This paper is an attempt to share with a larger audience some modern developments in the theory of finite automata. It is written for the mathematician who has a background in semigroup theory but knows next to nothing on automata and languages. No proofs are given, but the main results are illustrated by several examples and counterexamples

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

Abstract not found

Abstract not found

We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series - the semilinear and the bounded series - both natural generalizations of the semilinear languages, and we study their behaviour under rational operations, morphisms, Hadamard product, and difference. Turning to commutation on sets of words, we then study the notions of centralizer of a language - the largest set commuting with a language -, of root and of primitive root of a set of words. We answer a question raised by Conway more than thirty years ago - asking whether or not the centralizer of any rational language is rational - in the case of periodic, binary, and ternary sets of words, as well as for rational c-codes, the most general results on this problem. We also prove that any code has a unique primitive root and that two codes commute if and only if they have the same primitive root, thus solving two conjectures of Ratoandromanana, 1989. Moreover, we prove that the commutation with an c-code X can be characterized similarly as in free monoids: a language commutes with X if and only if it is a union of powers of the primitive root of X.

Abstract. We give an overview of some structural complexity measures and width parameters on digraphs, relate them to each other, and discuss their computational complexity aspects. Particular attention is given to cycle rank, perhaps the oldest of these measures to be studied in the literature, which turns out to be intimately related to structural and descriptional complexity aspects of regular expressions. Some new results on cycle rank thus have immediate implications in formal language theory. 1

Abstract. We continue our work [H. Gruber, M. Holzer: Provably shorter regular expressions from deterministic finite automata (extended abstract). In Proc. DLT, LNCS 5257, 2008] on the problem of finding good elimination orderings for the state elimination algorithm, one of the most popular algorithms for the conversion of finite automata into equivalent regular expressions. Here we tackle this problem both from the theoretical and from the practical side. First we show that the problem of finding optimal elimination orderings can be used to estimate the cycle rank of the underlying automata. This gives good evidence that the problem under consideration is difficult, to a certain extent. Moreover, we conduct experiments on a large set of carefully chosen instances for five different strategies to choose elimination orderings, which are known from the literature. Perhaps the most surprising result is that a simple greedy heuristic by [M. Delgado, J. Morais: Approximation to the smallest regular expression for a given regular language. In Proc. CIAA, LNCS 3317, 2004] almost always outperforms all other strategies, including those with a provable performance guarantee. 1

this paper, we establish a property of series of star-height 1 which is sufficient for deciding the star-height of a particular class of series. Unlike Soittola's theorem, the condition that we give not only concerns the greatest real positive root of the series, but the set of all its real positive roots. The star-height of the series in one variable is linked with various notions, in particular with that of the representation of the series by a matrix with polynomial entries. Each series of the class which particulary interests us in the following (the N-rational series having a dominating root) can be represented by a matrix over zN [z] of size 2. We give also a characterization of the series of star-height 1 that uses their representation by matrices with polynomial entries (for a general overview of matrices with polynomial entries, one can refer to the survey written by Boyle [3]). This paper is divided into three sections. In the first one, we present, by analogy with regular languages, some definitions and results about K-rational

Abstract. In this paper we lift the result of Hashiguchi of decidability of the restricted star-height problem for words to the level of finite trees. Formally, we show that it is decidable, given a regular tree language L and a natural number k whether L can be described by a disjunctive µ-calculus formula with at most k nesting of fixpoints. We show the same result for disjunctive µ-formulas allowing substitution. The latter result is equivalent to deciding if the language is definable by a regular expression with nesting depth at most k of Kleene-stars. The proof, following the approach of Kirsten in the word case, goes by reduction to the decidability of the limitedness problem for nondeterministic nested distance desert automata over trees. We solve this problem in the more general framework of alternating tree automata. 1