Relativized Counting Classes: Relations among Thresholds, Parity, and Mods (1991)
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| Venue: | Journal of Computer and System Sciences |
| Citations: | 20 - 5 self |
BibTeX
@ARTICLE{Beigel91relativizedcounting,
author = {Richard Beigel},
title = {Relativized Counting Classes: Relations among Thresholds, Parity, and Mods},
journal = {Journal of Computer and System Sciences},
year = {1991},
volume = {42},
pages = {76--96}
}
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Abstract
Well known complexity classes such as NP, co-NP, \PhiP (PARITY-P), and PP are produced by considering a nondeterministic polynomial time Turing machine N and defining acceptance in terms of the number of accepting paths in N . That is, they are subclasses of P #P[1] . Other interesting classes such as MOD k P and C=P are also subclasses of P #P[1] . Many relations among these classes are unresolved. Of course, these classes coincide if P = PSPACE. However, we develop a simple combinatorial technique for constructing oracles that separate counting classes. Our results suggest that it will be difficult to resolve the unknown relationships among different counting classes. In addition to presenting new oracle separations, we simplify several previous constructions. 1. Introduction In [26], Valiant defined the class #P of counting functions. Definition 1 [Valiant] #P is the class of functions for which there exists a nondeterministic polynomial-time Turing machine N such that f(x) is...
