Counting Classes

UR-CS Participating Faculty: Lane A. Hemaspaandra (= Lane A. Hemachandra) and Mitsunori Ogihara

Project Description

This project studies counting classes. The term ``counting classes'' has come to refer to a certain collection of classes--such as #P, SPP, probabilistic classes, parity-based classes, etc.--that are defined in terms of the number of accepting paths of nondeterministic machines.

Among the questions central to this project are:

  1. How powerful are counting classes?
  2. What are the properties of counting classes?
  3. How robust are the definitions of counting classes?

Bibliography

1
This is a list of selected journal (except when the work has not yet appeared in journal/book form) papers, from or related to this project, by University of Rochester authors. Essentially all the papers listed below can be found, in their full technical report versions, in the UR-CS Technical Report Archive's theory section. Lane Hemaspaandra's complete publication list can always be found at http://www.cs.rochester.edu/u/lane/publist.pdf.

2
E. Allender, R. Beals, and M. Ogihara.
The complexity of matrix rank and feasible systems of linear equations.
Computational Complexity, 8(2):99-126, 1999.

3
E. Allender and M. Ogihara.
Relationships among PL, #L, and the determinant.
RAIRO Theoretical Informatics and Applications, 30(1):1-21, 1996.

4
R. Beigel, L. Hemachandra, and G. Wechsung.
Probabilistic polynomial time is closed under parity reductions.
Information Processing Letters, 37(2):91-94, 1991.

5
B. Borchert, L. Hemaspaandra, and J. Rothe.
Restrictive acceptance suffices for equivalence problems.
London Mathematical Society Journal of Computation and Mathematics, 3:86-95, 2000.

6
J. Cai and L. Hemachandra.
On the power of parity polynomial time.
Mathematical Systems Theory, 23(2):95-106, 1990.

7
J. Cai and L. Hemachandra.
A note on enumerative counting.
Information Processing Letters, 38(4):215-219, 1991.

8
D. Eppstein, L. Hemachandra, J. Tisdall, and B. Yener.
Simultaneous strong separations of probabilistic and unambiguous complexity classes.
Mathematical Systems Theory, 25(1):23-36, 1992.

9
S. Fischer, L. Hemaspaandra, and L. Torenvliet.
Witness-isomorphic reductions and local search.
In A. Sorbi, editor, Complexity, Logic, and Recursion Theory, pages 207-223. Marcel Dekker, Inc., 1997.

10
J. Goldsmith, L. Hemachandra, D. Joseph, and P. Young.
Near-testable sets.
SIAM Journal on Computing, 20(3):506-523, 1991.

11
J. Goldsmith, L. Hemachandra, and K. Kunen.
Polynomial-time compression.
Computational Complexity, 2(1):18-39, 1992.

12
L. Hemachandra and A. Hoene.
On sets with efficient implicit membership tests.
SIAM Journal on Computing, 20(6):1148-1156, 1991.

13
L. Hemachandra and M. Ogiwara.
Is #P closed under subtraction?
In G. Rozenberg and A. Salomaa, editors, Current Trends in Theoretical Computer Science: Essays and Tutorials, pages 523-536. World Scientific, 1993.

14
L. Hemachandra and S. Rudich.
On the complexity of ranking.
Journal of Computer and System Sciences, 41(2):251-271, 1990.

15
L. Hemaspaandra, H. Hempel, and G. Wechsung.
Self-specifying machines.
International Journal of Foundations of Computer Science, 10(3):263-276, 1999.

16
L. Hemaspaandra, C. Homan, and S. Kosub.
Cluster computing and the power of edge recognition.
In Proceedings of the 3rd Annual Conference on Computation and Logic: Theory and Applications of Models of Computation. Springer-Verlag Lecture Notes in Computer Science.
To appear, 2006.

17
L. Hemaspaandra, S. Kosub, and K. Wagner.
The complexity of computing the size of an interval.
In Proceedings of the 28th International Colloquium on Automata, Languages, and Programming, pages 1040-1051. Springer-Verlag Lecture Notes in Computer Science #2076, July 2001.

18
L. Hemaspaandra, A. Naik, M. Ogihara, and A. Selman.
Computing solutions uniquely collapses the polynomial hierarchy.
SIAM Journal on Computing, 25(4):697-708, 1996.

19
L. Hemaspaandra and M. Ogihara.
Universally serializable computation.
Journal of Computer and System Sciences, 55(3):547-560, 1997.

20
L. Hemaspaandra, M. Ogihara, and G. Wechsung.
Reducing the number of solutions of NP functions.
Journal of Computer and System Sciences, 64(2):311-328, 2002.

21
L. Hemaspaandra, M. Ogihara, M. Zaki, and M. Zimand.
The complexity of finding top-Toda-equivalence-class members.
Theory of Computing Systems.
In press. Preliminary version available in Proceedings of 6th Latin American Symposium on Theoretical Informatics (Springer-Verlag, 2004).

22
L. Hemaspaandra and J. Rothe.
A second step towards complexity-theoretic analogs of Rice's Theorem.
Theoretical Computer Science, 244(1-2):205-217, 2000.

23
L. Hemaspaandra and M. Thakur.
Lower bounds and the hardness of counting properties.
Theoretical Computer Science, 326(1-3):1-28, 2004.

24
L. Hemaspaandra and H. Vollmer.
The Satanic notations: Counting classes beyond #P and other definitional adventures.
SIGACT News, 26(1):2-13, 1995.

25
L. Hemaspaandra, M. Zaki, and M. Zimand.
Polynomial-time semi-rankable sets.
In Journal of Computing and Information, 2(1), Special Issue: Proceedings of the 8th International Conference on Computing and Information, pages 50-67, 1996.
CD-ROM ISSN 1201-8511/V2/#1.

26
M. Liskiewicz, M. Ogihara, and S. Toda.
The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes.
Theoretical Computer Science, 304(1-3):129-156, 2003.

27
M. Liskiewicz, M. Ogihara, and S. Toda.
Counting self-avoiding walks in some regular graphs.
SIGACT News, 34(3):26-39, 2003.

28
M. Ogihara.
Equivalence of NC$^k$ and AC$^{k-1}$ closures of NP and other classes.
Information and Computation, 120(1):55-58, 1995.

29
M. Ogihara.
The PL hierarchy collapses.
SIAM Journal on Computing, 27(5):1430-1437, 1998.

30
M. Ogiwara.
A characterization of P $^{{\rm {C}}_={\rm {P}}}$.
IEICE Transactions on Communications, Electronics, Information, and Systems, E75-D(1):44-49, 1992.

31
M. Ogiwara.
Generalized theorems on the relationships among reducibility notions to certain complexity classes.
Mathematical Systems Theory, 27(3):189-200, 1994.

32
M. Ogiwara and L. Hemachandra.
A complexity theory for feasible closure properties.
Journal of Computer and System Sciences, 46(3):295-325, 1993.

33
M. Ogiwara, T. Thierauf, S. Toda, and O. Watanabe.
On closure properties of #P in the context of PF$\circ$#P.
Journal of Computer and System Sciences, 53(2):171-179, 1996.

34
H. Spakowski, M. Thakur, and R. Tripathi.
Quantum and classical complexity classes: Separations, collapses, and closure properties.
Information and Computation, 200(1):1-34, 2005.

35
S. Toda and M. Ogiwara.
Counting classes are at least as hard as the polynomial-time hierarchy.
SIAM Journal on Computing, 21(2):316-328, 1992.



Lane A. Hemaspaandra