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Definability and synthesis of transductions

Abstract : In the first part of this manuscript we focus on the study of rational functions, functions defined by one-way transducers.Our goal is to extend to transductions the many logic-algebra correspondences that have been established for languages, such as the celebrated Schützenberger-McNaughton-Papert Theorem. In the case of rational functions over finite words, we obtain a Myhill-Nerode-like characterization in terms of congruences of finite index. This characterization allows us to obtain a transfer result from logic-algebra equivalences for languages to logic-algebra equivalences for transductions. In particular, we show that one can decide if a rational function can be defined in first-order logic.Over infinite words, we obtain weaker results but are still able to decide first-order definability.In the second part we introduce a logic for transductions and solve the regular synthesis problem: given a formula in the logic, can we obtain a two-way deterministic transducer satisfying the formula?More precisely, we give an algorithm that always produces a regular function satisfying a given specification.We also exhibit an interesting link between transductions and words with ordered data. Thus we obtain as a side result an expressive logic for data words with decidable satisfiability.
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Submitted on : Wednesday, December 19, 2018 - 4:14:07 PM
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Nathan Lhote. Definability and synthesis of transductions. Other [cs.OH]. Université de Bordeaux; Université libre de Bruxelles (1970-..), 2018. English. ⟨NNT : 2018BORD0185⟩. ⟨tel-01960958⟩

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