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-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#
-
-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
-
-; 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
-
-
-
-Then `length mylist` evaluates to 3.
-
-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 n-1.)
-
-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 `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. (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 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.
-Then we have the following representations:
-
-
- (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.
-
-1. Write a function that sums the number of leaves in a tree.
-
-Expected behavior:
-
-
-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
-
-
-2. Write a function that counts the number of leaves.
-