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Machines de Mealy, (semi-)groupes d'automate, problèmes de décision et génération aléatoire

Abstract : In this thesis, we study Mealy automata, i.e. complete, deterministic, letter-to-letter transducers which have same input and output alphabet. These automata have been used since the 60s to generate (semi)groups that sometimes have remarkable properties, that were used to solve several open problems in (semi)group theory. In this work, we focus more specifically on the possible contributions that theoretical computer science can bring to the study of these automaton (semi)groups.The thesis consists of two main axis. The first one, which corresponds to the Chapters II and III, deals with decision problems and more precisely with the Burnside problem in Chapter II and with singular points in Chapter III. In these two chapters, we link structural properties of the automaton with properties of the generated group or of its action. The second axis, which comprises the Chapter IV, is related with random generation of finite groups. We seek, by drawing random Mealy automata in specific classes, to generate finite groups, and obtain a convergence result for the obtained distribution. This result echoes Dixon's theorem on random permutation groups
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Thibault Godin. Machines de Mealy, (semi-)groupes d'automate, problèmes de décision et génération aléatoire. Informatique et langage [cs.CL]. Université Sorbonne Paris Cité, 2017. Français. ⟨NNT : 2017USPCC172⟩. ⟨tel-02089159⟩

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