peano Arithmetic: Universal Property

17. Alonzo Church
●
●
●
Peano arithmetics is undecidable
Every effectively calculable function is a
computable function
Invented λ-calculus which is equivalent to
effectively calculable functions
Is it so that all that machines can do is equivalent to
lambdas?
Church-Rosser thesis: YES!
https://files.nyu.edu/cb125/public/Lambda/barendregt.94.pdf

18. So, can we build Peano arithmetic?
They are called Church Numerals
0:
1:
2:
…
n:
f → id
f → f
f → f∘f
f → f∘…∘f
}
●
●
●
●
●
n times

19. Simplest Language for Calculations
●
●
●
●
<λ
<λ
<λ
<λ
term>::=<variable>
term>::=<λ term> <λ term>
term>::=(λ <variable> <λ term>)
term>::=(<λ term>)
application
abstraction
http://www.itu.dk/people/sestoft/lamreduce/lamframes.html

20. Javascript Interpretation
● we have as many variables as we like
● λ x N ≡ function(x) {return N}
● (M N) ≡ M(N)
http://myjavatools.com/js.html

21. Examples of λ terms
●
●
●
●
●
x
λ x x
(x y)
((λ x x)(z t))
λ x ((x x) (x x))
Does not make much sense so far, right?

22. Same things in Javascript
●
●
●
●
●
x
function(x) {return x}
x(y)
(function(x) {return x})(z(t))
function(x) {return x(x)(x(x))}

23. Rules of Transformation
● α-equivalence, renaming bound(*) variables
● β-conversion (substitution/λ intro)
● η-conversion

24. Substitutions
1.
2.
3.
4.
5.
x[x:=N]
y[x:=N]
(M1 M2)[x:=N]
(λx M)[x:=N]
(λy M)[x:=N]
≡
≡
≡
≡
≡
N
y, if x ≠ y
(M1[x:=N]) (M2[x:=N])
λx M
λy (M[x:=N])//if x≠y and
y∉FV(N)

25. α-equivalence rule
Bound variable in a lambda expression - the one
which name is found right after λ.
● λx (λy (x (y z)))
Two lambda expressions differing only in names of
bound variables are equivalent.
λx (λy (x (y z))) ⇔ λt (λw (t (w z)))
λx (λy (x (y z))) ⇔ λz (λw (z (w z)))

26. Like in Javascript
function(x) {
return function(y) {
return y(function(z) { return z(x) }
}
}
// same as
function(a) {
return function(b) {
return b(function(c) { return c(a) }
}
}

27. β-conversion rule
(λx E) E’ ⇔ E[x := E’]
●
●
●
●
(λx
(λx
(λx
(λy
x) (y z) ⇔ (y z)
x) (λy y) ⇔ λy y ⇔ λx x
x) (λy y y) ⇔ λy y y
y y) (λx x) ⇔ (λx x) (λx x) ⇔ (λx x)

28. β-conversion is
β-reduction + β-abstraction
β-reduction
(λx E) E’ ⇒ E[x := E’]
β-abstraction
(λx E) E’ ⇐ E[x := E’]

29. β-reduction in Javascript
(function(x) {return M})(N)
Substitute all x in M with N

30. β-reduction - tautological case
●
●
●
●
t =
t t
(λx
(λx
λx x // identity function, right?
= ?
x) M = M
x) (λa a) = λa a

31. β-reduction - tautology, Javascript
id = function(x) { println(“<<x>>”);
return x }
id(id)

32. β-reduction - weird case
●
●
●
●
t =
t t
(λx
(λx
λx (x x)
= ?
(x x)) (λx (x x)) =...
(x x)) (λx (x x)) - oops.

33. That’s it for today