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Introduction to semiring in Computer Science

From Nanyang Technological University ( 南洋理工大学) in Singapore lecture

By Lim Zhi Hao

semiring in computer science call tuple is a system

Tropical semiring is a system (ℝ ∪ {+∞},min{x,y},x+y,+∞,0)

Log semiring is a system (ℝ ∪ {-∞,∞},x(⊕log)y,x+y,+∞,0)

where x(⊕log)y = -log((e^-x)+(e^-y))

Boolean semiring is a system ({0,1},x or y,x and y,0,1)

Proability semiring is a system (ℝ+ ∪ {+∞},x+y,x*y,0,1)

from Minesutsu lab at University of Tokyo (東京大学)[ 東大] in Japan

by Josef R. Novak suggest that NLP and ASP application interrest in Tropical semiring and Log semiring

with condition for ringoid

2.⊕ and ⊗ are binary operations on K

3. the operation ⊗ distributes over ⊕ both right and left.

∀a,b,c ∈ K : a⊗(b⊕c) = (a⊗b)⊕(a⊗c)…………left distribution

∀a,b,c ∈ K : (a⊕b)⊗c =(a⊗c)⊕(b⊗c)………….right distribution

which a⊗(b⊕c) we require that K is closed under ⊕.

and (a⊗b)⊕(a⊗c) we require that K is closed under ⊗. repectively

where ∀ b,c ∈ K×K : b⊕c ∈ K

∀ a,b ∈ K×K : a⊗b ∈ K and ∀ b,c ∈ K×K : b⊗c ∈ K

4. have Closure property

other refrence Closure definition
(K,*) be an algebraic structure.

Then K has the property of closure under * if and only if

: ∀x,y ∈ K×K : x*y ∈ K

K is said to be closed under *

or (K,*) is closed

a system (ℝ,+, ×) is distributive law in Algebra

and

A magma is an algebraic structure (K,*)such that K is closed under *.

That is, a magma definition

is a pair (K,*) where:

K is a set

* : K×K→K is a binary operation on K

∀a,b ∈ K : a⊕b ∈ K

∀a,b,c ∈ K : a⊕(b⊕c) = (a⊕b)⊕c

∀a,b ∈ K : a⊗b ∈ K

∀a,b,c ∈ K : a⊗(b⊗c) = (a⊗b)⊗c

∀a,b,c ∈ K : a⊗(b⊕c) = (a⊗b)⊕(a⊗c)…………left distribution

∀a,b,c ∈ K : (a⊕b)⊗c =(a⊗c)⊕(b⊗c)………….right distribution

from Signal Processing on Databases Lincoln Laboratory at Massachusetts Institute of Technology in USA

can be use semiring in product formation in this way

In computer science it hard to explain with semiring with mathematical way.

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