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README
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. "starsemiring") 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/thenetworkreliabilityproblemandstarsemirings/
additional credit
Some of the code in this library was lifted directly from the Haskell library 'semiringnum'.