XGitUrl: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment3.mdwn;h=e240b732b1405e176d696bbab3c5974a1d125f35;hp=f93afd6110ce1e32550d37639f6f7ac7f3a77bc4;hb=aab06ee4eafe7ac322521e4cef1b1704a9258900;hpb=6495d054c734d27c17123d04d84f3b3cf798b743;ds=sidebyside
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@@ 1,97 +1,151 @@
Assignment 3

Once again, the lambda evaluator will make working through this
assignment much faster and more secure.
+Erratum corrected 11PM Sun 3 Oct: the following line
##Writing recursive functions on version 1 style lists##
+ let tb = (make_list t12 (make_list t3 empty)) in
Recall that version 1 style lists are constructed like this:
+originally read
; 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 fixedpoint 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

+ 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]].
+
+
+
+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 (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 fixedpoint 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. Warmup: What does `head (tail (tail mylist))` evaluate to?
+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 n1.)
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).

+ Warning: it takes a long time for my browser to compute factorials larger than 4!
3. Write a function `listLenEq` that returns true just in case two lists have the
same length. That is,
+3. (Easy) Write a function `equal_length` that returns true just in case
+two lists have the same length. That is,
 listLenEq mylist (makeList meh (makeList meh (makeList meh nil))) ~~> true
+ equal_length mylist (make_list junk (make_list junk (make_list junk empty))) ~~> true
 listLenEq mylist (makeList meh (makeList meh nil))) ~~> false
+ equal_length mylist (make_list junk (make_list junk empty))) ~~> false
4. Now write the same function, but don't use the length function (hint: use `leq` as a model).
##Trees##
+4. (Still easy) Now write the same function, but don't use the length
+function.
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 (makepair A B) is a tree.
+5. In assignment 2, we discovered that version 3type 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.
[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
+Then we have the following representations:
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)))
+ (a) (b) (c)
+ .
+ /\ /\ /\
+ /  \ /\ 3 1 /\
+ 1 2 3 1 2 2 3
+
+[[1];[2];[3]] [[[1];[2]];[3]] [[1];[[2];[3]]]
It may require more resources than my browser is willing to devote to
JavaScript.]
+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.
+
+
+ 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
+
+ sumleaves t1 ~~> 1
+ sumleaves t2 ~~> 2
+ sumleaves t3 ~~> 3
+ sumleaves t12 ~~> 3
+ sumleaves t23 ~~> 5
+ sumleaves ta ~~> 6
+ sumleaves tb ~~> 6
+ sumleaves tc ~~> 6
+
+
+
 Write a function that counts the number of leaves.
+
+