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The Inherent Dimension of Bounded Counting Classes

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Computing and Combinatorics (COCOON 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1449))

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Abstract

Cronauer et al. [2] introduced the chain method to separate counting classes by oracles. Among the classes, for which this method is applicable, are NP, coNP, MOD-classes, . . . As these counting classes are defined via subsets of ℕk, it is natural to ask for the minimum value of k, such that a given class can be defined via such a set. We call this value the inherent dimension of the respective class.

The inherent dimension is a very natural concept, but it is quite hard to check the value for a given bounded counting class. Thus, we complement this notion by the notion of type-3 dimension, which is less natural than inherent dimension, but very easy to check. We compare type-3 dimension and inherent dimension, with the result that for classes of inherent dimension less than 3, both notions coincide, and generally the inherent dimension is never greater than the type-3 dimension.

For k ≤ 2 we can completely solve the questions, whether a given class has inherent dimension k, and which are the minimal classes with that dimension. For k ≥ 3 we give a sufficient condition for a class being of dimension at least k. We disprove the conjecture that this is also a necessary condition by a counterexample.

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Author information

Authors and Affiliations

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

    Ulrich Hertrampf

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  1. Ulrich Hertrampf
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Editors and Affiliations

  1. Institute of Information Science, Academia Sinica, 11529, Nankang, Taipei, Taiwan

    Wen-Lian Hsu

  2. Department of Computer Science, Yale University, 51 Prospect Street, New Haven, CT, 06520, USA

    Ming-Yang Kao

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© 1998 Springer-Verlag Berlin Heidelberg

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Hertrampf, U. (1998). The Inherent Dimension of Bounded Counting Classes. In: Hsu, WL., Kao, MY. (eds) Computing and Combinatorics. COCOON 1998. Lecture Notes in Computer Science, vol 1449. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-68535-9_19

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  • DOI: https://doi.org/10.1007/3-540-68535-9_19

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64824-6

  • Online ISBN: 978-3-540-68535-7

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