Data.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.

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.

Inspecting symbolic traces

ladnerFischerTrace :: Int -> IO ()

A symbolic trace can help illustrate the action of Ladner-Fischer. This generator produces the actions of Ladner-Fischer for addition, showing how the computation proceeds:

>>> ladnerFischerTrace 8
INPUTS
  s0 :: SWord8
  s1 :: SWord8
  s2 :: SWord8
  s3 :: SWord8
  s4 :: SWord8
  s5 :: SWord8
  s6 :: SWord8
  s7 :: SWord8
CONSTANTS
  s_2 = False :: Bool
  s_1 = True :: Bool
TABLES
ARRAYS
UNINTERPRETED CONSTANTS
USER GIVEN CODE SEGMENTS
AXIOMS
TACTICS
GOALS
DEFINE
  s8 :: SWord8 = s0 + s1
  s9 :: SWord8 = s2 + s8
  s10 :: SWord8 = s2 + s3
  s11 :: SWord8 = s8 + s10
  s12 :: SWord8 = s4 + s11
  s13 :: SWord8 = s4 + s5
  s14 :: SWord8 = s11 + s13
  s15 :: SWord8 = s6 + s14
  s16 :: SWord8 = s6 + s7
  s17 :: SWord8 = s13 + s16
  s18 :: SWord8 = s11 + s17
CONSTRAINTS
ASSERTIONS
OUTPUTS
  s0
  s8
  s9
  s11
  s12
  s14
  s15
  s18

scanlTrace :: Int -> IO ()

Trace generator for the reference spec. It clearly demonstrates that the reference implementation fewer operations, but is not parallelizable at all:

>>> scanlTrace 8
INPUTS
  s0 :: SWord8
  s1 :: SWord8
  s2 :: SWord8
  s3 :: SWord8
  s4 :: SWord8
  s5 :: SWord8
  s6 :: SWord8
  s7 :: SWord8
CONSTANTS
  s_2 = False :: Bool
  s_1 = True :: Bool
TABLES
ARRAYS
UNINTERPRETED CONSTANTS
USER GIVEN CODE SEGMENTS
AXIOMS
TACTICS
GOALS
DEFINE
  s8 :: SWord8 = s0 + s1
  s9 :: SWord8 = s2 + s8
  s10 :: SWord8 = s3 + s9
  s11 :: SWord8 = s4 + s10
  s12 :: SWord8 = s5 + s11
  s13 :: SWord8 = s6 + s12
  s14 :: SWord8 = s7 + s13
CONSTRAINTS
ASSERTIONS
OUTPUTS
  s0
  s8
  s9
  s10
  s11
  s12
  s13
  s14