X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment1.mdwn;h=cc535c2bc24edca5bb2b46f0e152f2a0dce3956b;hp=313db95633e36c857b5e6afe417a9ffb03c33b9c;hb=96bd066e0d1adcd0cacb08a55b6aa6c20541952a;hpb=da04ac1d11456ea14ab5c07a1ba8a16e45ffcd58 diff --git a/assignment1.mdwn b/assignment1.mdwn index 313db956..cc535c2b 100644 --- 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): @@ -12,7 +13,8 @@ them): 7. (\x (x x x)) (\x (x x x)) -**Booleans** +Booleans +-------- Recall our definitions of true and false. @@ -24,22 +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). 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. 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. + + 9. Define an "and" operator. + + 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 -capable of representing one of the two values "black" or "white". If we have + 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: @@ -74,28 +89,32 @@ 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 @@ -106,30 +125,24 @@ 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.