module Data.Semiring
( Semiring(..)
, StarSemiring(..)
, HasPositiveInfinity(..)
, HasNegativeInfinity(..)
, PositiveInfinite(..)
, NegativeInfinite(..)
, Infinite(..)
, Add(..)
, Mul(..)
, add
, mul
, Max(..)
, Min(..)
) where
import Data.Functor.Const (Const (..))
import Data.Functor.Identity (Identity (..))
import Data.Complex (Complex)
import Data.Fixed (Fixed, HasResolution)
import Data.Ratio (Ratio)
import Numeric.Natural (Natural)
import Data.Int (Int16, Int32, Int64, Int8)
import Data.Word (Word16, Word32, Word64, Word8)
import Foreign.C.Types (CChar, CClock, CDouble, CFloat, CInt,
CIntMax, CIntPtr, CLLong, CLong,
CPtrdiff, CSChar, CSUSeconds, CShort,
CSigAtomic, CSize, CTime, CUChar, CUInt,
CUIntMax, CUIntPtr, CULLong, CULong,
CUSeconds, CUShort, CWchar)
import Foreign.Ptr (IntPtr, WordPtr)
import System.Posix.Types (CCc, CDev, CGid, CIno, CMode, CNlink,
COff, CPid, CRLim, CSpeed, CSsize,
CTcflag, CUid, Fd)
import Data.Semigroup hiding (Max (..), Min (..))
import Control.Applicative (liftA2)
import Data.Coerce (coerce)
import GHC.Generics (Generic, Generic1)
import Data.Typeable (Typeable)
import Foreign.Storable (Storable)
import Data.Semiring.Infinite
import Data.Semiring.TH
class Semiring a where
zero
:: a
one
:: a
infixl 7 <.>
(<.>) :: a -> a -> a
infixl 6 <+>
(<+>) :: a -> a -> a
default zero :: Num a => a
default one :: Num a => a
default (<+>) :: Num a => a -> a -> a
default (<.>) :: Num a => a -> a -> a
zero = 0
one = 1
(<+>) = (+)
(<.>) = (*)
class Semiring a =>
StarSemiring a where
star :: a -> a
plus :: a -> a
star x = one <+> plus x
plus x = x <.> star x
type CoerceBinary a b = (a -> a -> a) -> b -> b -> b
instance Semiring Bool where
one = True
zero = False
(<+>) = (||)
(<.>) = (&&)
instance StarSemiring Bool where
star _ = True
plus = id
instance Semiring a =>
Semiring (NegativeInfinite a) where
one = pure one
zero = pure zero
(<+>) =
(coerce :: CoerceBinary (NegativeInfinite (Add a)) (NegativeInfinite a))
mappend
(<.>) = liftA2 (<.>)
instance Semiring a =>
Semiring (PositiveInfinite a) where
one = pure one
zero = pure zero
(<+>) =
(coerce :: CoerceBinary (PositiveInfinite (Add a)) (PositiveInfinite a))
mappend
(<.>) = liftA2 (<.>)
instance Semiring a =>
Semiring (Infinite a) where
one = pure one
zero = pure zero
(<+>) = (coerce :: CoerceBinary (Infinite (Add a)) (Infinite a)) mappend
(<.>) = liftA2 (<.>)
instance (Eq a, Semiring a) =>
StarSemiring (PositiveInfinite a) where
star (PosFinite x)
| x == zero = one
star _ = PositiveInfinity
instance Semiring () where
one = ()
zero = ()
_ <+> _ = ()
_ <.> _ = ()
instance StarSemiring () where
star _ = ()
plus _ = ()
instance Semiring a =>
Semiring [a] where
one = [one]
zero = []
[] <+> ys = ys
xs <+> [] = xs
(x:xs) <+> (y:ys) = (x <+> y) : (xs <+> ys)
[] <.> _ = []
_ <.> [] = []
(x:xs) <.> (y:ys) =
(x <.> y) : (map (x <.>) ys <+> map (<.> y) xs <+> (xs <.> ys))
type WrapBinary f a = (a -> a -> a) -> f a -> f a -> f a
newtype Add a = Add
{ getAdd :: a
} deriving (Eq,Ord,Read,Show,Bounded,Generic,Generic1,Num,Enum,Typeable
,Storable,Fractional,Real,RealFrac,Functor,Foldable,Traversable
,Semiring,StarSemiring)
newtype Mul a = Mul
{ getMul :: a
} deriving (Eq,Ord,Read,Show,Bounded,Generic,Generic1,Num,Enum,Typeable
,Storable,Fractional,Real,RealFrac,Functor,Foldable,Traversable
,Semiring,StarSemiring)
instance Semiring a =>
Semigroup (Add a) where
(<>) = (coerce :: WrapBinary Add a) (<+>)
instance Semiring a =>
Semigroup (Mul a) where
(<>) = (coerce :: WrapBinary Mul a) (<.>)
instance Semiring a =>
Monoid (Add a) where
mempty = Add zero
mappend = (<>)
instance Semiring a =>
Monoid (Mul a) where
mempty = Mul one
mappend = (<>)
add
:: (Foldable f, Semiring a)
=> f a -> a
add = getAdd . foldMap Add
mul
:: (Foldable f, Semiring a)
=> f a -> a
mul = getMul . foldMap Mul
newtype Min a = Min
{ getMin :: a
} deriving (Eq,Ord,Read,Show,Bounded,Generic,Generic1,Num,Enum,Typeable
