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 (see
-[[lists and numbers]]):
+Recall that version 1 style lists are constructed like this:
<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 pred = \n. isZero n 0 (length (tail (n (\p. makeList meh p) nil))) in
-let leq = \m n. isZero(n pred m) 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 eq = \m n. and (leq m n)(leq n m) in
-eq 2 2 yes no
+eq 3 3
</pre>
Then `length mylist` evaluates to 3.
-1. What does `head (tail (tail mylist))` evaluate to?
+1. Warm-up: What does `head (tail (tail mylist))` evaluate to?
2. Using the `length` function as a model, and using the predecessor
function, write a function that computes factorials. (Recall that n!,
interpreter; web pages are not supposed to be that computationally
intensive).
-3. (Easy) Write a function `listLenEq` that returns true just in case two lists have the
+
+3. 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.
-
-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.
+##Trees##
-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:
+Since we'll be working with linguistic objects, let's approximate
+trees as follows: a tree is a version 1 list
+a Church number is a tree, and
+if A and B are trees, then (make-pair A B) is a tree.
-<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 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 these
-trees.
+[The following should be correct, but won't run in my browser:
-#Write a function that sums the number of leaves in a tree.#
-Expected behavior:
+let factorial = Y (\fac n. isZero n 1 (mult n (fac (predecessor n)))) in
<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 t3 ~~> 3
-count-leaves t12 ~~> 3
-count-leaves t23 ~~> 5
-count-leaves ta ~~> 6
-count-leaves tb ~~> 6
-count-leaves tc ~~> 6
+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>
-#Write a function that counts the number of leaves.#
+It may require more resources than my browser is willing to devote to
+JavaScript.]