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 mul = \m n s. m (n s) in
- let pred = \n. iszero n 0 (length (tail (n (\p. make_list junk p) empty))) 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
-
- eq 2 2 yes no
+##Writing recursive functions on version 1 style lists##
+
+Recall that version 1 style lists are constructed like this:
+
+<pre>
+; 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
+</pre>
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:
-
-<pre>
- (a) (b) (c)
- .
- /|\ /\ /\
- / | \ /\ 3 1 /\
- 1 2 3 1 2 2 3
-
-[[1];[2];[3]] [[[1];[2]];[3]] [[1];[[2];[3]]]
-</pre>
-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.
+[The following should be correct, but won't run in my browser:
-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 factorial = Y (\fac n. isZero n 1 (mult n (fac (predecessor n)))) in
-<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 t3) 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.
+<pre>
+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)))
+</pre>
-</OL>
+It may require more resources than my browser is willing to devote to
+JavaScript.]