Documentation.SBV.Examples.BitPrecise.PrefixSum

The PrefixSum algorithm over power-lists and proof of the Ladner-Fischer implementation. See http://dl.acm.org/citation.cfm?id=197356 and http://www.cs.utexas.edu/~plaxton/c/337/05f/slides/ParallelRecursion-4.pdf.

Formalizing power-lists

type PowerList a = [a]

A poor man's representation of powerlists and basic operations on them: http://dl.acm.org/citation.cfm?id=197356 We merely represent power-lists by ordinary lists.

tiePL :: PowerList a -> PowerList a -> PowerList a

The tie operator, concatenation.

zipPL :: PowerList a -> PowerList a -> PowerList a

The zip operator, zips the power-lists of the same size, returns a powerlist of double the size.

unzipPL :: PowerList a -> (PowerList a, PowerList a)

Inverse of zipping.

Reference prefix-sum implementation

ps :: (a, a -> a -> a) -> PowerList a -> PowerList a

Reference prefix sum (ps) is simply Haskell's scanl1 function.

The Ladner-Fischer parallel version

lf :: (a, a -> a -> a) -> PowerList a -> PowerList a

The Ladner-Fischer (lf) implementation of prefix-sum. See http://www.cs.utexas.edu/~plaxton/c/337/05f/slides/ParallelRecursion-4.pdf or pg. 16 of http://dl.acm.org/citation.cfm?id=197356

Sample proofs for concrete operators

flIsCorrect :: Int -> (forall a. (OrdSymbolic a, Num a, Bits a) => (a, a -> a -> a)) -> Symbolic SBool

Correctness theorem, for a powerlist of given size, an associative operator, and its left-unit element.

thm1 :: IO ThmResult

Proves Ladner-Fischer is equivalent to reference specification for addition. 0 is the left-unit element, and we use a power-list of size 8. We have:

>>> thm1
Q.E.D.

thm2 :: IO ThmResult

Proves Ladner-Fischer is equivalent to reference specification for the function max. 0 is the left-unit element, and we use a power-list of size 16. We have:

>>> thm2
Q.E.D.