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Commutation Problems on Sets of Words and Formal Power Series
, 2002
"... 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 generaliza ..."
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Cited by 4 (3 self)
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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 ccodes, 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 ccode 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.
StarHeight of an NRational Series
, 1996
"... We prove a new result on INrational series in one variable. This result gives, under an appropriate hypothesis, a necessary and sufficient condition for an INrational series to be of starheight 1. The proof uses a theorem of Handelman on integral companion matrices. ..."
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Cited by 2 (1 self)
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We prove a new result on INrational series in one variable. This result gives, under an appropriate hypothesis, a necessary and sufficient condition for an INrational series to be of starheight 1. The proof uses a theorem of Handelman on integral companion matrices.
Deciding the weak definability of Büchi definable tree languages
"... Weakly definable languages of infinite trees are an expressive subclass of regular tree languages definable in terms of weak monadic secondorder logic, or equivalently weak alternating automata. Our main result is that given a Büchi automaton, it is decidable whether the language is weakly definabl ..."
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Weakly definable languages of infinite trees are an expressive subclass of regular tree languages definable in terms of weak monadic secondorder logic, or equivalently weak alternating automata. Our main result is that given a Büchi automaton, it is decidable whether the language is weakly definable. We also show that given a parity automaton, it is decidable whether the language is recognizable by a nondeterministic coBüchi automaton. The decidability proofs build on recent results about cost automata over infinite trees. These automata use counters to define functions from infinite trees to the natural numbers extended with infinity. We reduce to testing whether the functions defined by certain “quasiweak ” cost automata are bounded by a finite value.
PROGRAMME BLANC ANR FREC
"... Summary. One of the challenges of computer science is to manipulate objects from an infinite set using finitary means. All data processing problems have an infinite number of potential input data. All but the simplest specifications of computer systems talk about an infinite set of possible behavior ..."
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Summary. One of the challenges of computer science is to manipulate objects from an infinite set using finitary means. All data processing problems have an infinite number of potential input data. All but the simplest specifications of computer systems talk about an infinite set of possible behaviors, be it, for example, as input/output relation or as infinite sequences of possible actions. Of course mathematics is well accustomed to deal with infinite sets. But it is computer science that brings a completely new dimension to the picture, namely that of effectiveness. One of the central concepts that have emerged from computer science in response to this challenge is that of recognizability, whose combination with logic and automata has proved incredibly fruitful. Both logic and automata theory have then seen their areas of applications extend far beyond what could be imagined at their creation. One can for example refer to an essay “On the Unusual Effectiveness of Logic in Computer Science ” [HHI01] whose title appropriately summarizes this phenomenon and draws a comparison with the role of mathematics in physics. The theory of automata and recognizability has developed in two main directions: as an ever more sophisticated and efficient tool to handle finite, sequential and discrete behaviors (languages of finite words); and through a number of extensions of the theory aiming at the analysis of more complex,