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 make_pair = \f s g. g f s in
let fst = true in
let snd = false in
let empty = make_pair true junk in
let isempty = \x. x fst in
let make_list = \h t. make_pair false (make_pair h t) in
let head = \l. isempty l err (l snd fst) in
let tail = \l. isempty l err (l snd snd) in
; a list of numbers to experiment on
let mylist = make_list 1 (make_list 2 (make_list 3 empty)) 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. isempty l 0 (succ (length (tail l)))) 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
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: it takes a long time for my browser to compute factorials larger than 4!
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
equal_length mylist (make_list junk (make_list junk empty))) ~~> 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 = (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
2. Write a function that counts the number of leaves.