add damn4.rkt

[lambda.git] / assignment1.mdwn

diff --git a/assignment1.mdwn b/assignment1.mdwn
index dd1fa12..1c5dc98 100644 (file)
--- 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
 
     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))
 
 
     7. (\x (x x x)) (\x (x x x))
 
 

-**Booleans**
+Booleans
+--------

 
 Recall our definitions of true and false.
 
 
 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)))
 
        (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: 
 
 Expected behavior: 
 

-    (((neg true) 10) 20) 
+    (((neg true) 10) 20)

 
 evaluates to 20, and 
 
 
 evaluates to 20, and 
 

-    (((neg false) 10) 20) 
+    (((neg false) 10) 20)

 
 evaluates to 10.
 
 
 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:
 
 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.)
 
 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.)
 
 "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)
 
 
         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 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:
 
 
 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
 
         (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:
 
 
 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.
 
 
 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))
 Expected behavior:
 
         (define p ((make-pair 10) 20))


@@ -115,27 +121,27 @@ Expected behavior:
 Write out the definition of swap in Racket.
 
 
 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
 
 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).
 
 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)
 
 
         (((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))))
 
 
 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.]