# Relationships Among PL, #L, and the Determinant

@article{Allender1996RelationshipsAP, title={Relationships Among PL, \#L, and the Determinant}, author={Eric Allender and Mitsunori Ogihara}, journal={RAIRO Theor. Informatics Appl.}, year={1996}, volume={30}, pages={1-21} }

Recent results by Toda, Vinay, Damm, and Valiant have shown that the complexity of the determinant is characterized by the complexity ofcounting the number ofaccepting computations ofa 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 theorem of Jung, showing that probabilistic logspace-bounded (PL) machines lose none of their computational…

## 121 Citations

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It is shown that the perfect matching problem is in the complexity class SPL (in the nonuniform setting), and if there are problems in DSPACE(n) requiring exponential-size circuits, then all of the results hold in the uniform setting.

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- Mathematics, Computer ScienceMFCS
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The structure of classes in between NC1 and L based on counting functions or, equivalently, based on arithmetic circuits is examined, and the upper bound L is obtained for all these hierarchies.

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It is shown that noncommutative polynomial identity testing for algebraic branching programs over rationals is complete for the logspace counting class $\ceql$, and over fields of characteristic $p$ the problem is in $\ModpL/\Poly$.

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