-Assignment 3
-------------
+This lambda evaluator will allow you to write lambda terms and evaluate (that is, normalize) them, and inspect the results.
+(This won't work in Racket, because Racket doesn't even try to represent the internal structure of a function in a human-readable way.)
-Once again, the lambda evaluator will make working through this
-assignment much faster and more secure.
+*Lambda terms*: lambda terms are written with a backslash, thus: `((\x (\y x)) z)`.
-#Writing recursive functions on version 1 style lists#
+If you click "Normalize", the system will try to produce a normal-form lambda expression that your original term reduces to (~~>). So `((\x (\y x)) z)` reduces to `(\y z)`.
-Recall that version 1 style lists are constructed like this (see
-[[lists and numbers]]):
+*Let*: in order to make building a more elaborate set of terms easier, it is possible to define values using `let`.
+In this toy system, `let`s should only be used at the beginning of a file. If we have, for intance,
-<pre>
-; booleans
+ let true = (\x (\y x)) in
+ let false = (\x (\y y)) in
+ ((true yes) no)
+
+the result is `yes`.
+
+*Comments*: anything following a semicolon to the end of the line is ignored.
+Blank lines are fine.
+
+*Abbreviations*: In an earlier version, you couldn't use abbreviations. `\x y. y x x` had to be written `(\x (\y ((y x) x)))`. We've upgraded the parser though, so now it should be able to understand any lambda term that you can.
+
+*Constants*: The combinators `S`, `K`, `I`, `C`, `B`, `W`, and `T` are pre-defined to their standard values. Also, integers will automatically be converted to Church numerals. (`0` is `\s z. z`, `1` is `\s z. s z`, and so on.)
+
+
+
+<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 and = \l r. l (r true false) false in
-
-; version 1 lists
-let makePair = \f s g. g f s 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
-
-; 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
-
-eq 2 2 yes no
+let and = \l r. l r false in
+(
+ (and true true yes no)
+ (and true false yes no)
+ (and false true yes no)
+ (and false false yes no)
+)
+</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">
</pre>
+<script>
+/*jslint evil: true */
+/*members create, error, message, name, prototype, stringify, toSource,
+ toString, write
+*/
-Then `length mylist` evaluates to 3.
+/*global JSON, make_parse, parse, source, tree */
-1. What does `head (tail (tail mylist))` evaluate to?
+// Make a new object that inherits members from an existing object.
-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.)
+if (typeof Object.create !== 'function') {
+ Object.create = function (o) {
+ function F() {}
+ F.prototype = o;
+ return new F();
+ };
+}
-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).
+// Transform a token object into an exception object and throw it.
-3. (Easy) Write a function `listLenEq` that returns true just in case two lists have the
-same length. That is,
+Object.prototype.error = function (message, t) {
+ t = t || this;
+ t.name = "SyntaxError";
+ t.message = message;
+ throw t;
+};
- listLenEq mylist (makeList meh (makeList meh (makeList meh nil))) ~~> true
- listLenEq mylist (makeList meh (makeList meh nil))) ~~> false
+(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, '&')
+ .replace(/[<]/g, '<');
+ }
-4. (Still easy) Now write the same function, but don't use the length function.
+ document.getElementById('PARSE').onclick = function (e) {
+ go(document.getElementById('INPUT').value);
+ };
+}());
-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.
+</script>
-#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
+Under the hood
+---------------
-[[1];[2];[3]] [[[1];[2]];[3]] [[1];[[2];[3]]]
-</pre>
+The interpreter is written in JavaScript and runs inside your browser.
+So if you decide to reduce a term that does not terminate (such as `((\x (x x)) (\x (x x)))`), it will be your
+browser that stops responding, not the wiki server.
-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.
-
-1. Write a function that sums the number of leaves in a tree.
-
-Expected behavior:
-
-<pre>
-let t1 = (makeList 1 nil) in
-let t2 = (makeList 2 nil) in
-let t3 = (makeList 3 nil) in
-let t12 = (makeList t1 (makeList t2 nil)) in
-let t23 = (makeList t2 (makeList t3 nil)) in
-let ta = (makeList t1 t23) in
-let tb = (makeList t12 t3) in
-let tc = (makeList t1 (makeList t23 nil)) 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
-</pre>
+The main code is [here](http://lambda.jimpryor.net/code/lambda.js). Suggestions for improvements welcome.
+
+The code is based on:
+
+* Chris Barker's JavaScript lambda calculator
+* [Oleg Kiselyov's Haskell lambda calculator](http://okmij.org/ftp/Computation/lambda-calc.html#lambda-calculator-haskell).
+* The top-down JavaScript lexer and parser at <http://javascript.crockford.com/tdop/index.html>.
+
+Improvements we hope to add soon: the ability to reduce Combinatory Logic combinators and report the result as combinators, rather than in lambda forms.
-2. Write a function that counts the number of leaves.