semiring-num-0.1.0.2: src/Data/Semiring.hs
{-# LANGUAGE DefaultSignatures #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE StandaloneDeriving #-}
{-|
Module: Data.Semiring
Description: Haskell semirings
License: MIT
Maintainer: mail@doisinkidney.com
Stability: experimental
-}
module Data.Semiring
( Semiring(..)
, Add(..)
, Mul(..)
) where
import Data.Coerce (coerce)
import Data.Functor.Const (Const (..))
import Data.Functor.Identity (Identity (..))
import Data.Monoid
import Data.Semigroup (Max (..), Min (..))
import Data.Complex (Complex)
import Data.Fixed (Fixed, HasResolution)
import Data.Ratio (Ratio)
import Numeric.Natural (Natural)
import Data.Set (Set)
import qualified Data.Set as Set
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 GHC.Generics (Generic, Generic1)
import Test.QuickCheck (Arbitrary)
-- | A <https://en.wikipedia.org/wiki/Semiring Semiring> is like the
-- the combination of two 'Data.Monoid.Monoid's. The first
-- is called '<+>'; it has the identity element 'zero', and it is
-- commutative. The second is called '<.>'; it has identity element 'one',
-- and it must distribute over '<+>'.
--
-- = Laws
-- == Normal 'Precursor.Algebra.Monoid.Monoid' laws
-- * @(a '<+>' b) '<+>' c = a '<+>' (b '<+>' c)@
-- * @'zero' '<+>' a = a '<+>' 'zero' = a@
-- * @(a '<.>' b) '<.>' c = a '<.>' (b '<.>' c)@
-- * @'one' '<.>' a = a '<.>' 'one' = a@
--
-- == Commutativity of '<+>'
-- * @a '<+>' b = b '<+>' a@
--
-- == Distribution of '<.>' over '<+>'
-- * @a '<.>' (b '<+>' c) = (a '<.>' b) '<+>' (a '<.>' c)@
-- * @(a '<+>' b) '<.>' c = (a '<.>' c) '<+>' (b '<.>' c)@
--
-- Another useful law, annihilation, may be deduced from the axioms
-- above:
--
-- * @'zero' '<.>' a = a '<.>' 'zero' = 'zero'@
class Semiring a where
-- | The identity of '<+>'.
zero :: a
-- | The identity of '<.>'.
one :: a
-- | An associative binary operation, which distributes over '<+>'.
infixl 7 <.>
(<.>) :: a -> a -> a
-- | An associative, commutative binary operation.
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
(<+>) = (+)
(<.>) = (*)
------------------------------------------------------------------------
-- Instances
------------------------------------------------------------------------
instance Semiring Bool where
one = True
zero = False
(<+>) = (||)
(<.>) = (&&)
instance Semiring () where
one = ()
zero = ()
_ <+> _ = ()
_ <.> _ = ()
cartProd :: (Ord a, Monoid a) => Set a -> Set a -> Set a
cartProd xs ys =
Set.foldl' (\a x ->
Set.foldl' (flip (Set.insert . mappend x)) a ys)
Set.empty xs
-- | The 'Set' 'Semiring' is 'Data.Set.union' for '<+>', and a Cartesian
-- product for '<.>'.
instance (Ord a, Monoid a) => Semiring (Set a) where
(<.>) = cartProd
(<+>) = Set.union
zero = Set.empty
one = Set.singleton mempty
------------------------------------------------------------------------
-- Addition and multiplication newtypes
------------------------------------------------------------------------
type WrapBinary f a = (a -> a -> a) -> f a -> f a -> f a
-- | Monoid under '<+>'. Analogous to 'Data.Monoid.Sum', but uses the
-- 'Semiring' constraint, rather than 'Num'.
newtype Add a = Add
{ getAdd :: a
} deriving (Eq, Ord, Read, Show, Bounded, Generic, Generic1, Num
,Arbitrary)
-- | Monoid under '<.>'. Analogous to 'Data.Monoid.Product', but uses the
-- 'Semiring' constraint, rather than 'Num'.
