X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment3.mdwn;h=775eb79e571e78cfdca489121751b0262f9e4c07;hp=71960dc8429ebf75a81638f7c6664fe2dfc17c99;hb=101b2a435c05ae480eaefaf30348e80bd2d3de5d;hpb=f3400cc9679e5f4f78cee730349d74e465648eac;ds=sidebyside

diff --git a/assignment3.mdwn b/assignment3.mdwn
index 71960dc8..775eb79e 100644
--- a/assignment3.mdwn
+++ b/assignment3.mdwn
@@ -6,27 +6,127 @@ assignment much faster and more secure.
 
 *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 true = \x y. x in
 let false = \x y. y in
+let and = \l r. l (r true false) false in
+
+; version 1 lists
 let makePair = \f s g. g f s in
-let nil = makePair true meh in
-let makeList = \h t. makePair false (makePair h t) in
-let mylist = makeList 1 (makeList 2 (makeList 3 nil)) in
 let fst = true in
 let snd = false in
+let nil = makePair true meh in
 let isNil = \x. x fst in
+let makeList = \h t. makePair false (makePair h t) in
 let head = \l. isNil l err (l snd fst) in
 let tail = \l. isNil l err (l snd snd) in
-let succ = \n s z. s (n s z) in
+
+; a list of numbers to experiment on
+let mylist = makeList 1 (makeList 2 (makeList 3 nil)) in
+
+; a fixed-point combinator for defining recursive functions 
 let Y = \f. (\h. f (h h)) (\h. f (h h)) in
+
+; church numerals
+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 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
 
-length mylist
+eq 2 2 yes no
 </pre>
 
+
 Then `length mylist` evaluates to 3.
 
-What does `head (tail (tail mylist))` evaluate to?
+1. 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!,
+the factorial of n, is n times the factorial of n-1.)
+
+Warning: my browser isn't able to compute factorials of numbers
+greater than 2 (it does't provide enough resources for the JavaScript
+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
+same length.  That is,
+
+     listLenEq mylist (makeList meh (makeList meh (makeList meh nil))) ~~> true
+
+     listLenEq mylist (makeList meh (makeList meh nil))) ~~> false
+
+
+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.
+
+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 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.
+
+#Write a function that sums the number of leaves in a tree.#
+Expected behavior:
+
+<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
+<pre>
+
+#Write a function that counts the number of leaves.#
+