On Determinism and Unambiguity of Weighted Two-way Automata

In this paper, we first study the conversion of weighted two-way automata to one-way automata. We show that this conversion preserves the unambiguity but does not preserve the determinism. Yet, we prove that the conversion of an unambiguous weighted one-way automaton into a two-way automaton leads to a deterministic two-way automaton. As a consequence, we prove that unambiguous weighted two-way automata are equivalent to deterministic weighted two-way automata in commutative semirings.Comment: In Proceedings AFL 2014, arXiv:1405.527

Unambiguous Separators for Tropical Tree Automata

In this paper we show that given a max-plus automaton (over trees, and with real weights) computing a function f and a min-plus automaton (similar) computing a function g such that f ? g, there exists effectively an unambiguous tropical automaton computing h such that f ? h ? g. This generalizes a result of Lombardy and Mairesse of 2006 stating that series which are both max-plus and min-plus rational are unambiguous. This generalization goes in two directions: trees are considered instead of words, and separation is established instead of characterization (separation implies characterization). The techniques in the two proofs are very different

On Hadamard Series and Rotating Q-Automata

In this paper, we study rotating Q-automata, which are (memoryless) automata with weights in Q, that can read the input tape from left to right several times. We show that the series realized by valid rotating Q-automata are Q-Hadamard series (which are the closure of Q-rational series by pointwise inverse), and that every Q-Hadamard series can be realized by such an automaton. We prove that, although validity of rotating Q-automata is undecidable, the equivalence problem is decidable on rotating Q-automata. Finally, we prove that every valid two-way Q-automaton admits an equivalent rotating Q-automaton. The conversion, which is effective, implies the decidability of equivalence of two-way Q-automata

Regular Temporal Cost Functions

International audienceRegular cost functions have been introduced recently as an extension to the notion of regular languages with counting capabilities. The specificity of cost functions is that exact values are not considered, but only estimated. In this paper, we study the strict subclass of regular temporal cost functions. In such cost functions, it is only allowed to count the number of occurrences of consecutive events. For this reason, this model intends to measure the length of intervals, i.e., a discrete notion of time. We provide various equivalent representations for functions in this class, using automata, and 'clock based' reduction to regular languages. We show that the conversions are much simpler to obtain, and much more efficient than in the general case of regular cost functions. Our second aim in this paper is to use temporal cost function as a test-case for exploring the algebraic nature of regular cost functions. Following the seminal ideas of Schützenberger, this results in a decidable algebraic characterization of regular temporal cost functions inside the class of regular cost functions

Series which are both max-plus and min-plus rational are unambiguous

Consider partial maps from the free monoid into the field of real numbers with a rational domain. We show that two families of such series are actually the same: the unambiguous rational series on the one hand, and the max-plus and min-plus rational series on the other hand. The decidability of equality was known to hold in both families with different proofs, so the above unifies the picture. We give an effective procedure to build an unambiguous automaton from a max-plus automaton and a min-plus one that recognize the same series

Approche structurelle de quelques problèmes de la théorie des automates

This thesis deals wih three may problems. First, we define and study the Universl automaton based on Conway's factorizations. It is then proved that the universal automaton of a reversible language contains a subautomaton that corresponds to the minimal star height and which is quasi-reversible. Second, max-plus automata are studied and an algorithm for the decision of the sequentiality of a power series realized by such an automaton is explained. Finally, derivatives of regular expressions with multiplicities are defined; this extension of Antimirov's derivatives allows to build a weighted automaton from a weighted regular expression.Les travaux développés dans cette thèse empruntent trois directions principales. D'une part, une étude attentive des propriétés de l'automate universel d'un langage rationnel a été menée. Cet automate fini (introduit sous une forme sensiblement différente par J.H. Conway) accepte le langage et a la particularité de contenir l'image par morphisme de n'importe quel automate équivalent. Nous donnons un algorithme pour le construire à partir de l'automate minimal. L'exploitation des propriétés de l'automate universel d'un langage réversible nous a permis de montrer qu'il existe un sous-automate quasi-réversible (à partir duquel on peut facilement construire un automate réversible) de l'automate universel équivalent. De plus, il existe un tel sous-automate sur lequel on peut calculer une expression rationnelle qui représente le langageavec une hauteur d'étoile minimale. D'autre part, nous donnons un algorithme pour décider la séquentialité d'une série (max,+) ou (min,+) réalisée par par un automate sur un alphabet à une lettre. La complexité de cet algorithme ne dépend que de la structure de l'automate et non des valeurs des coefficients. Nous présentons aussi un algorithme qui permet de procéder directement à la déterminisation d'un automate réalisant une série séquentielle et, si ce n'est pas le cas, à l'obtention d'un automate équivalent non ambigu. Ce dernier point rejoint le résultat de Stéphane Gaubert qui montre qu'on peut obtenir une expression (et donc un automate) non ambiguë pour n'importe quel série (max,+) rationnelle sur une lettre. Enfin, nous proposons un algorithme pour construire, à partir d'une expression rationnelle avec multiplicité, un automate qui représente la même série. Cet algorithme, qui est la généralisation des travaux d'Antimirov, permet d'obtenir explicitement un ensemble fini d'expressions qui représentent un ensemble générateur du semi-module auquel appartiennent les quotients de la série rationnelle

On the size of the universal automaton of a regular language

International audienceThe universal automaton of a regular language is the maximal NFA without merging states that recognizes this language. This automaton is directly inspired by the factor matrix defined by Conway thirty years ago. We prove in this paper that a tight bound on its size with respect to the size of the smallest equivalent NFA is given by Dedekind's numbers. At the end of the paper, we deal with the unary case. Chrobak has proved that the size of the minimal deterministic automaton with respect to the smallest NFA is tightly bounded by the Landau's function; we show that the size of the universal automaton is in this case an exponential of the Landau's function

On the construction of reversible automata for reversible languages

International audienceReversible languages occur in many different domains. Although the decision for the membership of reversible languages was solved in 1992 by Pin, an effective construction of a reversible automaton for a reversible language was still unknown. We give in this paper a method to compute a reversible automaton from the minimal automaton of a reversible language. With this intention, we use the universal automaton of the language that can be obtained from the minimal automaton and that contains an equivalent automaton which is quasi-reversible. This quasi-reversible automaton has nearly the same properties as a reversible one and can easily be turnes into a reversible automaton

Sequentialization and unambiguity of (max,+) rational series over one letter

International audienceWe present an algorithm to decide whether a (max,+)-rational series over one letter is sequential. We discuss the relation between sequentiality and unambiguity of rational series