XGitUrl: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment1.mdwn;h=fa02cb83634bd078c64230e2b5b97f244a68e460;hp=9c7eaf552677f56c997890a4c2a29b8ea5340a0f;hb=b77bb678d30bbc972f6daf6e6152640faf8f5894;hpb=aa7244316365eb60642e4ab70bee7cfcc427593b
diff git a/assignment1.mdwn b/assignment1.mdwn
index 9c7eaf55..fa02cb83 100644
 a/assignment1.mdwn
+++ b/assignment1.mdwn
@@ 1,16 +1,15 @@
Reduction

Find "normal forms" for the following (that is, reduce them as far as it's possible to reduce
them):
+Find "normal forms" for the followingthat is, reduce them until no more reductions are possible. We'll write λx
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))
+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
@@ 18,15 +17,16 @@ 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:
(define true (lambda (t) (lambda (f) t)))
(define false (lambda (t) (lambda (f) f)))
* Define a "neg" operator that negates "true" and "false".
+
+ Define a `neg` operator that negates `true` and `false`.
Expected behavior:
@@ 38,35 +38,33 @@ evaluates to 20, and
evaluates to 10.
* Define an "and" operator.
+
 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".
+
 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 blackorwhitevalue, we should be able to
+one of those values, call it a "blackorwhite value", we should be able to
write:
 theblackorwhitevalue 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
a definition for each of "black" and "white". (Do it in both lambda calculus
+(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
+a definition for each of `black` and `white`. (Do it in both lambda calculus
and also in Racket.)
* 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
+
 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.)
+
@@ 75,74 +73,80 @@ 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 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.)
+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 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)

However, the latter is still what's going on under the hood.
+ (p getfirst)
+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
+
 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
+
 Define a `sixteen` function that makes
sixteen copies of its argument (and stores them in a data structure of
your choice).
* Inspired by our definition of ordered pairs, propose a data structure capable of representing ordered tripes. That is,
+
 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.
+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.
* Write a function secondplusthird that when given to your triple, returns the result of adding the second and third members of the triple.
+
 Write a function `secondplusthird` 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))))
+ (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.]