University of Virginia, Department of Computer Science CS200: Computer Science, Spring 2004

Notes: Wednesday 7 April 2004

Exam 2
Exam 2 will be handed out Wednesday, April 14 and due Monday, April 19. It will cover everything in the course through the end of Friday's lecture, but will focus on material that was not covered on Exam 1. Exam 2 will be open book and open notes.

The Exam 2 Study Guide contains sample questions from previous Exams on the topics your Exam 2 will cover.

Notes

How do we prove a model of computation is equivalent to a Turing Machine?

Lambda Calculus

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

Alpha Reduction: (renaming variables)

λ y . Mα 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)

x . M) Nβ M [ x |→ N ]
Example

f . ((λ x . f (xx)) (λ x . f (xx)))) λ x . x

Making "Primitives" out of nothing but Glue

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

cons ≡ λ xy .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)

zero?null?
predcdr
succ ≡ λ x . cons F x

0null
1succ 0cons F null
2succ 1cons F (cons F null)