{-# Language AllowAmbiguousTypes #-} -- | See the /connections/ package for idempotent & selective semirings, and lattices. module Data.Semiring.Property ( -- * Required properties of pre-semirings nonunital_on , morphism_presemiring , associative_addition_on , commutative_addition_on , associative_multiplication_on , distributive_on , distributive_finite1_on , morphism_distribitive_on -- * Required properties of semirings , morphism_semiring , neutral_addition_on , neutral_multiplication_on , annihilative_multiplication_on , distributive_finite_on -- * Left-distributive presemirings and semirings , distributive_xmult_on , distributive_xmult1_on -- * Commutative presemirings & semirings , commutative_multiplication_on -- * Cancellative presemirings & semirings , cancellative_addition_on , cancellative_multiplication_on ) where import Data.Semiring import Test.Logic (Rel) import Data.Foldable (Foldable) import Data.Functor.Apply (Apply) import Data.Semigroup.Foldable (Foldable1) import Data.Semigroup.Property import qualified Test.Operation as Prop import Prelude hiding (Num(..), sum) ------------------------------------------------------------------------------------ -- Required properties of pre-semirings & semirings -- | \( \forall a, b \in R: a * b \sim a * b + b \) -- -- If /R/ is non-unital (i.e. /one/ is not distinct from /zero/) then it will instead satisfy -- a right-absorbtion property. -- -- This follows from right-neutrality and right-distributivity. -- -- Compare 'codistributive' and 'closed_stable'. -- -- When /R/ is also left-distributive we get: \( \forall a, b \in R: a * b = a + a * b + b \) -- -- See also 'Data.Warning' and < https://blogs.ncl.ac.uk/andreymokhov/united-monoids/#whatif >. -- nonunital_on :: Presemiring r => Rel r b -> r -> r -> b nonunital_on (~~) a b = (a * b) ~~ (a * b + b) -- | Presemiring morphisms are distributive semigroup morphisms. -- -- This is a required property for presemiring morphisms. -- morphism_presemiring :: Eq s => Presemiring r => Presemiring s => (r -> s) -> r -> r -> r -> Bool morphism_presemiring f x y z = morphism_additive_on (==) f x y && morphism_multiplicative_on (==) f x y && morphism_distribitive_on (==) f x y z ------------------------------------------------------------------------------------ -- Required properties of semirings -- | \( \forall a, b, c \in R: (a + b) * c \sim (a * c) + (b * c) \) -- -- /R/ must right-distribute multiplication. -- -- When /R/ is a functor and the semiring structure is derived from 'Alternative', -- this translates to: -- -- @ -- (a '<|>' b) '*>' c = (a '*>' c) '<|>' (b '*>' c) -- @ -- -- See < https://en.wikibooks.org/wiki/Haskell/Alternative_and_MonadPlus >. -- -- This is a required property. -- distributive_on :: Presemiring r => Rel r b -> r -> r -> r -> b distributive_on (~~) = Prop.distributive_on (~~) (+) (*) -- | \( \forall M \geq 1; a_1 \dots a_M, b \in R: (\sum_{i=1}^M a_i) * b \sim \sum_{i=1}^M a_i * b \) -- -- /R/ must right-distribute multiplication over finite (non-empty) sums. -- -- For types with exact arithmetic this follows from 'distributive' and the universality of 'fold1'. -- distributive_finite1_on :: Presemiring r => Foldable1 f => Rel r b -> f r -> r -> b distributive_finite1_on (~~) as b = (sum1 as * b) ~~ (sumWith1 (* b) as) -- | \( \forall a, b, c \in R: f ((a + b) * c) \sim f (a * c) + f (b * c) \) -- -- Presemiring morphisms must be compatible with right-distribution. -- morphism_distribitive_on :: Presemiring r => Presemiring s => Rel s b -> (r -> s) -> r -> r -> r -> b morphism_distribitive_on (~~) f x y z = (f $ (x + y) * z) ~~ (f (x * z) + f (y * z)) ------------------------------------------------------------------------------------ -- Required properties of semirings -- | Semiring morphisms are monoidal presemiring morphisms. -- -- This is a required property for semiring morphisms. -- morphism_semiring :: Eq s => Semiring r => Semiring s => (r -> s) -> r -> r -> r -> Bool morphism_semiring f x y z = morphism_presemiring f x y z && morphism_additive_on' (==) f && morphism_multiplicative_on' (==) f -- | \( \forall a \in R: (z * a) \sim u \) -- -- A /R/ is semiring then its addititive one must be right-annihilative, i.e.: -- -- @ -- 'zero' '*' a ~~ 'zero' -- @ -- -- For 'Alternative' instances this property translates to: -- -- @ -- 'empty' '*>' a ~~ 'empty' -- @ -- -- All right semirings must have a right-absorbative addititive one, -- however note that depending on the 'Prd' instance this does not preclude -- IEEE754-mandated behavior such as: -- -- @ -- 'zero' '*' NaN ~~ NaN -- @ -- -- This is a required property. -- annihilative_multiplication_on :: Semiring r => Rel r b -> r -> b annihilative_multiplication_on (~~) r = Prop.annihilative_on (~~) (*) zero r -- | \( \forall M \geq 0; a_1 \dots a_M, b \in R: (\sum_{i=1}^M a_i) * b \sim \sum_{i=1}^M a_i * b \) -- -- /R/ must right-distribute multiplication between finite sums. -- -- For types with exact arithmetic this follows from 'distributive' & 'neutral_multiplication'. -- distributive_finite_on :: Semiring r => Foldable f => Rel r b -> f r -> r -> b distributive_finite_on (~~) as b = (sum as * b) ~~ (sumWith (* b) as) ------------------------------------------------------------------------------------ -- Left-distributive presemirings and semirings -- | \( \forall M,N \geq 0; a_1 \dots a_M, b_1 \dots b_N \in R: (\sum_{i=1}^M a_i) * (\sum_{j=1}^N b_j) \sim \sum_{i=1 j=1}^{i=M j=N} a_i * b_j \) -- -- If /R/ is also left-distributive then it supports xmult-multiplication. -- distributive_xmult_on :: Semiring r => Applicative f => Foldable f => Rel r b -> f r -> f r -> b distributive_xmult_on (~~) as bs = (sum as * sum bs) ~~ (xmult as bs) -- | \( \forall M,N \geq 1; a_1 \dots a_M, b_1 \dots b_N \in R: (\sum_{i=1}^M a_i) * (\sum_{j=1}^N b_j) = \sum_{i=1 j=1}^{i=M j=N} a_i * b_j \) -- -- If /R/ is also left-distributive then it supports (non-empty) xmult-multiplication. -- distributive_xmult1_on :: Presemiring r => Apply f => Foldable1 f => Rel r b -> f r -> f r -> b distributive_xmult1_on (~~) as bs = (sum1 as * sum1 bs) ~~ (xmult1 as bs)