Richard Beigel

On ACC

AbstractWe show that every languageL in the class ACC can be recognized by depth-two deterministic circuits with a symmetric-function gate at the root and

The polynomial method in circuit complexity

2^{\log ^{O(1)} n}

Counting classes: thresholds, parity, mods, and fewness

AND gates of fan-in logO(1)n at the leaves, or equivalently, there exist polynomialspn(x1,..., xn) overZ of degree logO(1)n and with coefficients of magnitude

Some connections between bounded query classes and nonuniform complexity

2^{\log ^{O(1)} n}

Representing Boolean functions as polynomials modulo composite numbers

and functionshn:Z→{0,1} such that for eachn and eachx∈{0,1}n,XL(x)=hn(pn(x1,...,xn)). This improves an earlier result of Yao (1985). We also analyze and improve modulus-amplifying polynomials constructed by Toda (1991) and Yao (1985).

Perceptrons, PP, and the polynomial hierarchy

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.