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Decidable Theories of Cayley-Graphs

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STACS 2003 (STACS 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2607))

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Abstract

We prove that a connected graph of bounded degree with only finitely many orbits has a decidable MSO-theory if and only if it is context-free. This implies that a group is context-free if and only if its Cayley-graph has a decidable MSO-theory. On the other hand, the first-order theory of the Cayley-graph of a group is decidable if and only if the group has a decidable word problem. For Cayley-graphs of monoids we prove the following closure properties. The class of monoids whose Cayley-graphs have decidable MSO-theories is closed under free products. The class of monoids whose Cayley-graphs have decidable firstorder theories is closed under general graph products. For the latter result on first-order theories we introduce a new unfolding construction, the factorized unfolding, that generalizes the tree-like structures considered by Walukiewicz. We show and use that it preserves the decidability of the first-order theory. p]Most of the proofs are omitted in this paper, they can be found in the full version [17].

This work was done while the first author worked at University of Leicester, parts of it were done while the second author was on leave at IRISA, Campus de Beaulieu, 35042 Rennes Cedex, France and supported by the INRIA cooperative research action FISC.

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Author information

Authors and Affiliations

  1. Institut für Algebra, Technische Universität Dresden, D-01062, Dresden, Germany

    Dietrich Kuske

  2. Institut für Informatik, Universität Stuttgart, Breitwiesenstr. 20-22, D-70565, Stuttgart, Germany

    Markus Lohrey

Authors
  1. Dietrich Kuske
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  2. Markus Lohrey
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Editors and Affiliations

  1. Institut für Informatik, Freie Universität Berlin, Takusstr. 9, 14195, Berlin, Germany

    Helmut Alt

  2. LIRMM, 161, Rue Ada, 34392, Montpellier Cedex 5, France

    Michel Habib

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Kuske, D., Lohrey, M. (2003). Decidable Theories of Cayley-Graphs. In: Alt, H., Habib, M. (eds) STACS 2003. STACS 2003. Lecture Notes in Computer Science, vol 2607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36494-3_41

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  • DOI: https://doi.org/10.1007/3-540-36494-3_41

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