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Summary: Relationships Among PL, L, and the Determinant
Eric Allendery
Department of Computer Science
Hill Center, Busch Campus, P.O. Box 1179
Piscataway, NJ 08855-1179, USA
allender@cs.rutgers.edu
Mitsunori Ogiharaz
Department of Computer Science
University of Rochester
Rochester, NY 14627
ogihara@cs.rochester.edu
Abstract
Recent results by Toda, Vinay, Damm, and Valiant have shown that the complexity of the
determinant is characterized by the complexity of counting the number of accepting compu-
tations of a nondeterministic logspace-bounded machine. This class of functions is known as
L. By using that characterization and by establishing a few elementary closure properties,
we give a very simple proof of a theorem of Jung, showing that probabilistic logspace-bounded
PL machines lose none of their computationalpower if they are restricted to run in polynomial
time.
We also present new results comparing and contrasting the classes of functions reducible to
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