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The Inherent Dimension of Bounded Counting Classes

  • Ulrich Hertrampf
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1449)

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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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Ulrich Hertrampf
    • 1
  1. 1.Institut für InformatikUniversität StuttgartStuttgartGermany

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