# semirings

two monoids as one, in holy haskimony

http://github.com/chessai/semirings

LTS Haskell 21.12: | 0.6@rev:1 |

Stackage Nightly 2023-09-22: | 0.6@rev:1 |

Latest on Hackage: | 0.6@rev:1 |

**chessai**

**chessai**

`semirings-0.6@sha256:cc8767bc62b9ff7bbef070ac206b070b173a8612a781eff2d87a14a01cf406b0,2668`

#### Module documentation for 0.6

*(full list with versions)*:

**nightly-2023-09-22**

*(full list with versions)*:

# 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.

Some relevant (informal) reading material:

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/

# additional credit

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

## Changes

## 0.6: [2021-01-07]

- Remove hashable flag (only necessary was unordered-containers flag)
- Drop redundant
`Eq`

constraint on default definition of`coprime`

- Document (lack of guaranteed) rounding behaviour of quotRem
- Fix totally broken Ord instance for Tropical
- Stop depending on integer-gmp

## 0.5.4: [2020.07.13]

- Drop support for GHCs prior to 7.10
- Add default quotRem implementation
- Expose Data.Semiring.Generic.gfromNatural

## 0.5.3: [2020.02.18]

- Fix non-terminating GenericSemiring instances
- Fix incorrect implementation of gtimes’ for product types in GSemiring
- Implement GcdDomain.divide explicitly
- Remove redundant imports
- Disambiguate all haddock identifiers

## 0.5.2: [2019.11.01]

- Add
`gcdExt`

function - Bump upper bound on base
- Add GcdDomain/Euclidean instances for
`Mod2`

- Add GcdDomain/Euclidean instances for {Int|Word}{8|16|32|64}
- Mention
`RebindableSyntax`

in haddocks

rev: b4334fe06635f106b1f08bac127c1ae259cddae6

## 0.5.1: [2019.09.13]

- Bump upper bound on containers to 0.7
- Bump upper bound on hashable to 1.4
- Remove redundant constraints from WrappedFractional instances
- Add lower bound on semigroups

rev: 7e6f5e312bec5495ce9390664578bfb09d6e3eb9

## 0.5: [2019.09.05]

- Add
`Field`

typeclass, instances, and functions. - Add
`Euclidean`

and`GcdDomain`

instances for`()`

,`CDouble`

,`CFloat`

, and`Complex`

. - Add
`Ring`

and`Bits`

instances for`WrappedFractional`

and`WrappedIntegral`

. - Add
`fromInteger`

and`fromIntegral`

functions for`Ring`

.

rev: eb2617d93d354085fe5b706a145108d090dbc027

## 0.4.2: [2019.06.06]

- Add
`GcdDomain`

and`Euclidean`

typeclasses. - Add
`Mod2`

, the integers modulo 2, along with its Semiring/Ring/Star instances.

rev: b5af2fa403c68a66a3282b2a452b9be1c98e3fd6

## 0.4.1: [2019.05.04]

- Remove unlawful and useless
`Ring`

instance for`GHC.Natural.Natural`

. - Correct behaviour/docs of Data.Semiring.(^)

rev: d6c42aeea602499e32081e84974910d0fe955db6

## 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.
- Add isZero/isOne predicates.

rev: 1285d3e42242db310083fbf78d2e611bccecc63a

## 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.

rev: 13d4b3920912f8030b5d47777fb57b6e0dd15c10

## 0.3.1.1: [2019.01.12]

- Fix build error caused by disabling building with containers.

rev: 5f02279613bfcd20c2e9d68f01d669e563540ced

## 0.3.1.0: [2019.01.12]

- Add Data.Semiring.Tropical
- 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

rev: 68c604250e2cf5688b3c641fd40b66fe7e1d45bf

## 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

rev: d7d47c3db82a8e85330bb138169b9783eb346f38

## 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
`^`

.

rev: e9b85d8aa6a238d07a061402f0ba365190eee7aa

## 0.2.1.0: [2018.09.26]

- Removed use of DefaultSignatures
- Removed free semiring

rev: 68e97e82280a50c374f50500a73222a5432cc45e

## 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.

rev: 60869059d2959676877c9661427814b2bafd5d97

## 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 Free semiring (Data.Semiring.Free)
- 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’

rev: b985dcf37b919facc2dfbec66ea923ca5427c9f6

## 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.