Separating the polynomial-time hierarchy by oracles
Proc. 26th annual symposium on Foundations of computer science
Almost optimal lower bounds for small depth circuits
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
On the power of uniform families of constant depth threshold circuits
MFCS '90 Proceedings on Mathematical foundations of computer science 1990
PP is closed under intersection
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
The expressive power of voting polynomials
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Randomized polynomials, threshold circuits, and the polynomial hierarchy
STACS 91 Proceedings of the 8th annual symposium on Theoretical aspects of computer science
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
Two remarks on the power of counting
Proceedings of the 6th GI-Conference on Theoretical Computer Science
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Suppose that f is computed by a constant depth circuit with 2m AND-, OR-, and NOT-gates, and m majority-gates. We prove that f is computed by a constant depth circuit with 2mo(1) AND-, OR-, and NOT-gates, and a single majority-gate, which is at the root.One consequence is that if f is computed by and AC0 circuit plus polylog majority-gates, then f is computed by a probabilistic perceptron having polylog order. Another consequence is that if f agrees with the parity function of three-fourths of all inputs, then fcannot be computed by a constant depth circuit with 2no(1) AND-, OR-, and NOT-gates, and no(1) majority-gates.