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
Communication complexity and quasi randomness
SIAM Journal on Discrete Mathematics
n&OHgr;(logn) lower bounds on the size of depth-3 threshold circuits with AND gates at the bottom
Information Processing Letters
Threshold circuits of bounded depth
Journal of Computer and System Sciences
Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs
Journal of Computer and System Sciences
Journal of Computer and System Sciences
PP is closed under intersection
Selected papers of the 23rd annual ACM symposium on Theory of computing
When do extra majority gates help?: polylog(N) majority gates are equivalent to one
Computational Complexity - Special issue on circuit complexity
Complex polynomials and circuit lower bounds for modular counting
Computational Complexity - Special issue on circuit complexity
On the computational power of depth-2 circuits with threshold and modulo gates
Theoretical Computer Science
Upper and lower bounds for some depth-3 circuit classes
Computational Complexity
A lower bound on the MOD 6 degree of the or function
Computational Complexity
A complex-number fourier technique for lower bounds on the mod-m degree
Computational Complexity
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Pseudorandom Bits for Constant Depth Circuits with Few Arbitrary Symmetric Gates
CCC '05 Proceedings of the 20th Annual IEEE Conference on Computational Complexity
Separating the polynomial-time hierarchy by oracles
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Depth Reduction for Circuits with a Single Layer of Modular Counting Gates
CSR '09 Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science - Theory and Applications
Cracks in the defenses: scouting out approaches on circuit lower bounds
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
Correlation bounds for poly-size AC0 circuits with n1-o(1) symmetric gates
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Exponential lower bound for bounded depth circuits with few threshold gates
Information Processing Letters
Computing symmetric boolean functions by circuits with few exact threshold gates
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
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We consider constant depth circuits augmented with few modular, or more generally, arbitrary symmetric gates. We prove that circuits augmented with o(log2n) symmetric gates must have size n$^{\Omega({\rm log}\ {\it n})}$ to compute a certain (complicated) function in ACC0. This function is also hard on the average for circuits of size n$^{\epsilon log {\it n}}$ augmented with o(log n) symmetric gates, and as a consequence we can get a pseudorandom generator for circuits of size m containing $o(\sqrt{{\rm log} \ m})$ symmetric gates. For a composite integer m having r distinct prime factors, we prove that circuits augmented with sMODm gates must have size ${\it n}^{\Omega(\frac{1}{s}{\rm log}^{\frac{1}{r-1}}n)}$ to compute MAJORITY or MODl, if l has a prime factor not dividing m. For proving the latter result we introduce a new notion of representation of boolean function by polynomials, for which we obtain degree lower bounds that are of independent interest.