4 Once again, the lambda evaluator will make working through this
5 assignment much faster and more secure.
7 #Writing recursive functions on version 1 style lists#
9 Recall that version 1 style lists are constructed like this (see
10 [[lists and numbers]]):
14 let false = \x y. y in
15 let and = \l r. l (r true false) false in
17 let make_pair = \f s g. g f s in
19 let get_snd = false in
20 let empty = make_pair true junk in
21 let isempty = \x. x get_fst in
22 let make_list = \h t. make_pair false (make_pair h t) in
23 let head = \l. isempty l err (l get_snd get_fst) in
24 let tail = \l. isempty l err (l get_snd get_snd) in
26 ; a list of numbers to experiment on
27 let mylist = make_list 1 (make_list 2 (make_list 3 empty)) in
30 let iszero = \n. n (\x. false) true in
31 let succ = \n s z. s (n s z) in
32 let mul = \m n s. m (n s) in
33 let pred = (\shift n. n shift (make\_pair 0 0) get\_snd) (\p. p (\x y. make\_pair (succ x) x)) in
34 let leq = \m n. iszero(n pred m) in
35 let eq = \m n. and (leq m n)(leq n m) in
37 ; a fixed-point combinator for defining recursive functions
38 let Y = \f. (\h. f (h h)) (\h. f (h h)) in
39 let length = Y (\length l. isempty l 0 (succ (length (tail l)))) in
44 Then `length mylist` evaluates to 3.
46 1. What does `head (tail (tail mylist))` evaluate to?
48 2. Using the `length` function as a model, and using the predecessor
49 function, write a function that computes factorials. (Recall that n!,
50 the factorial of n, is n times the factorial of n-1.)
52 Warning: it takes a long time for my browser to compute factorials larger than 4!
54 3. (Easy) Write a function `equal_length` that returns true just in case
55 two lists have the same length. That is,
57 equal_length mylist (make_list junk (make_list junk (make_list junk empty))) ~~> true
59 equal_length mylist (make_list junk (make_list junk empty))) ~~> false
62 4. (Still easy) Now write the same function, but don't use the length
65 5. In assignment 2, we discovered that version 3-type lists (the ones
67 work like Church numerals) made it much easier to define operations
68 like `map` and `filter`. But now that we have recursion in our
70 reasonable map and filter functions for version 1 lists are within our
71 reach. Give definitions for `map` and a `filter` for verson 1 type
74 #Computing with trees#
76 Linguists analyze natural language expressions into trees.
78 We'll need trees in future weeks, and tree structures provide good
79 opportunities for learning how to write recursive functions.
80 Making use of the resources we have at the moment, we can approximate
81 trees as follows: instead of words, we'll use Church numerals.
82 Then a tree will be a (version 1 type) list in which each element is
83 itself a tree. For simplicity, we'll adopt the convention that
84 a tree of length 1 must contain a number as its only element.
86 Then we have the following representations:
95 [[1];[2];[3]] [[[1];[2]];[3]] [[1];[[2];[3]]]
98 Limitations of this scheme include the following: there is no easy way
99 to label a constituent with a syntactic category (S or NP or VP,
100 etc.), and there is no way to represent a tree in which a mother has a
103 When processing a tree, you can test for whether the tree contains
104 only a numeral (in which case the tree is leaf node) by testing for
105 whether the length of the list is less than or equal to 1. This will
106 be your base case for your recursive functions that operate on these
110 <LI>Write a function that sums the values at the leaves in a tree.
114 let t1 = (make_list 1 empty) in
115 let t2 = (make_list 2 empty) in
116 let t3 = (make_list 3 empty) in
117 let t12 = (make_list t1 (make_list t2 empty)) in
118 let t23 = (make_list t2 (make_list t3 empty)) in
119 let ta = (make_list t1 t23) in
120 let tb = (make_list t12 t3) in
121 let tc = (make_list t1 (make_list t23 empty)) in
133 <LI>Write a function that counts the number of leaves.