COCOON 1998: Computing and Combinatorics pp 157-166

# 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.

## References

1. 1.
Bovet, D. P., Crescenzi, P., Silvestri, R.: A Uniform Approach to Define Complexity Classes. Theoretical Computer Science 104 (1992) 263–283.
2. 2.
Cronauer, K., Hertrampf, U., Vollmer, H., Wagner, K.W.: The Chain Method to Separate Counting Classes. Theory of Computing Systems 31 (1998) 93–108.
3. 3.
Fenner, S., Fortnow, L. Kurtz, S.: Gap-definable Counting Classes. Journal of Computer and System Sciences 48 (1994) 116–148.
4. 4.
Gundermann, T., Nasser, N.A., Wechsung, G.: A Survey on Counting Classes. Proceedings of the 5th Structure in Complexity Theory Conference. IEEE (1990) 140–153.Google Scholar
5. 5.
Hertrampf, U.: Complexity Classes with Finite Acceptance Types. Proceedings of the 11th Symp. on Theoretical Aspects of Computer Science. LNCS 775 (1994) 543–553.Google Scholar
6. 6.
Hertrampf, U.: Classes of Bounded Counting Type and Their Inclusion Relations. Proceedings of the 12th Symp. on Theoretical Aspects of Computer Science. LNCS 900 (1995) 60–70.Google Scholar
7. 7.
Hertrampf, U.: The Inherent Dimension of Bounded Counting Classes. Technical Report. University of Stuttgart (1998).Google Scholar
8. 8.
Hertrampf, U., Lautemann, C., Schwentick, T., Vollmer, H., Wagner, K.W.: On the Power of Polynomial Time Bit-Reductions. Proceedings of the 8th Structure in Complexity Theory Conference. IEEE (1993) 200–207.Google Scholar
9. 9.
Hertrampf, U., Vollmer, H., Wagner, K.W.: On Balanced vs. Unbalanced Computation Trees. Mathematical Systems Theory 29 (1996) 411–421.
10. 10.
Jenner, B., McKenzie, P., Thérien, D.: Logspace and Logtime Leaf Languages. Proceedings of the 9th Structure in Complexity Theory Conference. IEEE (1994) 242–254.Google Scholar
11. 11.
12. 12.
Valiant, L.G.: The Complexity of Computing the Permanent. Theoretical Computer Science 8 (1979) 189–201.
13. 13.
Vereshchagin, N.K.: Relativizable and Non-relativizable Theorems in the Polynomial Theory of Algorithms. (In Russian.) Izvestija Rossijskoj Akademii Nauk 57 (1993) 51–90.Google Scholar