X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment1.mdwn;h=fa02cb83634bd078c64230e2b5b97f244a68e460;hp=fad7c8c4a6b223ae68ae0cc3f89f2e6fa23fde35;hb=0514157d311eae9ec04606dc4b3e6a68038f3186;hpb=c1302edddd57fadcf94648c9173613b2273552a0 diff --git a/assignment1.mdwn b/assignment1.mdwn index fad7c8c4..fa02cb83 100644 --- a/assignment1.mdwn +++ b/assignment1.mdwn @@ -1,58 +1,70 @@ -**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. We'll write `λx` as `\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)) +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. - "true" defined to be `\t \f. t` - "false" defined to be `\t \f. f` +> **true** is defined to be `\t \f. t` +> **false** is defined to be `\t \f. f` In Racket, these can be defined like this: (define true (lambda (t) (lambda (f) t))) (define false (lambda (t) (lambda (f) f))) -test -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. +
+
1. 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: - true xor true = false - true xor false = true - false xor true = true - false xor false = false -) + (((neg true) 10) 20) -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 +evaluates to 20, and + + (((neg false) 10) 20) + +evaluates to 10. + +
2. Define an `and` operator. + +
3. 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 + + +
4. 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 + the-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 +(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 and also in Racket.) -12. 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 +
5. 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.) +
@@ -61,75 +73,80 @@ Pairs Recall our definitions of ordered pairs. - the pair (x,y) is defined as `\f. f x y` +> the pair **(**x**,**y**)** is defined to be `\f. f x y` To extract the first element of a pair p, you write: - p (\fst \snd. fst) + 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 make-pair (lambda (fst) (lambda (snd) (lambda (f) ((f fst) snd))))) + (define get-first (lambda (fst) (lambda (snd) fst))) + (define get-second (lambda (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. + (define p ((make-pair 10) 20)) + (p get-first) ; will evaluate to 10 + (p get-second) ; will evaluate to 20 + +If you're puzzled 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 elements with that +function. In other words, the pair is a higher-order function. (Consider the similarities between this definition of a pair and a generalized quantifier.) 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))) + + (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: - (p get-first) -However, the latter is still what's going on under the hood. + (p get-first) -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 +However, the latter is still what's going on under the hood. (Remark: `(lifted-f ((make-pair 10) 20))` stands to `(((make-pair 10) 20) f)` as `(((make-pair 10) 20) f)` stands to `((f 10) 20)`.) + + +
+
1. 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 -Write out the definition of swap in Racket. +Write out the definition of `swap` in Racket. -14. Define a "dup" function that duplicates its argument to form a pair +
2. 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 + + ((dup 10) get-first) ; evaluates to 10 + ((dup 10) get-second) ; evaluates to 10 + +
3. 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. +
4. Inspired by our definition of ordered pairs, propose a data structure capable of representing ordered triples. 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 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 add\ -ing the second and third members of the triple. +
5. 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)))) + + (define add (lambda (x) (lambda (y) (+ x y)))) + + + +