Bounded-depth, polynomial-size circuits for symmetric functions
Theoretical Computer Science
Computational limitations of small-depth circuits
Computational limitations of small-depth circuits
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
The complexity of symmetric functions in bounded-depth circuits
Information Processing Letters
The expressive power of voting polynomials
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Complex Polynomials and Circuit Lower Bounds for Modular Counting
LATIN '92 Proceedings of the 1st Latin American Symposium on Theoretical Informatics
Computing Symmetric Functions with AND/OR Circuits and a Single MAJORITY Gate
STACS '93 Proceedings of the 10th Annual Symposium on Theoretical Aspects of Computer Science
Separating the polynomial-time hierarchy by oracles
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Symmetric functions capture general functions
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
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qAC0[2] is the class of languages computable by circuits of constant depth and quasi-polynomial (2logO(1) n) size with unbounded fan-in AND, OR, and PARITY gates. Symmetric functions are those functions that are invariant under permutations of the input variables. Thus a symmetric function fn : {0, 1}n → {0, 1} can also be seen as a function fn : {0, 1, ..., n} → {0, 1}. We give the following characterization of symmetric functions in qAC0[2], according to how fn(x) changes as x grows from 0 to n. A symmetric function f = (fn) is in qAC0[2] if and only if fn has period 2t(n) = logO(1) n except within both ends of length logO(1) n.