**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.