# Counting classes: Thresholds, parity, mods, and fewness

- First Online:

## Abstract

Counting classes are classes of languages defined in terms of the number of accepting computations of non-deterministic polynomial-time Turing machines. Well known examples of counting classes are NP, co-NP, ⊕P, and PP. Every counting class consists of languages in P^{#P[1]}, the class of languages computable in polynomial time using a single call to an oracle capable of determining the number of accepting paths of an NP machine.

We perform an in-depth investigation of counting classes defined in terms of thresholds and moduli. We show that the computational power of a threshold machine is a monotone function of the threshold. Then we show that the class MODZ_{k}P is at least as large as FewP. Finally, we improve a result of Cai and Hemachandra by showing that recognizing languages in the class Few is as easy as distinguishing uniquely satisfiable formulas from unsatisfiable formulas (or detecting unique solutions, as in [21]).

## Preview

Unable to display preview. Download preview PDF.

### References

- [1]L. Adleman and K. Manders. Reducibility, randomness, and intractibility. In
*Proceedings of the 9th Annual ACM Symposium on Theory of Computing*, pages 151–153, 1977.Google Scholar - [2]E. W. Allender. The complexity of sparse sets in P. In
*Structure in Complexity Theory*, pages 1–11. Springer-Verlag, June 1986. Volume 223 of*Lecture Notes in Computer Science*.Google Scholar - [3]T. Baker, J. Gill, and R. Solovay. Relativizations of the
*P*=?*NP*question.*SIAM J. Comput.*, 4:431–442, 1975.CrossRefGoogle Scholar - [4]R. Beigel. Relativized counting classes: Relations among thresholds, parity, and mods.
*J. Comput. Syst. Sci.*To appear.Google Scholar - [5]A. Blass and Y. Gurevich. On the unique satisfiability problem.
*Information and Control*, 55:80–88, 1982.CrossRefGoogle Scholar - [6]J. Cai and L. A. Hemachandra. On the power of parity. In
*Proceedings of the 6th Annual Symposium on Theoretical Aspects of Computer Science*, pages 229–240. Springer-Verlag, 1989. Lecture Notes in Computer Science.Google Scholar - [7]J. Gill. Computational complexity of probabilistic Turing machines.
*SIAM J. Comput.*, 6:675–695, 1977.CrossRefGoogle Scholar - [8]L. M. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of Boolean functions.
*Theoretical Comput. Sci.*, 43:43–58, 1986.CrossRefGoogle Scholar - [9]J. Köbler, U. Schöning, S. Toda, and J. Torán. Turing machines with few accepting computations and low sets for PP. In
*Proceedings of the 4th Annual Conference on Structure in Complexity Theory*, pages 208–215. IEEE Computer Society Press, June 1989.Google Scholar - [10]A. Meyer and L. J. Stockmeyer. The equivalence problem for regular expressions with squaring requires exponential space. In
*Proceedings of the 13th Annual IEEE Symposium on Switching and Automata Theory*, pages 125–129, 1972.Google Scholar - [11]C. H. Papadimitriou and S. K. Zachos. Two remarks on the complexity of counting. In
*Proceedings of the 6th GI Conference of Theoretical Computer Science*, pages 269–276. Springer-Verlag, 1983. Volume 145 of*Lecture Notes in Computer Science*.Google Scholar - [12]C. Rackoff. Relativized questions involving probabilistic algorithms.
*Journal of the Association for Computing Machinery*, 29(1):261–268, Jan. 1982.Google Scholar - [13]U. Schöning. The power of counting. In
*Proceedings of the 3rd Annual Conference on Structure in Complexity Theory*, pages 2–18. IEEE Computer Society Press, June 1988.Google Scholar - [14]J. Simon.
*On Some Central Problems In Computational Complexity*. PhD thesis, Cornell University, Ithaca, New York, 1975. Dept. of Computer Science, TR 75–224.Google Scholar - [15]R. Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In
*Proceedings of the 19th Annual ACM Symposium on Theory of Computing*, pages 77–82, 1987.Google Scholar - [16]L. J. Stockmeyer. The polynomial-time hierarchy.
*Theoretical Comput. Sci.*, 3:1–22, 1977.CrossRefGoogle Scholar - [17]J. Torán. An oracle characterization of the counting hierarchy. In
*Proceedings of the 3rd Annual Conference on Structure in Complexity Theory*, pages 213–223. IEEE Computer Society Press, June 1988.Google Scholar - [18]J. Torán.
*Structural Properties of The Counting Hierarchies*. PhD thesis, Facultat d'Informàtica de Barcelona, 1988.Google Scholar - [19]L. G. Valiant. The relative complexity of checking and evaluating.
*Inf. Process. Lett.*, 5:20–23, 1976.CrossRefGoogle Scholar - [20]L. G. Valiant. The complexity of computing the permanent.
*Theoretical Comput. Sci.*, 8:189–201, 1979.CrossRefGoogle Scholar - [21]L. G. Valiant and V. V. Vazirani. NP is as easy as detecting unique solutions. In
*Proceedings of the 17th Annual ACM Symposium on Theory of Computing*, 1985.Google Scholar - [22]K. W. Wagner. The complexity of combinatorial problems with succinct input representation.
*Acta Inf.*, 23:325–356, 1986.CrossRefGoogle Scholar