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On First-Order Fragments for Words and Mazurkiewicz Traces

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Developments in Language Theory (DLT 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4588))

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Abstract

We summarize several characterizations, inclusions, and separations on fragments of first-order logic over words and Mazurkiewicz traces. The results concerning Mazurkiewicz traces can be seen as generalizations of those for words. It turns out that over traces it is crucial, how easy concurrency can be expressed. Since there is no concurrency in words, this distinction does not occur there. In general, the possibility of expressing concurrency also increases the complexity of the satisfiability problem.

In the last section we prove an algebraic and a language theoretic characterization of the fragment Σ 2[E] over traces. Over words the relation E is simply the order of the positions. The algebraic characterization yields decidability of the membership problem for this fragment. For words this result is well-known, but although our proof works in a more general setting it is quite simple and direct. An essential step in the proof consists of showing that every homomorphism from a free monoid to a finite aperiodic monoid M admits a factorization forest of finite height. We include a simple proof that the height is bounded by 3|M|.

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Authors and Affiliations

  1. Universität Stuttgart, FMI, Germany

    Volker Diekert

  2. Université Bordeaux 1, LaBRI, France

    Manfred Kufleitner

Authors
  1. Volker Diekert
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  2. Manfred Kufleitner
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Tero Harju Juhani Karhumäki Arto Lepistö

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Diekert, V., Kufleitner, M. (2007). On First-Order Fragments for Words and Mazurkiewicz Traces. In: Harju, T., Karhumäki, J., Lepistö, A. (eds) Developments in Language Theory. DLT 2007. Lecture Notes in Computer Science, vol 4588. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73208-2_1

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  • DOI: https://doi.org/10.1007/978-3-540-73208-2_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73207-5

  • Online ISBN: 978-3-540-73208-2

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