X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment3.mdwn;h=f93afd6110ce1e32550d37639f6f7ac7f3a77bc4;hp=d00dba0b2cbbfa44a6adc84668e145aa6e96a7b6;hb=b059b718b62f3b4beffb3bd7fbe66af01069f9c9;hpb=97cbee46cb35e1eb7aa5ea5f8b1af3583f5b7522

diff --git a/assignment3.mdwn b/assignment3.mdwn
index d00dba0b..f93afd61 100644
--- a/assignment3.mdwn
+++ b/assignment3.mdwn
@@ -4,92 +4,94 @@ Assignment 3
 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:
 
-<textarea id="INPUT" style="border: 2px solid black; color: black; font-family: monospace; height: 3in; overflow: auto; padding: 0.5em; width: 100%;">
+<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 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 3 3
+</pre>
+
+
+Then `length mylist` evaluates to 3.
+
+1. Warm-up: What does `head (tail (tail mylist))` evaluate to?
 
-length mylist
-</textarea>
-<input id="PARSE" value="Normalize" type="button">
-<input id="ETA" type="checkbox">do eta-reductions too
-<noscript><p>You may not see it because you have JavaScript turned off. Uffff!</p></noscript>
-<script src="/code/lambda.js"></script>
-<script src="/code/tokens.js"></script>
-<script src="/code/parse.js"></script>
-<script src="/code/json2.js"></script>
-<pre id="OUTPUT">
+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. 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).
+
+##Trees##
+
+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.
+
+
+
+
+[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
+
+<pre>
+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>
-<script>
-/*jslint evil: true */
-
-/*members create, error, message, name, prototype, stringify, toSource,
-    toString, write
-*/
-
-/*global JSON, make_parse, parse, source, tree */
-
-// Make a new object that inherits members from an existing object.
-
-if (typeof Object.create !== 'function') {
-    Object.create = function (o) {
-        function F() {}
-        F.prototype = o;
-        return new F();
-    };
-}
-
-// Transform a token object into an exception object and throw it.
-
-Object.prototype.error = function (message, t) {
-    t = t || this;
-    t.name = "SyntaxError";
-    t.message = message;
-    throw t;
-};
-
-
-(function () {
-    var parse = make_parse();
-
-    function go(source) {
-        var string, tree, expr, eta;
-        try {
-            tree = parse(source);
- //           string = JSON.stringify(tree, ['key', 'name', 'message', 'value', 'arity', 'first', 'second', 'third', 'fourth'], 4);
-			expr = tree.handler();
-            // string = JSON.stringify(expr, ['key', 'name', 'message', 'value', 'arity', 'first', 'second', 'tag', 'variable', 'left', 'right', 'bound', 'body' ], 4);
-//			string = expr.to_string() + "\n\n~~>\n\n";
-			string = '';
-			eta = document.getElementById('ETA').checked;
-			string = string + reduce(expr, eta, false).to_string();
-        } catch (e) {
-            string = JSON.stringify(e, ['name', 'message', 'from', 'to', 'key',
-                    'value', 'arity', 'first', 'second', 'third', 'fourth'], 4);
-        }
-        document.getElementById('OUTPUT').innerHTML = string
-            .replace(/&/g, '&amp;')
-            .replace(/[<]/g, '&lt;');
-    }
-
-    document.getElementById('PARSE').onclick = function (e) {
-        go(document.getElementById('INPUT').value);
-    };
-}());
-
-</script>
+
+It may require more resources than my browser is willing to devote to
+JavaScript.]
+