-**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
7. (\x (x x x)) (\x (x x x))
-**Booleans**
+Booleans
+--------
Recall our definitions of true and false.
(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".
-(9). Define an "and" operator.
+Expected behavior:
-10. Define an "xor" operator. (If you haven't seen this term before, here's a truth table:
+ (((neg true) 10) 20)
+
+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:
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.)
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:
(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:
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))
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.]