X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment1.mdwn;h=fa02cb83634bd078c64230e2b5b97f244a68e460;hp=7568717dcc67bf49a3775eb31573727bb58070f2;hb=aa5ce0c31af63013f3ce832c6fd3038b88b681c9;hpb=00ca9d64a3e95f9163545be1e0ae7845298a10c2 diff --git a/assignment1.mdwn b/assignment1.mdwn index 7568717d..fa02cb83 100644 --- a/assignment1.mdwn +++ b/assignment1.mdwn @@ -17,8 +17,8 @@ Booleans Recall our definitions of true and false. -> `true` defined to be `\t \f. t` -> `false` defined to be `\t \f. f` +> **true** is defined to be `\t \f. t` +> **false** is defined to be `\t \f. f` In Racket, these can be defined like this: @@ -40,16 +40,13 @@ evaluates to 10.
• Define an `and` operator. -
• Define an `xor` operator. - -(If you haven't seen this term before, here's a truth table: +
• 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 -)
• Inspired by our definition of boolean values, propose a data structure capable of representing one of the two values `black` or `white`. @@ -57,7 +54,7 @@ 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 + 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 @@ -76,66 +73,62 @@ Pairs Recall our definitions of ordered pairs. -> the pair (x,y) is defined as `\f. f x y` +> 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) + 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))) + (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 + (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 +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 elemens with that -function. In other words, the pair is also a function. (Of course, in the -untyped lambda calculus, absolutely *everything* is a function: functors, -arguments, abstracts, redexes, values---everything.) +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))) + (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) + (lifted-get-first p) instead of: - (p get-first) + (p get-first) -However, the latter is still what's going on under the hood. +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)`.)
-
1. Define a `swap` function that reverses the elements of a pair. - -Expected behavior: +
2. 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 + (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. +Write out the definition of `swap` in Racket.
3. 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 + ((dup 10) get-first) ; evaluates to 10 + ((dup 10) get-second) ; evaluates to 10
4. Define a `sixteen` function that makes sixteen copies of its argument (and stores them in a data structure of @@ -143,7 +136,7 @@ your choice).
5. Inspired by our definition of ordered pairs, propose a data structure capable of representing ordered triples. That is, - (((make-triple M) N) P) + (((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. @@ -151,7 +144,7 @@ should return an object that behaves in a reasonable way to serve as a triple. I You can help yourself to the following definition: - (define add (lambda (x) (lambda (y) (+ x y)))) + (define add (lambda (x) (lambda (y) (+ x y))))