# semirings

two monoids as one, in holy haskimony http://github.com/chessai/semirings

 Version on this page: 0.2.1.1 LTS Haskell 13.25: 0.2.1.1 Stackage Nightly 2019-06-12: 0.4.2 Latest on Hackage: 0.4.2

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# semirings

Haskellers are usually familiar with monoids and semigroups. A monoid has an appending operation `<>` or `mappend` and an identity element `mempty`. A semigroup has an append `<>`, but does not require an `mempty` element.

A Semiring has two appending operations, ‘plus’ and ‘times’, and two respective identity elements, ‘zero’ and ‘one’.

More formally, A semiring R is a set equipped with two binary relations + and *, such that:

• (R, +) is a commutative monoid with identity element 0:
• (a + b) + c = a + (b + c)
• 0 + a = a + 0 = a
• a + b = b + a
• (R, *) is a monoid with identity element 1:
• (a * b) * c = a * (b * c)
• 1 * a = a * 1 = a
• Multiplication left and right distributes over addition
• a * (b + c) = (a * b) + (a * c)
• (a + b) * c = (a * c) + (b * c)
• Multiplication by ‘0’ annihilates R:
• 0 * a = a * 0 = 0

# *-semirings

A *-semiring (pron. “star-semiring”) is any semiring with an additional operation ‘star’ (read as “asteration”), such that:

• star a = 1 + a * star a = 1 + star a * a

A derived operation called “aplus” can be defined in terms of star by:

• star :: a -> a
• star a = 1 + aplus a
• aplus :: a -> a
• aplus a = a * star a

As such, a minimal instance of the typeclass ‘Star’ requires only ‘star’ or ‘aplus’ to be defined.

# use cases

semirings themselves are useful as a way to express that a type that supports a commutative and associative operation. Some examples:

• Numbers {Int, Integer, Word, Double, etc.}:
• ‘plus’ is ‘Prelude.+’
• ‘times’ is ‘Prelude.*’
• ‘zero’ is 0.
• ‘one’ is 1.
• Booleans:
• ‘plus’ is ‘||’
• ‘times’ is ‘&&’
• ‘zero’ is ‘False’
• ‘one’ is ‘True’
• Set:
• ‘plus’ is ‘union’
• ‘times’ is ‘intersection’
• ‘zero’ is the empty Set.
• ‘one’ is the singleton Set containing the ‘one’ element of the underlying type.
• NFA:
• ‘plus’ unions two NFAs.
• ‘times’ appends two NFAs.
• ‘zero’ is the NFA that acceptings nothing.
• ‘one’ is the empty NFA.
• DFA:
• ‘plus’ unions two DFAs.
• ‘times’ intersects two DFAs.
• ‘zero’ is the DFA that accepts nothing.
• ‘one’ is the DFA that accepts everything.

*-semirings are useful in a number of applications; such as matrix algebra, regular expressions, kleene algebras, graph theory, tropical algebra, dataflow analysis, power series, and linear recurrence relations.

http://stedolan.net/research/semirings.pdf

http://r6.ca/blog/20110808T035622Z.html

https://byorgey.wordpress.com/2016/04/05/the-network-reliability-problem-and-star-semirings/

Some of the code in this library was lifted directly from the Haskell library ‘semiring-num’.

## 0.4.2: [2019.06.06]

• Add `Euclidean` typeclass.
• Add `Mod2`, the integers modulo 2, along with its Semiring/Ring/Star instances. 0.4.1: [2019.05.04]

• Remove unlawful and useless `Ring` instance for `GHC.Natural.Natural`.
• Correct behaviour/docs of Data.Semiring.(^)

## 0.4: [2019.05.01]

• Remove unlawful instances of `Ring` (thanks to @Bodigrim for noticing these)
• Add `fromNatural` to `Semiring` typeclass (thanks @Bodigrim)
• Remove Semiring/Ring instances for [] and Vector. (thanks @Bodigrim) These instances are better served by a dedicated polynomial package, which @Bodigrim has made at http://hackage.haskell.org/package/poly.

## 0.3.1.2: [2019.04.02]

• Fix build error on windows caused by providing instances to POSIX types. Thanks to @Bodigrim and @CarlEdman for reporting this.

## 0.3.1.1: [2019.01.12]

• Fix build error caused by disabling building with containers.

## 0.3.1.0: [2019.01.12]

• Fix build problem on GHC 7.4 caused by introduction of IntSetOf/IntMapOf
• Make sure there are no warnings when building with -Wall, for any GHC

## 0.3.0.0: [2019.01.01]

• Rename the test suite to make `stack` happy.
• Clarified documentation. See #26.
• Simplify implementation of `^`. See #24.
• Add ‘GenericSemiring’, a newtype wrapper meant to be used with `-XDerivingVia`, helping avoid ‘-XDefaultSignatures’.
• Add newtypes for `IntSet` and `IntMap`.
• Remove `Semiring` and `Ring` instances for `Product` and `Sum`.
• Make `sum` and `product` more efficient for base>=4.7

## 0.2.1.1: [2018.10.01]

• Fixed build on GHC-7.4
• Provide `Semiring` and `Ring` for an arbitrary `Num` via `WrappedNum` newtype.
• Make note of `Semiring` semantics for `Vector` and `[]` in the documentation.
• Require build script to ensure `semirings` builds with GHC-8.4.3 and GHC-8.6.1
• Fixed unlawful behaviour of `[]` `Semiring` instance.
• Improve performance of `^`.

## 0.2.1.0: [2018.09.26]

• Removed use of DefaultSignatures
• Removed free semiring

## 0.2.0.1: [2018.07.28]

• Add instances for `Op`, `Equivalence`, `Comparison`, and `Predicate` from Data.Functor.Contravariant (upcoming base 4.12.0.0)
• docfix for (prod -> product, prod’ -> product’) change that occured in version 0.2.0.0.

## 0.2.0.0: [2018.07.23]

• Fixed the `Semiring` instances of `Set`, `HashSet`, `Vector`, `Storable Vector`, `Unboxed Vector`.
• Removed the `Semiring` instances of `Seq`, `Alt`, `Endo`.
• Added comprehensive test suite that tests all `Semiring` instances defined in Data.Semiring
• Added newtypes: `Add`, `Mul`
• Bounds for containers: [0.3,0.6] -> [0.5.4,0.6.0.9]
• Add semiring instance for `Proxy`
• names changed: (prod -> product, prod’ -> product’)
• sum’ and product’ now use foldl’ instead of foldr’

## 0.1.2: [2018.05.04]

• `semirings` now builds back to GHC-7.4.1.
• many doc fixes.

## 0.1.1: [2018.04.20]

• Remove unused `coerce-util` dependency.

## 0.1.0:

• Initial version.
Depends on 5 packages:
Used by 10 packages: