X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment3.mdwn;h=c7c5be41755bf6c0f28e7ca860474d53884e0489;hp=f89f5017a3363236c5e4fc3ba52c45e7ba776818;hb=34983e83fd449a9856d571ddf4f5290ab6a25bad;hpb=5c7dbf2ff40b4cf646b9fc3bb9290d19e0a8e388
diff --git a/assignment3.mdwn b/assignment3.mdwn
index f89f5017..c7c5be41 100644
--- a/assignment3.mdwn
+++ b/assignment3.mdwn
@@ -4,9 +4,10 @@ Assignment 3
Once again, the lambda evaluator will make working through this
assignment much faster and more secure.
-*Writing recursive functions on version 1 style lists*
+#Writing recursive functions on version 1 style lists#
-Recall that version 1 style lists are constructed like this:
+Recall that version 1 style lists are constructed like this (see
+[[lists and numbers]]):
; booleans
@@ -35,12 +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
@@ -57,34 +57,78 @@ 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
+3. (Easy) 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).
+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,
+reasonable map and filter functions for version 3 lists are within our
+reach. Give definitions for `map` and a `filter` for verson 1 type lists.
+
+#Computing with trees#
+
+Linguists analyze natural language expressions into trees.
+We'll need trees in future weeks, and tree structures provide good
+opportunities for learning how to write recursive functions.
+Making use of the resources we have at the moment, we can approximate
+trees as follows: instead of words, we'll use Church numerals.
+Then a tree will be a (version 1 type) list in which each element is
+itself a tree. For simplicity, we'll adopt the convention that
+a tree of length 1 must contain a number as its only element.
+Then we have the following representations:
+
+
+ (a) (b) (c)
+ .
+ /|\ /\ /\
+ / | \ /\ 3 1/\
+ 1 2 3 1 2 2 3
+
+[[1];[2];[3]] [[[1];[2]];[3]] [[1];[[2];[3]]]
+
+
+Limitations of this scheme include the following: there is no easy way
+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.
- That is, (makeList 1 (makeList 2 (makeList 3 nil)))
+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 these
+trees.
-[The following should be correct, but won't run in my browser:
+1. Write a function that sums the number of leaves in a tree.
-let factorial = Y (\fac n. isZero n 1 (mult n (fac (predecessor n)))) in
+Expected behavior:
-let reverse =
- Y (\rev l. isNil l nil
- (isNil (tail l) l
- (makeList (head (rev (tail l)))
- (rev (makeList (head l)
- (rev (tail (rev (tail l))))))))) in
-
-reverse (makeList 1 (makeList 2 (makeList 3 nil)))
+let t1 = (make-list 1 nil) in
+let t2 = (make-list 2 nil) in
+let t3 = (make-list 3 nil) in
+let t12 = (make-list t1 (make-list t2 nil)) in
+let t23 = (make-list t2 (make-list t3 nil)) in
+let ta = (make-list t1 t23) in
+let tb = (make-list t12 t3) in
+let tc = (make-list t1 (make-list t23 nil)) in
+
+count-leaves t1 ~~> 1
+count-leaves t2 ~~> 2
+count-leaves t3 ~~> 3
+count-leaves t12 ~~> 3
+count-leaves t23 ~~> 5
+count-leaves ta ~~> 6
+count-leaves tb ~~> 6
+count-leaves tc ~~> 6
-It may require more resources than my browser is willing to devote to
-JavaScript.]
+2. Write a function that counts the number of leaves.