+; 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 nil = makePair true meh in -let makeList = \h t. makePair false (makePair h t) in -let mylist = makeList 1 (makeList 2 (makeList 3 nil)) 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 -let succ = \n s z. s (n s z) 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 -length mylist +eq 3 3+ Then `length mylist` evaluates to 3. -What does `head (tail (tail mylist))` evaluate to? +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. 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. Now write the same function (true iff two lists have the same +length) but don't use the length function (hint: use `leq` as a model). + + That is, (makeList 1 (makeList 2 (makeList 3 nil))) + +[The following should be correct, but won't run in my browser: + +let factorial = Y (\fac n. isZero n 1 (mult n (fac (predecessor n)))) in + +

+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))) ++ +It may require more resources than my browser is willing to devote to +JavaScript.] + diff --git a/test.mdwn b/test.mdwn deleted file mode 100644 index 12315d65..00000000 --- a/test.mdwn +++ /dev/null @@ -1,3 +0,0 @@ -A hook to get access to files still under construction. - -[[week2]] -- 2.11.0