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A Parameterized Halting Problem

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The Multivariate Algorithmic Revolution and Beyond

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7370))

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Abstract

The parameterized problem \(p\textsc{-Halt}\) takes as input a nondeterministic Turing machine \(\mathbb{M}\) and a natural number n, the size of \(\mathbb{M}\) being the parameter. It asks whether every accepting run of \(\mathbb{M}\) on empty input tape takes more than n steps. This problem is in the class XPuni, the class “uniform XP,” if there is an algorithm deciding it, which for fixed machine \(\mathbb{M}\) runs in time polynomial in n. It turns out that various open problems of different areas of theoretical computer science are related or even equivalent to \(p{\rm \textsc{-Halt}\in{XP}_{uni}}\). Thus this statement forms a bridge which allows to derive equivalences between statements of different areas (proof theory, complexity theory, descriptive complexity, …) which at first glance seem to be unrelated. As our presentation shows, various of these equivalences may be obtained by the same method.

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Author information

Authors and Affiliations

  1. Shanghai Jiaotong University, China

    Yijia Chen

  2. Albert-Ludwigs-Universität Freiburg, Germany

    Jörg Flum

Authors
  1. Yijia Chen
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  2. Jörg Flum
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Editor information

Editors and Affiliations

  1. Department of Information and Computing Sciences, Utrecht University, P.O. Box 80.089, 3508 TB, Utrecht, The Netherlands

    Hans L. Bodlaender

  2. School of Mathematics, Statistics and Operations Research, Victoria University, P.O. Box 600, Wellington, New Zealand

    Rod Downey

  3. Institute of Informatics, University of Bergen, Postboks 7800, 5020, Bergen, Norway

    Fedor V. Fomin

  4. Computer and Automation Research Institute, Hungarian Academy of Sciences (MTA SZTAKI), Pf. 63, 1518, Budapest, Hungary

    Dániel Marx

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Chen, Y., Flum, J. (2012). A Parameterized Halting Problem. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds) The Multivariate Algorithmic Revolution and Beyond. Lecture Notes in Computer Science, vol 7370. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30891-8_17

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  • DOI: https://doi.org/10.1007/978-3-642-30891-8_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-30890-1

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