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Abstract: We study the polynomial time counting hierarchy, a hierarchy of complexity classes related to the notion of counting. We investigate some of their structural properties, settling many open questions dealing with oracle characterizations, closure under boolean operations, and relations with other complexity classes. We develop a new combinatorial technique to obtain relativized separations for some of the studied classes, which imply absolute separations for some logarithmic time bounded... (Update)
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.... and the oracle hierarchy built using PP as elementary class, defined by Wagner [24, 25] in 1986, has been called the counting hierarchy [20, 21]. Since then, a lot of other classes which also have some counting process inherent in their definition, appeared; for an overview see...
.... 9f 2 GapL; 8x 2 f0; 1g x 2 A ( f(x) 0g: These classes were defined as the analogs of the well known classes GapP and and C=P (Tor an[20]) by replacing polynomial time with logarithmic space. Many properties of C=P remain true for C=L, including closure under AC 0 conjunctive,...
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PP is Closed Under Intersection - Beigel, Reingold, Spielman (1991) (Correct)
When Plans Distinguish Bayes Nets - Dekhtyar, Goldsmith, Pearce (2001) (Correct)
Time-Space Tradeoffs in the Counting Hierarchy - Allender, Kouck, Ronneburger (2001) (Correct)
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27: The complexity of combinatorial problems with succinct input representation (context) - Wagner - 1986
18: PP is as hard as the polynomial-time hierarchy (context) - Toda - 1991
14: Gap-definable counting classes - Fenner, Fortnow et al. - 1991
BibTeX entry: (Update)
Jacobo Tor'an. Complexity classes defined by counting quantifiers. J. Assoc. Comput. Mach., 38(3):753--774, 1991. http://citeseer.ist.psu.edu/toran91complexity.html More
@article{ toran91complexity,
author = "Jacobo Toran",
title = "Complexity Classes Defined by Counting Quantifiers",
journal = "Journal of the ACM",
volume = "38",
number = "3",
pages = "753-774",
year = "1991",
url = "citeseer.ist.psu.edu/toran91complexity.html" }
Citations (may not include all citations):9 Gabarr'o: Structural Complexity (context) - Balc'azar, Diaz - 1987
8 Theoretical Computer Science (context) - Angluin, problems et al. - 1980
2 Solovay: Relativizations of the P (context) - Baker, Gill - 1975
1 Gill: Relative to a random oracle A p (context) - Bennet
The graph only includes citing articles where the year of publication is known.
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Computing Functions with Parallel Queries to NP - Jenner, Torán (1993) (Correct)
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On Finding the Number of Graph Automorphisms - Robert Beals, Richard Chang.. (1996) (Correct)
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