PP Is Closed under Intersection

In this 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 incl...

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Bibliographic Details
Published inJournal of computer and system sciences Vol. 50; no. 2; pp. 191 - 202
Main Authors Beigel, R., Reingold, N., Spielman, D.
Format Journal Article Conference Proceeding
LanguageEnglish
Published Brugge Elsevier Inc 01.04.1995
Academic Press
Subjects
Algorithmics. Computability. Computer arithmetics
Applied sciences
Automata. Abstract machines. Turing machines
Computer science; control theory; systems
Exact sciences and technology
Language theory and syntactical analysis
Theoretical computing
Intersection
Lower bound
Probabilistic approach
Closure
Turing machine
Computational complexity
Rational function
Mathematical programming
Polynomial time
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Summary:In this 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 that are needed to compute the parity function.
ISSN:0022-0000
1090-2724
DOI:10.1006/jcss.1995.1017