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, 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:
+
+<pre>
+ (a) (b) (c)
+ .
+ /|\ /\ /\
+ / | \ /\ 3 1/\
+ 1 2 3 1 2 2 3
+
+[[1];[2];[3]] [[[1];[2]];[3]] [[1];[[2];[3]]]
+</pre>
+
+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:
+<pre>
let factorial = Y (\fac n. isZero n 1 (mult n (fac (predecessor n)))) in
-<pre>
let reverse =
Y (\rev l. isNil l nil
(isNil (tail l) l