X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment3.mdwn;h=e240b732b1405e176d696bbab3c5974a1d125f35;hp=d00dba0b2cbbfa44a6adc84668e145aa6e96a7b6;hb=dc9298c6718ec9bc63550ec6bc6b5a187f235e50;hpb=97cbee46cb35e1eb7aa5ea5f8b1af3583f5b7522

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 Assignment 3
 ------------
 
+Erratum corrected 11PM Sun 3 Oct: the following line
+
+	let tb = (make_list t12 (make_list t3 empty)) in
+
+originally read 
+
+	let tb = (make_list t12 t3) in
+
+This has been corrected below, and in the preloaded evaluator for 
+working on assignment 3, available here: [[assignment 3 evaluator]].
+
+<hr>
+
 Once again, the lambda evaluator will make working through this
 assignment much faster and more secure.
 
-*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%;">
-let true = \x y. x in
-let false = \x y. y in
-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 isNil = \x. x fst 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
-let Y = \f. (\h. f (h h)) (\h. f (h h)) in
-let length = Y (\length l. isNil l 0 (succ (length (tail l)))) in
-
-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">
+#Writing recursive functions on version 1 style lists#
+
+Recall that version 1 style lists are constructed like this (see
+[[lists and numbers]]):
+
+	; booleans
+	let true = \x y. x in
+	let false = \x y. y in
+	let and = \l r. l (r true false) false in
+
+	let make_pair = \f s g. g f s in
+	let get_fst = true in
+	let get_snd = false in
+	let empty = make_pair true junk in
+	let isempty = \x. x get_fst in
+	let make_list = \h t. make_pair false (make_pair h t) in
+	let head = \l. isempty l err (l get_snd get_fst) in
+	let tail = \l. isempty l err (l get_snd get_snd) in
+	
+	; a list of numbers to experiment on
+	let mylist = make_list 1 (make_list 2 (make_list 3 empty)) in
+	
+	; church numerals
+	let iszero = \n. n (\x. false) true in
+	let succ = \n s z. s (n s z) in
+	let add = \l r. l succ r in
+	let mul = \m n s. m (n s) in
+	let pred = (\shift n. n shift (make\_pair 0 0) get\_snd) (\p. p (\x y. make\_pair (succ x) x)) in
+	let leq = \m n. iszero(n pred m) in
+	let eq = \m n. and (leq m n)(leq n m) in
+	
+	; a fixed-point combinator for defining recursive functions
+	let Y = \f. (\h. f (h h)) (\h. f (h h)) in
+	let length = Y (\length l. isempty l 0 (succ (length (tail l)))) in
+	let fold = Y (\f l g z. isempty l z (g (head l)(f (tail l) g z))) in
+	
+	eq 2 2 yes no
+
+
+Then `length mylist` evaluates to 3.
+
+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: it takes a long time for my browser to compute factorials larger than 4!
+
+3. (Easy) Write a function `equal_length` that returns true just in case
+two lists have the same length.  That is,
+
+		equal_length mylist (make_list junk (make_list junk (make_list junk empty))) ~~> true
+
+		equal_length mylist (make_list junk (make_list junk empty))) ~~> 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 1 lists are within our
+reach.  Give definitions for `map` and a `filter` for verson 1 type
+lists.
+
+#Computing with trees#
+
+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>
-<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>
+
+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.
+
+<OL start=6>
+<LI>Write a function that sums the values at the leaves in a tree.
+
+Expected behavior:
+
+	let t1 = (make_list 1 empty) in
+	let t2 = (make_list 2 empty) in
+	let t3 = (make_list 3 empty) in
+	let t12 = (make_list t1 (make_list t2 empty)) in
+	let t23 = (make_list t2 (make_list t3 empty)) in
+	let ta = (make_list t1 t23) in
+	let tb = (make_list t12 (make_list t3 empty)) in
+	let tc = (make_list t1 (make_list t23 empty)) in
+
+	sum-leaves t1 ~~> 1
+	sum-leaves t2 ~~> 2
+	sum-leaves t3 ~~> 3
+	sum-leaves t12 ~~> 3
+	sum-leaves t23 ~~> 5
+	sum-leaves ta ~~> 6
+	sum-leaves tb ~~> 6
+	sum-leaves tc ~~> 6
+
+
+<LI>Write a function that counts the number of leaves.
+
+</OL>
+