Notes: 22 April 2002

Schedule

• Today and tomorrow: Problem Set 8 (Progress Meetings)
• Monday, 29 April: Problem Set 8 (Final)
• Monday, 6 May: Final Due (will be handed out Monday 29 April)

Lambda Calculus

Lambda Calculus Term Grammar

term ::= variable | term term | ( term ) | λ variable . term

Rules

Alpha Reduction: (renaming variables)

λ y . M ⇒_α λ v . M [y |→ v]) where v does not occur in M.

We can can change the name of a lambda variable, but replacing all occurances of the variable in the body term with a new name that does not appear in the body term.

Beta Reduction: (substitution)

(λ x . M) N ⇒_β M [ x |→ N ]

Making "Primitives" out of nothing but Glue

T ≡ λ x (λ y . x)
F ≡ λ xy . y
if ≡ λ pca . pca

cons ≡ λ xy . (λ z . zxy)
car ≡ λ p . p T
cdr ≡ λ p . p F

null ≡ λ p . T
null? ≡ λ x . (x λ y . λ z . F)

0 ≡ null
1 ≡ cons F null
2 ≡ cons 1 null

succ ≡ λ x . cons F x
pred ≡ λ x cdr x
zero? ≡ null?

Fixed Points

What is a fixed point?

All Lambda Calculus Terms Have Fixed Points

For any Lambda Calculus term F, there exists a Lambda Calculus Term X such that FX = X.

Proof: W = λ x. F(xx)
X = WW
X = (λ x. F (xx)) (λ x. F (xx))
       ⇒_β F ((λ x. F (xx)) (λ x. F (xx))) = FX

Y-Operator

Y ≡ λ f. (λ x.f (xx)) (λ x.f (xx))

The Y-Operator calculates the fixed point of any Lambda Calculus term!

Try this in Scheme:

(define Y
  (lambda (f)
    ((lambda (x) (f (lambda (y) ((x x) y))))
     (lambda (x) (f (lambda (y) ((x x) y)))))))
(define f (lambda (g) (lambda (n) (if (= n 0) 1 (* n (g (- n 1)))))))
((Y f) 5)

Why do we need the lambda (y) to make this work in Scheme?

Hint: In LazyScheme, it would work with:

(define Y
  (lambda (f)
    ((lambda (x) (f (x x)))
     (lambda (x) (f (x x))))))

How do the evaluation rules of regular Scheme differ from those of Lambda Calculus?

University of Virginia Department of Computer Science CS 200: Computer Science

David Evans evans@virginia.edu Using these Materials