X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment3.mdwn;h=f93afd6110ce1e32550d37639f6f7ac7f3a77bc4;hp=e240b732b1405e176d696bbab3c5974a1d125f35;hb=b059b718b62f3b4beffb3bd7fbe66af01069f9c9;hpb=e17c03bd8ce5b051ce06e123d3ce65d6086591be diff --git a/assignment3.mdwn b/assignment3.mdwn index e240b732..f93afd61 100644 --- a/assignment3.mdwn +++ b/assignment3.mdwn @@ -1,151 +1,97 @@ 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]]. - -
+; 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 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 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 +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!, 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! +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 `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 +3. Write a function `listLenEq` that returns true just in case two lists have the +same length. That is, - equal_length mylist (make_list junk (make_list junk empty))) ~~> false + 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. +4. Now write the same function, but don't use the length function (hint: use `leq` as a model). -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. +##Trees## -#Computing with 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. -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: -
- (a) (b) (c) - . - /|\ /\ /\ - / | \ /\ 3 1 /\ - 1 2 3 1 2 2 3 +[The following should be correct, but won't run in my browser: -[[1];[2];[3]] [[[1];[2]];[3]] [[1];[[2];[3]]] -- -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. - -
+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))) +-