,Storable,Fractional,Real,RealFrac,Functor,Foldable,Traversable)
newtype Max a = Max
{ getMax :: a
} deriving (Eq,Ord,Read,Show,Bounded,Generic,Generic1,Num,Enum,Typeable
,Storable,Fractional,Real,RealFrac,Functor,Foldable,Traversable)
instance Ord a =>
Semigroup (Max a) where
(<>) = (coerce :: WrapBinary Max a) max
instance Ord a =>
Semigroup (Min a) where
(<>) = (coerce :: WrapBinary Min a) min
instance (Ord a, HasNegativeInfinity a) =>
Monoid (Max a) where
mempty = Max negativeInfinity
mappend = (<>)
instance (Ord a, HasPositiveInfinity a) =>
Monoid (Min a) where
mempty = Min positiveInfinity
mappend = (<>)
instance (Semiring a, Ord a, HasNegativeInfinity a) =>
Semiring (Max a) where
(<+>) = mappend
zero = mempty
(<.>) = (coerce :: WrapBinary Max a) (<+>)
one = Max zero
instance (Semiring a, Ord a, HasPositiveInfinity a) =>
Semiring (Min a) where
(<+>) = mappend
zero = mempty
(<.>) = (coerce :: WrapBinary Min a) (<+>)
one = Min zero
instance (Semiring a, Ord a, HasPositiveInfinity a, HasNegativeInfinity a) =>
StarSemiring (Max a) where
star (Max x)
| x > zero = Max positiveInfinity
| otherwise = Max zero
instance (Semiring a, Ord a, HasPositiveInfinity a, HasNegativeInfinity a) =>
StarSemiring (Min a) where
star (Min x)
| x < zero = Min negativeInfinity
| otherwise = Min zero
instance Semiring b =>
Semiring (a -> b) where
zero = const zero
one = const one
(f <+> g) x = f x <+> g x
(f <.> g) x = f x <.> g x
instance StarSemiring b =>
StarSemiring (a -> b) where
star f x = star (f x)
plus f x = plus (f x)
instance Monoid a =>
Semiring (Endo a) where
zero = Endo mempty
Endo f <+> Endo g = Endo (f `mappend` g)
one = mempty
(<.>) = mappend
instance (Monoid a, Eq a) =>
StarSemiring (Endo a) where
star (Endo f) = Endo converge
where
converge x = go x
where
go inp =
mappend
x
(if inp == next
then inp
else go next)
where
next = mappend x (f inp)
instance Semiring Any where
(<+>) = coerce (||)
zero = Any False
(<.>) = coerce (&&)
one = Any True
instance StarSemiring Any where
star _ = Any True
plus = id
instance Semiring All where
(<+>) = coerce (||)
zero = All False
(<.>) = coerce (&&)
one = All True
instance StarSemiring All where
star _ = All True
plus = id
instance Semiring Int
instance Semiring Int8
instance Semiring Int16
instance Semiring Int32
instance Semiring Int64
instance Semiring Integer
instance Semiring Word
instance Semiring Word8
instance Semiring Word16
instance Semiring Word32
instance Semiring Word64
instance Semiring Float
instance Semiring Double
instance Semiring CUIntMax
instance Semiring CIntMax
instance Semiring CUIntPtr
instance Semiring CIntPtr
instance Semiring CSUSeconds
instance Semiring CUSeconds
instance Semiring CTime
instance Semiring CClock
instance Semiring CSigAtomic
instance Semiring CWchar
instance Semiring CSize
instance Semiring CPtrdiff
instance Semiring CDouble
instance Semiring CFloat
instance Semiring CULLong
instance Semiring CLLong
instance Semiring CULong
instance Semiring CLong
instance Semiring CUInt
instance Semiring CInt
instance Semiring CUShort
instance Semiring CShort
instance Semiring CUChar
instance Semiring CSChar
instance Semiring CChar
instance Semiring IntPtr
instance Semiring WordPtr
instance Semiring Fd
instance Semiring CRLim
instance Semiring CTcflag
instance Semiring CSpeed
instance Semiring CCc
instance Semiring CUid
instance Semiring CNlink
instance Semiring CGid
instance Semiring CSsize
instance Semiring CPid
instance Semiring COff
instance Semiring CMode
instance Semiring CIno
instance Semiring CDev
instance Semiring Natural
instance Integral a => Semiring (Ratio a)
deriving instance Semiring a => Semiring (Product a)
deriving instance Semiring a => Semiring (Sum a)
instance RealFloat a => Semiring (Complex a)
instance HasResolution a => Semiring (Fixed a)
deriving instance Semiring a => Semiring (Identity a)
deriving instance Semiring a => Semiring (Const a b)
$(traverse semiringIns [2..9])
$(traverse starIns [2..9])