Title: PP is closed under intersection

Authors: Richard Beigel, Nick Reingold, and Daniel Spielman

Abstract: In his seminal paper on probabilistic Turing machines, Gill asked whether the class PP is closed under intersection and union. We give a positive answer to this question. We also show that PP is closed under a variety of polynomial-time truth-table reductions. Consequences in complexity theory include the definite collapse and (assuming P ≠ PP) separation of certain query hierarchies over PP.

Similar techniques allow us to combine several threshold gates into a single threshold gate. Consequences in the study of circuits include the simulation of circuits with a small number of threshold gates by circuits having only a single threshold gate at the root (perceptrons), and a lower bound on the number of threshold gates needed to compute the parity function.

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