Tropical semirings  pp. 50-69

By Jean-Eric Pin

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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 named 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 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 have tried to keep the paper selfcontained as much as possible. Thus, in principle, there are no prerequisites for reading this survey, apart from a standard mathematical background. However, it was clearly not possible to give a full exposition of the theory of automata within 20 pages. Therefore, suitable references will be given for readers who would like to pursue the subject further and join the tropical community.

The paper is organized as follows. The main definitions are introduced in Section 2. Two apparently disconnected applications of 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.