X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment3.mdwn;h=775eb79e571e78cfdca489121751b0262f9e4c07;hp=342896c2c429ee51d798d943352d415d682aa7a5;hb=101b2a435c05ae480eaefaf30348e80bd2d3de5d;hpb=825c122aac6dd19e421ea72049b608615c00f46a diff --git a/assignment3.mdwn b/assignment3.mdwn index 342896c2..775eb79e 100644 --- a/assignment3.mdwn +++ b/assignment3.mdwn @@ -36,12 +36,11 @@ 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 pred = \n. isZero n 0 (length (tail (n (\p. makeList meh p) nil))) in +let leq = \m n. isZero(n pred m) in let eq = \m n. and (leq m n)(leq n m) in -eq 3 3 +eq 2 2 yes no @@ -66,13 +65,13 @@ same length. That is, listLenEq mylist (makeList meh (makeList meh nil))) ~~> false -4. (Still easy) Now write the same function, but don't use the length function (hint: use leq as a model). +4. (Still easy) Now write the same function, but don't use the length function. 5. In assignment 2, we discovered that version 3-type lists (the ones that work like Church numerals) made it much easier to define operations -like map and filter. But now that we have recursion in our toolbox, +like map and filter. But now that we have recursion in our toolbox, reasonable map and filter functions for version 3 lists are within our -reach. Give definitions for such a map and a filter. +reach. Give definitions for map and a filter for verson 1 type lists. 6. Linguists analyze natural language expressions into trees. We'll need trees in future weeks, and tree structures provide good @@ -95,16 +94,17 @@ Then we have the following representations: Limitations of this scheme include the following: there is no easy way -to label a constituent (typically a syntactic category, S or NP or VP, +to label a constituent with a syntactic category (S or NP or VP, etc.), and there is no way to represent a tree in which a mother has a single daughter. When processing a tree, you can test for whether the tree contains only a numeral (in which case the tree is leaf node) by testing for whether the length of the list is less than or equal to 1. This will -be your base case for your recursive functions that operate on trees. +be your base case for your recursive functions that operate on these +trees. -Write a function that sums the number of leaves in a tree. +#Write a function that sums the number of leaves in a tree.# Expected behavior:
@@ -128,35 +128,5 @@ count-leaves tb ~~> 6
count-leaves tc ~~> 6

-Write a function that counts the number of leaves.
+#Write a function that counts the number of leaves.#

-
-
-
-[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)))
-                                                   (rev (tail (rev (tail l))))))))) in
-
-reverse (makeList 1 (makeList 2 (makeList 3 nil)))
-
-
-It may require more resources than my browser is willing to devote to
-JavaScript.]
-
-; trees
-let t1 = (makeList 1 nil) in
-let t2 = (makeList 2 nil) in
-let t3 = (makeList 3 nil) in
-let t12 = (makeList t1 (makeList t2 nil)) in
-let t23 = (makeList t2 (makeList t3 nil)) in
-let ta = (makeList t1 t23) in
-let tb = (makeList t12 t3) in
-let tc = (makeList t1 (makeList t23 nil)) in