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Collapsing Exact Arithmetic Hierarchies

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Algorithms and Computation (WALCOM 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8344))

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Abstract

We provide a uniform framework for proving the collapse of the hierarchy \({\sf NC}^1(\mathcal{C})\) for an exact arithmetic class \(\mathcal{C}\) of polynomial degree. These hierarchies collapse all the way down to the third level of the AC 0-hierarchy, \({\sf AC^0_3}(\mathcal{C})\). Our main collapsing exhibits are the classes

$$\mathcal{C} \in \{{\sf C}_={{\sf NC}^1}, {\sf C}_={\sf L}, {\sf C}_={\sf SAC^1}, {\sf C}_={\sf P}\}.$$

NC 1(C = L) and NC 1(C = P) are already known to collapse [1,19,20].

We reiterate that our contribution is a framework that works for all these hierarchies. Our proof generalizes a proof from [9] where it is used to prove the collapse of the AC 0(C = NC 1) hierarchy. It is essentially based on a polynomial degree characterization of each of the base classes.

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Authors and Affiliations

  1. Chennai Mathematical Institute, India

    Nikhil Balaji & Samir Datta

Authors
  1. Nikhil Balaji
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  2. Samir Datta
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Editors and Affiliations

  1. Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur, 721302, Kharagpur, India

    Sudebkumar Prasant Pal

  2. National Institute of Informatics, 2-1-2 Hitotsubashi, 101-8430, Chiyoda-ku, Tokyo, Japan

    Kunihiko Sadakane

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Balaji, N., Datta, S. (2014). Collapsing Exact Arithmetic Hierarchies. In: Pal, S.P., Sadakane, K. (eds) Algorithms and Computation. WALCOM 2014. Lecture Notes in Computer Science, vol 8344. Springer, Cham. https://doi.org/10.1007/978-3-319-04657-0_26

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  • DOI: https://doi.org/10.1007/978-3-319-04657-0_26

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-04656-3

  • Online ISBN: 978-3-319-04657-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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