module Satchmo.Binary
( Number, width, number, fixed
, add, times
, equals
)
where
import Prelude hiding ( and, or, not )
import qualified Satchmo.Code as C
import Satchmo.Boolean
import Satchmo.Counting
type Booleans = [ Boolean ]
data Number = Number
{ encode :: Booleans
, decode :: C.Decoder Integer
}
instance C.Decode Number Integer where
decode = decode
width :: Number -> Int
width n = length $ encode n
number :: Int -> SAT Number
number w = do
xs <- sequence $ replicate w boolean
return $ make xs
make :: [ Boolean ] -> Number
make xs = Number
{ encode = xs
, decode = do ys <- mapM C.decode xs ; return $ fromBinary ys
}
fromBinary :: [ Bool ] -> Integer
fromBinary xs = foldr ( \ x y -> 2*y + if x then 1 else 0 ) 0 xs
toBinary :: Int -> Integer -> [ Bool ]
toBinary 0 0 = []
toBinary b n | b > 0 =
let (d,m) = divMod n 2
in toEnum ( fromIntegral m ) : toBinary (b1) d
fixed :: Int -> Integer -> SAT Number
fixed b n = do
xs <- mapM constant $ toBinary b n
return $ make xs
add :: Number -> Number -> SAT Number
add ( Number { encode = xs } ) ( Number { encode = ys } ) = do
false <- constant False
( zs, carry ) <- add_with_carry false xs ys
return $ make $ zs ++ [carry]
restricted_add :: Number -> Number -> SAT Number
restricted_add a b = do
c <- add a b
restricted ( max (width a) (width b)) c
restricted :: Int -> Number -> SAT Number
restricted w ( Number { encode = xs } ) = do
let ( low, high ) = splitAt w xs
sequence $ do x <- high ; return $ assert [ not x ]
return $ make low
add_with_carry :: Boolean
-> Booleans -> Booleans
-> SAT ( Booleans, Boolean )
add_with_carry cin [] [] = return ( [], cin )
add_with_carry cin (x:xs) [] = do
z <- xor [ cin, x ]
c <- and [ cin, x ]
( zs, cout ) <- add_with_carry c xs []
return ( z : zs, cout )
add_with_carry cin [] (y:ys) = do
add_with_carry cin (y:ys) []
add_with_carry cin (x:xs ) (y:ys) = do
z <- xor [ cin, x, y ]
c <- atleast 2 [ cin, x, y ]
( zs, cout ) <- add_with_carry c xs ys
return ( z : zs, cout )
times :: Number -> Number -> SAT Number
times ( Number { encode = [x] } ) ys = times1 x ys
times ( Number { encode = x:xs } ) ys = do
xys <- times1 x ys
xsys <- times (make xs) ys
zs <- shift xsys
add xys zs
shift :: Number -> SAT Number
shift ( Number { encode = xs } ) = do
false <- constant False
return $ make $ false : xs
times1 :: Boolean -> Number -> SAT Number
times1 x ( Number { encode = ys } ) = do
zs <- mapM ( \ y -> and [x,y] ) ys
return $ make zs
equals :: Number -> Number -> SAT Boolean
equals ( Number { encode = xs } ) ( Number { encode = ys } ) = do
equals' xs ys
equals' :: Booleans -> Booleans -> SAT Boolean
equals' [] [] = constant True
equals' (x:xs) (y:ys) = do
z <- xor [x, y]
rest <- equals' xs ys
and [ not z, rest ]
equals' xs [] = and $ map not xs
equals' [] ys = and $ map not ys