University of Virginia, Department of Computer Science CS200: Computer Science, Spring 2002

Notes: 19 April 2002

Schedule
Progress Meetings
Monday, 22 April and Tuesday, 23 April, each PS8 group will meet with me to demonstrate your progress and present your plans for completing the assignment. Here are the possible times:
• Monday, 3:30pm
• Monday, 4:30pm
• Tuesday, 10am
• Tuesday, 11am
• Tuesday, 3pm
Lambda Calculus

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

Lambda Calculus Term Grammar

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

Rules

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)

If you find the meaning of life from class unsatisfying, try Eliezer S. Yudkowsky's Frequently Asked Questions about the Meaning of Life.

William James attempted to explain The Meaning of Truth in 1911 (before Gödel proved it was impossible):

My account of truth is realistic, and follows the epistemological dualism of common sense. Suppose I say to you 'The thing exists' -- is that true or not? How can you tell? Not till my statement has developed its meaning farther is it determined as being true, false, or irrelevant to reality altogether. But if now you ask 'what thing?' and I reply 'a desk'; if you ask 'where?' and I point to a place; if you ask 'does it exist materially, or only in imagination?' and I say 'materially'; if moreover I say 'I mean that desk,' and then grasp and shake a desk which you see just as I have described it, you are willing to call my statement true. But you and I are commutable here; we can exchange places; and, as you go bail for my desk, so I can go bail for yours.
(We'll stick with the T ≡ λ xy . x) version.)