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Fields of Logic and Computation III pp 57–78Cite as

Parameterized Parallel Computing and First-Order Logic

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Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 12180))

Abstract

The relationship between the complexity class \(\mathrm{AC}^0\) and first-order logic transfers to the parameterized class \(\mathrm {para}\text {-}\mathrm{AC}^0\), the parameterized analogue of \(\mathrm{AC}^0\). In the last years this relationship has turned out to be very fruitful. In this paper we survey some of the results obtained, mainly applications of logic to complexity theory. However the last section presents a strict hierarchy theorem for first-order logic obtained by a result of complexity theory.

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Notes

  1. 1.

    The trivial no-instance will be useful in the description of the kernelization function, though in our formulas we will use \(\chi ^k_{\textsc {no}}\) to indicate the cases where the trivial no-instance would have to be output (rather than presenting the cumbersome formula that outputs the trivial no-instance).

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Acknowledgement

We thank an anonymous referee whose detailed review has improved the presentation considerably. The collaboration of the authors is funded by the Sino-German Center for Research Promotion (GZ 1518). Yijia Chen is also supported by National Natural Science Foundation of China (Project 61872092).

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Authors and Affiliations

  1. Fudan University, Shanghai, China

    Yijia Chen

  2. Albert-Ludwigs-Universität, Freiburg im Breisgau, Germany

    Jörg Flum

Authors
  1. Yijia Chen
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  2. Jörg Flum
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Corresponding author

Correspondence to Jörg Flum .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

    Andreas Blass

  2. Université Paris Est Créteil, Fontainebleau, France

    Patrick Cégielski

  3. School of Computer Science, Tel Aviv University, Tel Aviv, Israel

    Nachum Dershowitz

  4. Fakultät für Mathematik und Informatik, Universität Leipzig, Leipzig, Germany

    Manfred Droste

  5. Reactive Systems Group, Universitat des Saarlandes, Saarbrücken, Germany

    Bernd Finkbeiner

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Chen, Y., Flum, J. (2020). Parameterized Parallel Computing and First-Order Logic. In: Blass, A., Cégielski, P., Dershowitz, N., Droste, M., Finkbeiner, B. (eds) Fields of Logic and Computation III. Lecture Notes in Computer Science(), vol 12180. Springer, Cham. https://doi.org/10.1007/978-3-030-48006-6_5

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  • DOI: https://doi.org/10.1007/978-3-030-48006-6_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-48005-9

  • Online ISBN: 978-3-030-48006-6

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