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The last ten years saw the emergence of some results about recognizable subsets of a free monoid with multiplicities in the Min-Plus semiring. An interesting aspect of this theoretical body is that its discovery was motivated throughout by applications such as the finite power property, Eggan's classical star height problem and the measure of the nondeterministic complexity of finite automata. We review here these results, their applications and point out some open problems. 1 Introduction One of the richest extensions of finite automaton theory is obtained by associating multiplicities to words, edges and states. Perhaps the most intuitive appearence of this concept is obtained by counting for every word the number of successful paths spelling it in a (nondeterministic) finite automaton. This is motivated by the formalization of ambiguity in a finite automaton and leads to the theory of recognizable subsets of a free monoid with multiplicities in the semiring of natural numbers. This...

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

We show that the answer to the Burnside problem is positive for semigroups of matrices with entries in the (max,+)-algebra (that is, the semiring (R[ f\Gamma1g; max; +)), and also for semigroups of (max,+)-linear projective maps with rational entries. An application to the estimation of the Lyapunov exponent of certain products of random matrices is also discussed. 1. Introduction The "(max,+)-algebra" is a traditional name for the semiring (R[f\Gamma1g; max; +), denoted Rmax in the sequel. This is a particular example of idempotent semiring (that is a semiring whose additive law satisfies a \Phi a = a), also known as dioid [17, 18, 2]. This algebraic structure has been popularized by its applications to Graph Theory and Operations Research [17, 8]. Linear operators in this algebra are central in Hamilton-Jacobi theory and in the study of exponential asymptotics [33]. The study of automata and semigroups of matrices over the analogous "tropical" semiring (N [ f+1g;min;+) has been ...

this paper, we present the topological approach to the limitedness problem on distance automata. Our techniques has been improved so that the Brown's result is no longer needed. The main mathematical tools used are the local structure theory for nite semigroup [13] and some basic topological ideas. Most of the technical results in this paper except Lemma 3.9 and Lemma 3.10 were obtained in ([14], [15]). Besides for the sake of completeness, we include all the proofs because many of them have been re-worked for better presentations

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: In this paper we pursue the study of the decidability of the dot-depth hierarchy. We give an e#ective lower bound for the dot-depth of an aperiodic monoid. The main tool for this is the study of a certain operation on varieties of finite monoids in terms of Mal'cev product. We also prove the equality of two decidable varieties which were known to contain all dot-depth two monoids. Finally, we restrict our attention to inverse monoids, and we prove that the class of inverse dot-depth two monoids is locally finite. In this paper we pursue the study of the dot-depth hierarchy, a hierarchy of star-free (i.e. recognizable aperiodic) languages with connections to finite monoid theory, formal logic (Thomas [30]) and computational complexity (Barrington and Therien [1]). The dot-depth hierarchy was first introduced by Brzozowski and Cohen [3] in 1971 and was studied by numerous authors since. It consists in a strictly increasing sequence of classes of star-free languages whose union is the c...

We show that the answer to the Burnside problem is positive for semigroups of matrices with entries in the (max; +)-algebra (that is, the semiring (R[ f\Gamma1g; max; +)), and also for semigroups of (max; +)-linear projective maps with rational entries. An application to the estimation of the Lyapunov exponent of certain products of random matrices is also discussed.

infinite structures. Abstract. The theorem of factorisation forests shows the existence of nested factorisations — a la Ramsey — for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the importance of this approach in the context of automata over infinite words and trees. We extend the theorem of factorisation forest in two directions: we show that it is still valid for any word indexed by a linear ordering; and we show that it admits a deterministic variant for words indexed by well-orderings. A byproduct of this work is also an improvement on the known bounds for the original result. We apply the first variant for giving a simplified proof of the closure under complementation of rational sets of words indexed by countable scattered linear orderings. We apply the second variant in the analysis of monadic second-order logic over trees, yielding new results on monadic interpretations over trees. Consequences of it are new caracterisations of prefix-recognizable structures and of the Caucal hierarchy. 1

Simon [3] has proved that every morphism from a free semigroup to a finite semigroup S admits a Ramseyan factorization forest of height at most 9jSj. In this paper, we prove the same result of Simon with an improved bound of 7jSj. We provide a simple algorithm for constructing a factorization forest. In addition, we show that the algorithm cannot be improved significantly. We give examples of semigroup morphism such that any Ramseyan factorization forest for the morphism would require a height not less than jSj.