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[lambda.git] / assignment1.mdwn

diff --git a/assignment1.mdwn b/assignment1.mdwn
index 313db95..bfaafdc 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.
 


@@ -24,22 +26,32 @@ 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".
-Expected behavior: (((neg true) 10) 20) evaluates to 20,
-(((neg false) 10) 20) evaluates to 10.
+* Define a "neg" operator that negates "true" and "false".
+Expected behavior: 

 
 

-(9). Define an "and" operator.
+    (((neg true) 10) 20)

 
 

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


@@ -50,7 +62,7 @@ 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.)
 


@@ -74,28 +86,32 @@ 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 +122,24 @@ 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.