XGitUrl: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment1.mdwn;h=fa02cb83634bd078c64230e2b5b97f244a68e460;hp=7568717dcc67bf49a3775eb31573727bb58070f2;hb=d9ef81bf6980969f16aebfed4b6fda9c3c5463bf;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 "blackorwhite value", we should be able to
write:
 thevalue ifblack ifwhite
+ thevalue ifblack ifwhite
(where `ifblack` and `ifwhite` are anything), and get back one of `ifblack` or
`ifwhite`, depending on which of the blackorwhite 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 makepair (lambda (fst) (lambda (snd) (lambda (f) ((f fst) snd)))))
 (define getfirst (lambda (fst) (lambda (snd) fst)))
 (define getsecond (lambda (fst) (lambda (snd) snd)))
+ (define makepair (lambda (fst) (lambda (snd) (lambda (f) ((f fst) snd)))))
+ (define getfirst (lambda (fst) (lambda (snd) fst)))
+ (define getsecond (lambda (fst) (lambda (snd) snd)))
Now we can write:
 (define p ((makepair 10) 20))
 (p getfirst) ; will evaluate to 10
 (p getsecond) ; will evaluate to 20
+ (define p ((makepair 10) 20))
+ (p getfirst) ; will evaluate to 10
+ (p getsecond) ; 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, valueseverything.)
+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 higherorder 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 liftedgetfirst (lambda (p) (p getfirst)))
 (define liftedgetsecond (lambda (p) (p getsecond)))
+ (define liftedgetfirst (lambda (p) (p getfirst)))
+ (define liftedgetsecond (lambda (p) (p getsecond)))
Now you can write:
 (liftedgetfirst p)
+ (liftedgetfirst p)
instead of:
 (p getfirst)
+ (p getfirst)
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: `(liftedf ((makepair 10) 20))` stands to `(((makepair 10) 20) f)` as `(((makepair 10) 20) f)` stands to `((f 10) 20)`.)
 Define a `swap` function that reverses the elements of a pair.

Expected behavior:
+
 Define a `swap` function that reverses the elements of a pair. Expected behavior:
 (define p ((makepair 10) 20))
 ((p swap) getfirst) ; evaluates to 20
 ((p swap) getsecond) ; evaluates to 10
+ (define p ((makepair 10) 20))
+ ((p swap) getfirst) ; evaluates to 20
+ ((p swap) getsecond) ; evaluates to 10
Write out the definition of swap in Racket.
+Write out the definition of `swap` in Racket.
 Define a `dup` function that duplicates its argument to form a pair
whose elements are the same.
Expected behavior:
 ((dup 10) getfirst) ; evaluates to 10
 ((dup 10) getsecond) ; evaluates to 10
+ ((dup 10) getfirst) ; evaluates to 10
+ ((dup 10) getsecond) ; evaluates to 10
 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).
 Inspired by our definition of ordered pairs, propose a data structure capable of representing ordered triples. That is,
 (((maketriple M) N) P)
+ (((maketriple M) N) P)
should return an object that behaves in a reasonable way to serve as a triple. In addition to defining the `maketriple` function, you have to show how to extract elements of your triple. Write a `getfirstoftriple` function, that does for triples what `getfirst` does for pairs. Also write `getsecondoftriple` and `getthirdoftriple` 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))))