CS200: Computer Science, Spring 2003
Notes: Friday 11 April 2003
- Saturday-Wednesday: Design reviews
- Friday, 18 April: Exam 2 out
- Wednesday, 23 April: Exam 2 Due
- 28 April (9am): Problem Set 8 Due
NotesUniversal Turing Machine
A Turing Machine that can simulate any other Turing Machine on an input: TMU (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 Grammarterm ::= variable
term ::= term term
term ::= ( term )
term ::= λ variable . term
RulesAlpha Reduction: (renaming variables)Uninteresting Reduction Rulesλ 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!
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
λ f . ((λ x . f (xx)) (λ x . f (xx)))
Using these Materials