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Module documentation for 0.5.2

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semirings

Hackage Build Status

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

0.5.2: [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.
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