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Finding All Solutions of Equations in Free Groups and Monoids with Involution

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Computer Science - Theory and Applications (CSR 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8476))

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Abstract

The aim of this paper is to present a PSPACE algorithm which yields a finite graph of exponential size and which describes the set of all solutions of equations in free groups and monoids with involution in the presence of rational constraints. This became possible due to the recently invented recompression technique of the second author.

He successfully applied the recompression technique for pure word equations without involution or rational constraints. In particular, his method could not be used as a black box for free groups (even without rational constraints). Actually, the presence of an involution (inverse elements) and rational constraints complicates the situation and some additional analysis is necessary. Still, the recompression technique is powerful enough to simplify proofs for many existing results in the literature. In particular, it simplifies proofs that solving word equations is in PSPACE (Plandowski 1999) and the corresponding result for equations in free groups with rational constraints (Diekert, Hagenah and Gutiérrez 2001). As a byproduct we obtain a direct proof that it is decidable in PSPACE whether or not the solution set is finite.

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Author information

Authors and Affiliations

  1. Institut für Formale Methoden der Informatik, University of Stuttgart, Germany

    Volker Diekert

  2. Institute of Computer Science, University of Wroclaw, Poland

    Artur Jeż

  3. Max Planck Institute für Informatik, Saarbrücken, Germany

    Wojciech Plandowski

  4. Institute of Informatics, University of Warsaw, Poland

    Artur Jeż

Authors
  1. Volker Diekert
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  2. Artur Jeż
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  3. Wojciech Plandowski
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Editor information

Editors and Affiliations

  1. Steklov Institute of Mathematics at St. Petersburg, Russian Academy of Sciences, 27 Fontanka,, 191023, St. Petersburg, Russia

    Edward A. Hirsch

  2. Higher School of Economics, National Research University, 109028, Moscow, Russia

    Sergei O. Kuznetsov

  3. CNRS and University Paris Diderot, Paris, France

    Jean-Éric Pin

  4. Moscow State University, 119992, Moscow, Russia

    Nikolay K. Vereshchagin

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Diekert, V., Jeż, A., Plandowski, W. (2014). Finding All Solutions of Equations in Free Groups and Monoids with Involution. In: Hirsch, E.A., Kuznetsov, S.O., Pin, JÉ., Vereshchagin, N.K. (eds) Computer Science - Theory and Applications. CSR 2014. Lecture Notes in Computer Science, vol 8476. Springer, Cham. https://doi.org/10.1007/978-3-319-06686-8_1

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  • DOI: https://doi.org/10.1007/978-3-319-06686-8_1

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-06685-1

  • Online ISBN: 978-3-319-06686-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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