X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment1.mdwn;h=1c5dc9811536796dc52ebade7c949289ef5ebe89;hp=dd1fa12cd0b3c4f29c3d9fa92ed087d85a8870f2;hb=0bec426ca6b6e8672c487a1b9f0a3ef1823d267a;hpb=7e98785584b3e6bef8f331c032e2b41b0c92deea

diff --git a/assignment1.mdwn b/assignment1.mdwn
index dd1fa12c..1c5dc981 100644
--- a/assignment1.mdwn
+++ b/assignment1.mdwn
@@ -1,7 +1,7 @@
-**Reduction**
+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 following (that is, reduce them until no more reductions are possible):
 
     1. (\x \y. y x) z
     2. (\x (x x)) z
@@ -12,7 +12,8 @@ them):
     7. (\x (x x x)) (\x (x x x))
 
 
-**Booleans**
+Booleans
+--------
 
 Recall our definitions of true and false.
 
@@ -24,30 +25,34 @@ 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".
+* Define a "neg" operator that negates "true" and "false".
+
 Expected behavior: 
 
-    (((neg true) 10) 20) 
+    (((neg true) 10) 20)
 
 evaluates to 20, and 
 
-    (((neg false) 10) 20) 
+    (((neg false) 10) 20)
 
 evaluates to 10.
 
-* 9. Define an "and" operator.
+* Define an "and" operator.
+
+* Define an "xor" operator. 
 
-* 10. Define an "xor" operator. (If you haven't seen this term before, here's a truth table:
+(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
+    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
+* 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:
 
@@ -58,7 +63,7 @@ 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
+* 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.)
 
@@ -75,11 +80,11 @@ To extract the first element of a pair p, you write:
 
         p (\fst \snd. fst)
 
-Here are some defintions in Racket:
+Here are some definitions 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)))
+        (define get-first (lambda (fst) (lambda (snd) fst)))
+        (define get-second (lambda (fst) (lambda (snd) snd)))
 
 Now we can write:
 
@@ -87,7 +92,7 @@ Now we can write:
         (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 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.
+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, values---everything.)
 
 If you like, you can disguise what's going on like this:
 
@@ -105,7 +110,8 @@ instead of:
 However, the latter is still what's going on under the hood.
 
 
-13. Define a "swap" function that reverses the elements of a pair.
+* Define a "swap" function that reverses the elements of a pair.
+
 Expected behavior:
 
         (define p ((make-pair 10) 20))
@@ -115,27 +121,27 @@ Expected behavior:
 Write out the definition of swap in Racket.
 
 
-14. 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) get-first) ; evaluates to 10
         ((dup 10) get-second) ; evaluates to 10
 
-15. 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).
 
-16. 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 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 defining 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.
+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.
 
-17. 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.
+* 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))))
 
-18. [Super hard, unless you have lots of experience programming] Write a function that reverses the order of the elements in a list.
+* 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.]