+++ /dev/null
-*Reduction*
-
-Find "normal forms" for the following (that is, reduce them as far as it's possible to reduce
-them):
-
-1. (\x \y. y x) z
-2. (\x (x x)) z
-3. (\x (\x x)) z
-4. (\x (\z x)) z
-5. (\x (x (\y y))) (\z (z z))
-6. (\x (x x)) (\x (x x))
-7. (\x (x x x)) (\x (x x x))
-
-
-*Booleans*
-
-Recall our definitions of true and false.
-
- "true" defined to be `\t \f. t`
- "false" defined to be `\t \f. f`
-
-In Racket, these can be defined like this:
-
- (define true (lambda (t) (lambda (f) t)))
- (define false (lambda (t) (lambda (f) f)))
-
-
-8. Define a "neg" operator that negates "true" and "false".
-Expeceted behavior: (((neg true) 10) 20) evaluates to 20,
-(((neg false) 10) 20) evaluates to 10.
-
-9. Define an "and" operator.
-
-10. Define an "xor" operator. (If you haven't seen this term before, here's a truth table:
- true xor true = false
- true xor false = true
- false xor true = true
- false xor false = false
-)
-
-11. Inspired by our definition of boolean values, propose a data structure
-capable of representing one of the two values "black" or "white". If we have
-one of those values, call it a black-or-white-value, we should be able to
-write:
-
- the-black-or-white-value if-black if-white
-(where if-black and if-white are anything), and get back one of if-black or
-if-white, depending on which of the black-or-white values we started with. Give
-a definition for each of "black" and "white". (Do it in both lambda calculus
-and also in Racket.)
-
-12. Now propose a data structure capable of representing one of the three values
-"red" "green" or "blue," based on the same model. (Do it in both lambda
-calculus and also in Racket.)
-
-
-
-Pairs
------
-
-Recall our definitions of ordered pairs.
-
- the pair (x,y) is defined as `\f. f x y`
-
-To extract the first element of a pair p, you write:
-
- p (\fst \snd. fst)
-
-Here are some defintions in Racket:
-
- (define make-pair (lambda (fst) (lambda (snd) (lambda (f) ((f fst) snd)))))
- (define get-first (lamda (fst) (lambda (snd) fst)))
- (define get-second (lamda (fst) (lambda (snd) snd)))
-
-Now we can write:
- (define p ((make-pair 10) 20))
- (p get-first) ; will evaluate to 10
- (p get-second) ; will evaluate to 20
-
-If you're bothered by having the pair to the left and the function that operates on it come seco\
-nd, think about why it's being done this way: the pair is a package that takes a function for op\
-erating on its elements as an argument, and returns the result of operating on its elemens with \
-that function. In other words, the pair is also a function.
-
-If you like, you can disguise what's going on like this:
- (define lifted-get-first (lambda (p) (p get-first)))
- (define lifted-get-second (lambda (p) (p get-second)))
-
-Now you can write:
- (lifted-get-first p)
-instead of:
- (p get-first)
-However, the latter is still what's going on under the hood.
-
-
-13. Define a "swap" function that reverses the elements of a pair.
-Expected behavior:
- (define p ((make-pair 10) 20))
- ((p swap) get-first) ; evaluates to 20
- ((p swap) get-second) ; evaluates to 10
-
-Write out the definition of swap in Racket.
-
-
-14. Define a "dup" function that duplicates its argument to form a pair
-whose elements are the same.
-Expected behavior:
- ((dup 10) get-first) ; evaluates to 10
- ((dup 10) get-second) ; evaluates to 10
-15. Define a "sixteen" function that makes
-sixteen copies of its argument (and stores them in a data structure of
-your choice).
-
-16. Inspired by our definition of ordered pairs, propose a data structure capable of representin\
-g ordered tripes. That is,
- (((make-triple M) N) P)
-should return an object that behaves in a reasonable way to serve as a triple. In addition to de\
-fining the make-triple function, you have to show how to extraxt elements of your triple. Write \
-a get-first-of-triple function, that does for triples what get-first does for pairs. Also write \
-get-second-of-triple and get-third-of-triple functions.
-
-> I expect some to come back with the lovely
-> (\f. f first second third)
-> and others, schooled in a certain mathematical perversion, to come back
-> with:
-> (\f. f first (\g. g second third))
-
-
-17. Write a function second-plus-third that when given to your triple, returns the result of add\
-ing the second and third members of the triple.
-
-You can help yourself to the following definition:
- (define add (lambda (x) (lambda (y) (+ x y))))
-