X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment3.mdwn;h=9f64d808f637a1d3c67c1cfa8584e69d0f9e27bf;hp=4e3c4f3355d5f2f0f26521146759f777d29f87c0;hb=e09159ccfdc281101e7777af85e26d74cff95379;hpb=5fe94991f45f63c51ce3809cba60515f18163a65 diff --git a/assignment3.mdwn b/assignment3.mdwn index 4e3c4f33..9f64d808 100644 --- a/assignment3.mdwn +++ b/assignment3.mdwn @@ -57,8 +57,7 @@ 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 @@ -66,14 +65,75 @@ same length. That is, listLenEq mylist (makeList meh (makeList meh nil))) ~~> false -4. 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 (hint: use `leq` as a model). + +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 such a map and a filter. + +6. 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
+
+[;;]  [[;];]   [;[;]]
+```
+ +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, +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. + +Write a function that sums the number of leaves in a tree. +Expected behavior: + +let t1 = (make-list 1 nil) +let t2 = (make-list 2 nil) +let t3 = (make-list 3 nil) +let t12 = (make-list t1 (make-list t2 nil)) +let t23 = (make-list t2 (make-list t3 nil)) +let ta = (make-list t1 t23) +let tb = (make-list t12 t3) +let tc = (make-list t1 (make-list t23 nil)) + +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 + +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
```