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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 power if they are restricted to run in polynomial time. We also present new results comparing and contrasting the classes of functions reducible to PL, #L, and the determinant, using various notions of reducibility.

Relationships among $PL$, $\#L$, and the determinant