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[lambda.git] / assignment1.mdwn

diff --git a/assignment1.mdwn b/assignment1.mdwn
index fad7c8c..93136f2 100644 (file)
--- a/assignment1.mdwn
+++ b/assignment1.mdwn
@@ -1,4 +1,5 @@
-**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 as far as it's possible to reduce 
 them):


@@ -12,7 +13,8 @@ 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.
 


@@ -24,21 +26,29 @@ 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)))
 

-test
-8. Define a "neg" operator that negates "true" and "false".
-Expected behavior: (((neg true) 10) 20) evaluates to 20,
-(((neg false) 10) 20) evaluates to 10.
+8. [8] Define a "neg" operator that negates "true" and "false".
+Expected behavior: 

 
 

-9. Define an "and" operator.
+    (((neg true) 10) 20) 
+
+evaluates to 20, and 
+
+    (((neg false) 10) 20) 
+
+evaluates to 10.
+
+9. [9] Define an "and" operator.
+
+10. [10] Define an "xor" operator. (If you haven't seen this term before, here's a truth table:

 
 

-10. 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
        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
+* 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
 one of those values, call it a black-or-white-value, we should be able to
 write:
 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:


@@ -74,28 +84,32 @@ Here are some defintions in Racket:
         (define get-second (lamda (fst) (lambda (snd) snd)))
 
 Now we can write:
         (define get-second (lamda (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 operates on it come seco\
-nd, think about why it's being done this way: the pair is a package that takes a function for op\
-erating 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.

 
 If you like, you can disguise what's going on like this:
 
 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)))
 
 Now you can write:
         (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:
 instead of:

+

         (p get-first)
         (p get-first)

+

 However, the latter is still what's going on under the hood.
 
 
 13. Define a "swap" function that reverses the elements of a pair.
 Expected behavior:
 However, the latter is still what's going on under the hood.
 
 
 13. 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


@@ -106,30 +120,24 @@ Write out the definition of swap in Racket.
 14. Define a "dup" function that duplicates its argument to form a pair
 whose elements are the same.
 Expected behavior:
 14. 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

+

 15. Define a "sixteen" function that makes
 sixteen copies of its argument (and stores them in a data structure of
 your choice).
 
 15. 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 representin\
-g 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 de\
-fining 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.
+16. Inspired by our definition of ordered pairs, propose a data structure capable of representing ordered tripes. That is,

 
 

-> I expect some to come back with the lovely
->     (\f. f first second third)
-> and others, schooled in a certain mathematical perversion, to come back
-> with:
->     (\f. f first (\g. g second third))
+        (((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.

 
 

-17. Write a function second-plus-third that when given to your triple, returns the result of add\
-ing the second and third members of the triple.
+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.

 
 You can help yourself to the following definition:
 
 You can help yourself to the following definition:

+

     (define add (lambda (x) (lambda (y) (+ x y))))
 
     (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.