+++ /dev/null
-Reduction
----------
-
-Find "normal forms" for the following---that is, reduce them until no more reductions are possible. We'll write <code>λx</code> as `\x`.
-
-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** is defined to be `\t \f. t`
-> **false** is 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)))
-
-<OL start=8>
-<LI>Define a `neg` operator that negates `true` and `false`.
-
-Expected behavior:
-
- (((neg true) 10) 20)
-
-evaluates to 20, and
-
- (((neg false) 10) 20)
-
-evaluates to 10.
-
-<LI>Define an `and` operator.
-
-<LI>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
-
-
-<LI>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-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.)
-
-<LI>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.)
-</OL>
-
-
-
-Pairs
------
-
-Recall our definitions of ordered pairs.
-
-> the pair **(**x**,**y**)** is defined to be `\f. f x y`
-
-To extract the first element of a pair p, you write:
-
- p (\fst \snd. fst)
-
-Here are some definitions in Racket:
-
- (define make-pair (lambda (fst) (lambda (snd) (lambda (f) ((f fst) snd)))))
- (define get-first (lambda (fst) (lambda (snd) fst)))
- (define get-second (lambda (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 puzzled by having the pair to the left and the function that
-operates on it come second, think about why it's being done this way: the pair
-is a package that takes a function for operating on its elements *as an
-argument*, and returns *the result of* operating on its elements with that
-function. In other words, the pair is a higher-order function. (Consider the similarities between this definition of a pair and a generalized quantifier.)
-
-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. (Remark: `(lifted-f ((make-pair 10) 20))` stands to `(((make-pair 10) 20) f)` as `(((make-pair 10) 20) f)` stands to `((f 10) 20)`.)
-
-
-<OL start=13>
-<LI>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.
-
-
-<LI>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
-
-<LI>Define a `sixteen` function that makes
-sixteen copies of its argument (and stores them in a data structure of
-your choice).
-
-<LI>Inspired by our definition of ordered pairs, propose a data structure capable of representing ordered triples. 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 defining the `make-triple` function, you have to show how to extract 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.
-
-<LI>Write a function `second-plus-third` that when given to your triple, returns the result of adding the second and third members of the triple.
-
-You can help yourself to the following definition:
-
- (define add (lambda (x) (lambda (y) (+ x y))))
-
-<!-- Write a function that reverses the order of the elements in a list. [Only attempt this problem if you're feeling frisky, it's super hard unless you have lots of experience programming.] -->
-
-</OL>
-