# The Inherent Dimension of Bounded Counting Classes

## 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 ℕ^{k}, 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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