Notes: Monday 5 April 2004

Schedule

• Wednesday, 7 April: Problem Set 7
• Wednesday, 7 April: Problem Set 8 Team Requests
• Exam 2 will be handed out on 14 April and due on 19 April.

Notes
Universal Turing Machine

A Turing Machine that can simulate any other Turing Machine on an input: TM_U (P, I) = the output of running TM-P on input I

What is a calculus?

What properties make Scheme too complex to be a good model of computation?

Lambda Calculus is a set of rules for manipulating symbols. They can be given meanings that map well to computation.

Lambda Calculus Term Grammar

term ::= variable
term ::= term term
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 change the name of a lambda variable by 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 ]

All computation can be modeled using Beta Reduction!

Uninteresting Reduction Rules

M → M
PM → PN if M → N
MP → NP if M → N
λ x . M → λ x . N → if M → N
M → P if M → N and N → P

Identity and Composition

I ≡ λ x . x
C ≡ λ x . (λ y . yx)

CII = (λx.(λy. yx)) (λx.x) (λx.x)
      →_β

Example

λ f . ((λ x . f (xx)) (λ x . f (xx)))