Tropical Semirings
نویسنده
چکیده
It is a well-known fact that the boolean calculus is one of the mathematical foundations of electronic computers. This explains the important role of the boolean semiring in computer science. The aim of this paper is to present other semirings that occur in theoretical computer science. These semirings were baptized tropical semirings by Dominique Perrin in honour of the pioneering work of our brazilian colleague and friend Imre Simon, but are also commonly known as (min,+)-semirings. The aim of this paper is to present the tropical semirings and to survey a few problems relevant to them. We shall try to give an updated status of the different questions, but detailed solutions of most problems would be too long and technical for this survey. They can be found in the other papers of this volume or in the relevant literature. We tried to keep the paper self-contained as much as possible. Thus, in principle, there are no prerequisite to read this survey, besides a standard mathematical background. However, it was clearly not possible to give within 20 pages a full exposition of the theory of automata. Therefore, suitable references will be given for the readers who would like to elaborate and join the tropical community. The paper is organized as follows. The main definitions are introduced in Section 2. Two apparently disconnected applications of the tropical semirings are presented: the Burnside type problems in group and semigroup theory, in Section 3 and decidability problems in formal language theory, in Section 4. The connection between the two problems is explained in Section 5. A conclusion section ends the paper.
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تاریخ انتشار 2005