Haskell homework

Thue sequence

Write a Haskell program to produce the infinite Thue sequence, t, defined by the limit

    lim i->infinity: t_i
where
    t₀ = 0
    t_i+1 = t_i [0->01, 1->10]

The last expression means replace (simultaenously) each 0 in t_i by 01 and each 1 by 10. Thus

t = 01101001100101101001011001101001...

Some properties that may be verified:

t_i+1 = t_i ~t_i , the catenation of t_i and its complement.

t = t # ~t, where # is the interleave operator defined on equal-length sequences:

    infixl 6 #    -- 6 is the same precedence as +
    (#) :: [a] -> [a] -> [a]
    [] # [] = []
    (x:xs) # (y:ys) = x : y : xs#ys

The nth character of t (counted from 0) is the parity bit for the binary representation of n.
t contains no `cubes', substrings of the form www, where w is nonempty.
t_2i is a palindrome.

Your program should produce the first n characters of t in time O(n), and notionally run forever. Note that naive counting of bits per property 2 takes time O(n logn).

Nand expressions

It is well known that nand is universal, in the sense that all Boolean functions are expressible in terms of nand. For the record, nand may be defined thus:

    nand :: Bool -> Bool -> Bool
    nand True True = False
    nand _ _ = True

Write a Haskell program that finds a minimal nand-only expression for each of the 16 binary operations with signature Bool×Bool->Bool. For example, the binary operation f(x,y) whose truth table is

x y f(x,y)

False False True

False True True

True False False

True True False

is equivalent to nand x x.

Problems designed by Doug McIlroy

Last modified:
Tue May 20 12:03:50 EDT 2003

x	y	f(x,y)
`False`	`False`	`True`
`False`	`True`	`True`
`True`	`False`	`False`
`True`	`True`	`False`