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University of Virginia Computer Science CS150: Computer Science, Fall 2005 |
(none) 7 November 2005 |
The Exam 2 Study Guide contains sample questions from previous Exams on the topics your Exam 2 will cover.
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)
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.
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.
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CS 150: Computer Science University of Virginia |
evans@virginia.edu Using these Materials |