Notes: Wednesday 7 April 2004

Schedule

• Now: Problem Set 7
• Before 11:59pm tonight: Problem Set 8 Team Requests (optional)
• Exam 2 will be handed out on 14 April and due on 19 April.

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 . 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 ]

Example

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

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)

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

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

Links

Guy L. Steele, Growing a Language, OOPSLA 1998 Keynote. Guy Steele was a co-designer of Scheme, and one of the leading proponents of Java (and a co-author of the Java Language Specificaion). The second page shows a machine that is equivalent to a Turing Machine. If you are bothered by the sexist language in this paper (of course, unlike more authors Steele had a particularly good reason for writing this way), read Douglas Hofstadter's A Person Paper on Purity in Language. If you weren't bothered by the sexist language in this paper, you should especially read Douglas Hofstadter's A Person Paper on Purity in Language. Read the Post Scriptum on this essay before you find it offensive.

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

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.
William James' explanation of The Meaning of Truth in 1911.
We'll stick with the T ≡ λ x (λ y . x) version.