newtype Mul a = Mul
{ getMul :: a
} deriving (Eq, Ord, Read, Show, Bounded, Generic, Generic1, Num
,Arbitrary)
instance Functor Add where fmap = coerce
instance Functor Mul where fmap = coerce
instance Foldable Add where
foldr =
(coerce :: ((a -> b -> c) -> (b -> a -> c))
-> (a -> b -> c)
-> (b -> Add a -> c)) flip
foldl = coerce
foldMap = coerce
length = const 1
instance Foldable Mul where
foldr =
(coerce :: ((a -> b -> c) -> (b -> a -> c))
-> (a -> b -> c)
-> (b -> Mul a -> c)) flip
foldl = coerce
foldMap = coerce
length = const 1
instance Applicative Add where
pure = coerce
(<*>) =
(coerce :: ((a -> b) -> a -> b)
-> (Add (a -> b) -> Add a -> Add b)) ($)
instance Applicative Mul where
pure = coerce
(<*>) =
(coerce :: ((a -> b) -> a -> b)
-> (Mul (a -> b) -> Mul a -> Mul b)) ($)
instance Monad Add where
(>>=) = flip coerce
instance Monad Mul where
(>>=) = flip coerce
instance Semiring a => Monoid (Add a) where
mempty = Add zero
mappend = (coerce :: WrapBinary Add a) (<+>)
instance Semiring a => Monoid (Mul a) where
mempty = Mul one
mappend = (coerce :: WrapBinary Mul a) (<.>)
instance Semiring a => Semiring (Add a) where
zero = Add zero
one = Add one
(<+>) = (coerce :: WrapBinary Add a) (<+>)
(<.>) = (coerce :: WrapBinary Add a) (<.>)
instance Semiring a => Semiring (Mul a) where
zero = Mul zero
one = Mul one
(<+>) = (coerce :: WrapBinary Mul a) (<+>)
(<.>) = (coerce :: WrapBinary Mul a) (<.>)
------------------------------------------------------------------------
-- Ord wrappers
------------------------------------------------------------------------
-- | The 'Semiring' for 'Max' uses the 'max' operation for '<+>', and
-- normal '+' for '<.>'.
instance (Ord a, Bounded a, Semiring a) => Semiring (Max a) where
(<+>) = mappend
zero = mempty
(<.>) = (coerce :: WrapBinary Max a) (<+>)
one = Max zero
-- | The 'Semiring' for 'Min' uses the 'min' operation for '<+>', and
-- normal '+' for '<.>'.
instance (Ord a, Bounded a, Semiring a) => Semiring (Min a) where
(<+>) = mappend
zero = mempty
(<.>) = (coerce :: WrapBinary Min a) (<+>)
one = Min zero
------------------------------------------------------------------------
-- (->) instance
------------------------------------------------------------------------
-- | The @(->)@ instance is analogous to the one for 'Monoid'.
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
------------------------------------------------------------------------
-- Endo instance
------------------------------------------------------------------------
-- | The 'Endo' semiring uses function composition for '<.>', and
-- pointwise 'mappend' for '<+>'. The underlying 'Monoid' needs to be
-- commutative.
instance Monoid a => Semiring (Endo a) where
(<.>) = mappend
one = mempty
Endo f <+> Endo g = Endo (\x -> f x `mappend` g x)
zero = Endo (const mempty)
------------------------------------------------------------------------
-- Instances for Bool wrappers
------------------------------------------------------------------------
instance Semiring Any where
(<+>) = coerce (||)
zero = Any False
(<.>) = coerce (&&)
one = Any True
instance Semiring All where
(<+>) = coerce (||)
zero = All False
(<.>) = coerce (&&)
one = All True
------------------------------------------------------------------------
-- Boring instances
------------------------------------------------------------------------
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 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)
------------------------------------------------------------------------
-- Very boring instances
------------------------------------------------------------------------
instance (Semiring a, Semiring b) => Semiring (a,b) where
zero = (zero, zero)
(a1,b1) <+> (a2,b2) =
(a1 <+> a2, b1 <+> b2)
one = (one, one)
(a1,b1) <.> (a2,b2) =
(a1 <.> a2, b1 <.> b2)
instance (Semiring a, Semiring b, Semiring c) => Semiring (a,b,c) where
zero = (zero, zero, zero)
(a1,b1,c1) <+> (a2,b2,c2) =
(a1 <+> a2, b1 <+> b2, c1 <+> c2)
one = (one, one, one)
(a1,b1,c1) <.> (a2,b2,c2) =
(a1 <.> a2, b1 <.> b2, c1 <.> c2)
instance (Semiring a, Semiring b, Semiring c, Semiring d) => Semiring (a,b,c,d) where
zero = (zero, zero, zero, zero)
(a1,b1,c1,d1) <+> (a2,b2,c2,d2) =
(a1 <+> a2, b1 <+> b2,
c1 <+> c2, d1 <+> d2)
one = (one, one, one, one)
(a1,b1,c1,d1) <.> (a2,b2,c2,d2) =
(a1 <.> a2, b1 <.> b2, c1 <.> c2, d1 <.> d2)
instance (Semiring a, Semiring b, Semiring c, Semiring d, Semiring e) =>
Semiring (a,b,c,d,e) where
zero = (zero, zero, zero, zero, zero)
(a1,b1,c1,d1,e1) <+> (a2,b2,c2,d2,e2) =
(a1 <+> a2, b1 <+> b2, c1 <+> c2,
d1 <+> d2, e1 <+> e2)
one = (one, one, one, one, one)
(a1,b1,c1,d1,e1) <.> (a2,b2,c2,d2,e2) =
(a1 <.> a2, b1 <.> b2, c1 <.> c2,
d1 <.> d2, e1 <.> e2)