Hi i am planning to investigate the applications of semiring in decision making.i wish to have some clues/clarifications of basic queries as follows:
Q.1 what are the examples of semiring structure's real life situation? how do we describe some real life situation that could be modeled into a semiring structure?
Q.2 how can we use boolean lattices/boolean logic/ boolean search etc to solve certain practical problems in semiring structure to arrive at decision making?
Q.3 can we use graphs/vectors/matrices etc as tools in " Application of semiring in decision making" ?
Q.4 how to link semirings to :
(a) graph theory?
(b) vectors and matrices?
(c) boolean algebra and boolean logic etc ?
It may not be my field, but in trying to contribute.. let me ask you if you've already read about Algebraic Decision Diagrams (ADD's), also about Semiring Labelled Decision Diagrams (SLDD's).
Both are kind of extensions of Binary Decision Diagrams.
In their senses, are commonly used on logical synthesis and multivariate polinomials compilation.
Q.1. The usage may appear in real life problems, I can't clarify with eg. but in many articles you will see it arises in networks problems, minimal path algorithms implemented with these kind of diagrams abovementioned.
Q.2. Can't give you tips on that, but see related topics involving the boolean semiring $(\{0,1\},0,1,+, \cdot)$, and its BDD.
Q.3. Recall Q.1. and see that networks problems commonly are described over directed graphs. You'll find couple generics examples.
Q.4. This one have tons of different answers... let's say:
(a) - (b) you may build up a semiring from a valued graph and after use its L-matrix, associating symbols to elements referring to non-connected two vertex, with respect to those semiring operations predesired...
(c) recall Q.1 + Q.2.
If you haven't read about the Decision Diagrams yet, hope I've helped.
You can link semiring with graph theory by giving weights to vertices and edges of a graph G by the elements from a given semiring. Such a study have been initiated in the year 2015 by me with my scholars. The theory developed is called $S$ valued (Semiring valued) graphs. Lot of work is going on in this area, which you can use to study for decision making problems by using shortest path problem or network analysis.
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