From 8f5ac0c5b7e25c726f3afb65b1fc413af8a4fa52 Mon Sep 17 00:00:00 2001 From: Jim Pryor Date: Tue, 14 Sep 2010 11:47:08 -0400 Subject: [PATCH] reformat assignment1 Signed-off-by: Jim Pryor --- assignment1.mdwn | 74 ++++++++++++++++++++++++++++++++------------------------ 1 file changed, 43 insertions(+), 31 deletions(-) diff --git a/assignment1.mdwn b/assignment1.mdwn index 1c5dc981..9bb65b38 100644 --- a/assignment1.mdwn +++ b/assignment1.mdwn @@ -1,15 +1,15 @@ Reduction --------- -Find "normal forms" for the following (that is, reduce them until no more reductions are possible): +Find "normal forms" for the following---that is, reduce them until no more reductions are possible. We'll write λ`x` 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 @@ -17,15 +17,16 @@ Booleans Recall our definitions of true and false. - "true" defined to be `\t \f. t` - "false" defined to be `\t \f. f` +> "true" defined to be `\t \f. t` +> "false" 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))) -* Define a "neg" operator that negates "true" and "false". +
+
1. Define a "neg" operator that negates "true" and "false". Expected behavior: @@ -37,9 +38,9 @@ evaluates to 20, and evaluates to 10. -* Define an "and" operator. +
2. Define an "and" operator. -* Define an "xor" operator. +
3. Define an "xor" operator. (If you haven't seen this term before, here's a truth table: @@ -50,22 +51,23 @@ evaluates to 10. ) -* Inspired by our definition of boolean values, propose a data structure -capable of representing one of the two values "black" or "white". +
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 +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.) -* 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.) +
@@ -74,7 +76,7 @@ Pairs Recall our definitions of ordered pairs. - the pair (x,y) is defined as `\f. f x y` +> the pair (x,y) is defined as `\f. f x y` To extract the first element of a pair p, you write: @@ -92,7 +94,13 @@ 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. (Of course, in the untyped lambda calculus, absolutely *everything* is a function: functors, arguments, abstracts, redexes, values---everything.) +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: @@ -110,7 +118,8 @@ instead of: However, the latter is still what's going on under the hood. -* Define a "swap" function that reverses the elements of a pair. +
+
1. Define a `swap` function that reverses the elements of a pair. Expected behavior: @@ -121,27 +130,30 @@ Expected behavior: Write out the definition of swap in Racket. -* 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 -* Define a "sixteen" function that makes +
3. Define a `sixteen` function that makes sixteen copies of its argument (and stores them in a data structure of your choice). -* Inspired by our definition of ordered pairs, propose a data structure capable of representing ordered tripes. That is, +
4. Inspired by our definition of ordered pairs, propose a data structure capable of representing ordered triples. 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 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. +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. -* 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. +
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)))) -* 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.] + + +
+ -- 2.11.0