X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment3.mdwn;h=f93afd6110ce1e32550d37639f6f7ac7f3a77bc4;hp=f669cb1e6488906d7d2baf66d27927d91abf1ab0;hb=b059b718b62f3b4beffb3bd7fbe66af01069f9c9;hpb=c0674bbb2f8587d3fd625277628140a99b52a9d6 diff --git a/assignment3.mdwn b/assignment3.mdwn index f669cb1e..f93afd61 100644 --- a/assignment3.mdwn +++ b/assignment3.mdwn @@ -4,10 +4,9 @@ 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 (see -[[lists and numbers]]): +Recall that version 1 style lists are constructed like this:
; booleans @@ -36,17 +35,18 @@ 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 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 2 2 yes no +eq 3 3Then `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!, @@ -57,75 +57,41 @@ 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 + +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). -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. +##Trees## -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: +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. -
- (a) (b) (c) - . - /|\ /\ /\ - / | \ /\ 3 1/\ - 1 2 3 1 2 2 3 -[[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. +[The following should be correct, but won't run in my browser: -#Write a function that sums the number of leaves in a tree.# -Expected behavior: +let factorial = Y (\fac n. isZero n 1 (mult n (fac (predecessor n)))) in
-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 +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)))-#Write a function that counts the number of leaves.# +It may require more resources than my browser is willing to devote to +JavaScript.]