From: Jim Pryor
Date: Tue, 14 Sep 2010 15:47:08 +0000 (-0400)
Subject: reformat assignment1
X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=8f5ac0c5b7e25c726f3afb65b1fc413af8a4fa52
reformat assignment1
Signed-off-by: Jim Pryor
---
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".
+
+- 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.
+
- Define an "and" operator.
-* Define an "xor" operator.
+
- 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".
+
- 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
+
- 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.
+
+- 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
+
- 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
+
- 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,
+
- 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.
+
- 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.]
+
+
+
+