Assignment 3
------------
+Erratum corrected 11PM Sun 3 Oct: the following line
+
+ let tb = (make_list t12 (make_list t3 empty)) in
+
+originally read
+
+ 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]].
+
+<hr>
+
Once again, the lambda evaluator will make working through this
assignment much faster and more secure.
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 get_fst = true in
+ let get_snd = false in
let empty = make_pair true junk in
- let isempty = \x. x fst 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 snd fst) in
- let tail = \l. isempty l err (l snd snd) 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
-
- ; 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 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 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
+ let fold = Y (\f l g z. isempty l z (g (head l)(f (tail l) g z))) in
+
eq 2 2 yes no
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: 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 (make_list junk empty))) ~~> true
- equal_length mylist (make_list junk (make_list junk empty))) ~~> false
+ 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
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.
+<OL start=6>
+<LI>Write a function that sums the values at the leaves in a tree.
Expected behavior:
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 tb = (make_list t12 (make_list t3 empty)) in
let tc = (make_list t1 (make_list t23 empty)) in
sum-leaves t1 ~~> 1
sum-leaves tc ~~> 6
-2. Write a function that counts the number of leaves.
+<LI>Write a function that counts the number of leaves.
+
+</OL>