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diff --git a/assignment1.mdwn b/assignment1.mdwn
index 4980cc27..93136f22 100644
--- a/assignment1.mdwn
+++ b/assignment1.mdwn
@@ -1,18 +1,20 @@
-*Reduction*
+Reduction
+---------
Find "normal forms" for the following (that is, reduce them as far as it's possible to reduce
them):
-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*
+Booleans
+--------
Recall our definitions of true and false.
@@ -24,26 +26,35 @@ In Racket, these can be defined like this:
(define true (lambda (t) (lambda (f) t)))
(define false (lambda (t) (lambda (f) f)))
+8. [8] Define a "neg" operator that negates "true" and "false".
+Expected behavior:
-8. Define a "neg" operator that negates "true" and "false".
-Expeceted behavior: (((neg true) 10) 20) evaluates to 20,
-(((neg false) 10) 20) evaluates to 10.
+ (((neg true) 10) 20)
-9. Define an "and" operator.
+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
+
)
-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:
the-black-or-white-value if-black if-white
+
(where if-black and if-white are anything), and get back one of if-black or
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
@@ -73,28 +84,32 @@ Here are some defintions in Racket:
(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
-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:
+
(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)
+
instead of:
+
(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:
+
(define p ((make-pair 10) 20))
((p swap) get-first) ; evaluates to 20
((p swap) get-second) ; evaluates to 10
@@ -105,30 +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:
+
((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).
-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:
+
(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.