bit-stream
Lazy, infinite, compact stream of Bool
with O(1) indexing.
Most useful for memoization of predicates.
Example 1
Consider following predicate:
isOdd :: Word -> Bool
isOdd 0 = False
isOdd n = not (isOdd (n - 1))
Its computation is expensive, so we’d like to memoize its values into
BitStream
using tabulate
and access this stream via index
instead of recalculation of isOdd
:
isOddBS :: BitStream
isOddBS = tabulate isOdd
isOdd' :: Word -> Bool
isOdd' = index isOddBS
We can do even better by replacing part of recursive calls to isOdd
by indexing memoized values. Write isOddF
such that isOdd = fix isOddF
:
isOddF :: (Word -> Bool) -> Word -> Bool
isOddF _ 0 = False
isOddF f n = not (f (n - 1))
and use tabulateFix
:
isOddBS :: BitStream
isOddBS = tabulateFix isOddF
isOdd' :: Word -> Bool
isOdd' = index isOddBS
Example 2
Define a predicate, which checks whether its argument is
a prime number by trial division.
isPrime :: Word -> Bool
isPrime n
| n < 2 = False
| n < 4 = True
| even n = False
| otherwise = and [ n `rem` d /= 0 | d <- [3, 5 .. ceiling (sqrt (fromIntegral n))], isPrime d]
Convert it to unfixed form:
isPrimeF :: (Word -> Bool) -> Word -> Bool
isPrimeF f n
| n < 2 = False
| n < 4 = True
| even n = False
| otherwise = and [ n `rem` d /= 0 | d <- [3, 5 .. ceiling (sqrt (fromIntegral n))], f d]
Create its memoized version for faster evaluation:
isPrimeBS :: BitStream
isPrimeBS = tabulateFix isPrimeF
isPrime' :: Word -> Bool
isPrime' = index isPrimeBS