*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]]):
<pre>
; booleans
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
</pre>
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
</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,
+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:
-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))
+<pre>
+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 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
-
-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)))
</pre>
-It may require more resources than my browser is willing to devote to
-JavaScript.]
+#Write a function that counts the number of leaves.#