From 5c7dbf2ff40b4cf646b9fc3bb9290d19e0a8e388 Mon Sep 17 00:00:00 2001 From: Chris Barker Date: Sat, 25 Sep 2010 17:04:47 -0400 Subject: [PATCH] new text --- assignment3.mdwn | 70 +++++++++++++++++++++++++++++++++++++++++++++++++++----- test.mdwn | 3 --- 2 files changed, 64 insertions(+), 9 deletions(-) delete mode 100644 test.mdwn diff --git a/assignment3.mdwn b/assignment3.mdwn index 71960dc8..f89f5017 100644 --- a/assignment3.mdwn +++ b/assignment3.mdwn @@ -9,24 +9,82 @@ assignment much faster and more secure. Recall that version 1 style lists are constructed like this:
+; booleans
let true = \x y. x in
let false = \x y. y in
+let and = \l r. l (r true false) false in
+
+; version 1 lists
let makePair = \f s g. g f s in
-let nil = makePair true meh in
-let makeList = \h t. makePair false (makePair h t) in
-let mylist = makeList 1 (makeList 2 (makeList 3 nil)) in
let fst = true in
let snd = false in
+let nil = makePair true meh in
let isNil = \x. x fst in
+let makeList = \h t. makePair false (makePair h t) in
let head = \l. isNil l err (l snd fst) in
let tail = \l. isNil l err (l snd snd) in
-let succ = \n s z. s (n s z) in
+
+; a list of numbers to experiment on
+let mylist = makeList 1 (makeList 2 (makeList 3 nil)) in
+
+; a fixed-point combinator for defining recursive functions
let Y = \f. (\h. f (h h)) (\h. f (h h)) in
+
+; church numerals
+let isZero = \n. n (\x. false) true in
+let succ = \n s z. s (n s z) in
+let mult = \m n s. m (n s) in
let length = Y (\length l. isNil l 0 (succ (length (tail l)))) in
+let predecessor = \n. length (tail (n (\p. makeList meh p) nil)) in
+let leq = ; (leq m n) will be true iff m is less than or equal to n
+  Y (\leq m n. isZero m true (isZero n false (leq (predecessor m)(predecessor n)))) in
+let eq = \m n. and (leq m n)(leq n m) in

-length mylist
+eq 3 3

+ Then length mylist evaluates to 3. -What does head (tail (tail mylist)) evaluate to? +1. What does head (tail (tail mylist)) evaluate to? + +2. Using the length function as a model, and using the predecessor +function, write a function that computes factorials. (Recall that n!, +the factorial of n, is n times the factorial of n-1.) + +Warning: my browser isn't able to compute factorials of numbers +greater than 2 (it does't provide enough resources for the JavaScript +interpreter; web pages are not supposed to be that computationally +intensive). + + +3. Write a function listLenEq that returns true just in case two lists have the +same length. That is, + + listLenEq mylist (makeList meh (makeList meh (makeList meh nil))) ~~> true + listLenEq mylist (makeList meh (makeList meh nil))) ~~> false + + +4. Now write the same function (true iff two lists have the same +length) but don't use the length function (hint: use leq as a model). + + That is, (makeList 1 (makeList 2 (makeList 3 nil))) + +[The following should be correct, but won't run in my browser: + +let factorial = Y (\fac n. isZero n 1 (mult n (fac (predecessor n)))) in + +
+let reverse =
+  Y (\rev l. isNil l nil
+                   (isNil (tail l) l
+                          (makeList (head (rev (tail l)))
+