X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment1.mdwn;h=1c5dc9811536796dc52ebade7c949289ef5ebe89;hp=cc535c2bc24edca5bb2b46f0e152f2a0dce3956b;hb=0bec426ca6b6e8672c487a1b9f0a3ef1823d267a;hpb=96bd066e0d1adcd0cacb08a55b6aa6c20541952a diff --git a/assignment1.mdwn b/assignment1.mdwn index cc535c2b..1c5dc981 100644 --- a/assignment1.mdwn +++ b/assignment1.mdwn @@ -1,8 +1,7 @@ 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 @@ -26,21 +25,21 @@ 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". +* Define a "neg" operator that negates "true" and "false". Expected behavior: - (((neg true) 10) 20) + (((neg true) 10) 20) evaluates to 20, and - (((neg false) 10) 20) + (((neg false) 10) 20) evaluates to 10. - 9. Define an "and" operator. +* Define an "and" operator. - 10. Define an "xor" operator. +* Define an "xor" operator. (If you haven't seen this term before, here's a truth table: @@ -51,9 +50,8 @@ evaluates to 10. ) - 11. Inspired by our definition of boolean values, propose a data structure - capable of representing one of the two values "black" or "white". - +* 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: @@ -65,7 +63,7 @@ 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.) -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.) @@ -82,11 +80,11 @@ To extract the first element of a pair p, you write: 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: @@ -94,7 +92,7 @@ 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: @@ -112,7 +110,8 @@ instead of: 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)) @@ -122,27 +121,27 @@ Expected behavior: 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.]