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
Non-uniform automata over groups
Information and Computation
An oracle separating ⊕ P from PPPH
Information Processing Letters
Some notes on threshold circuits, and multiplication in depth 4
Information Processing Letters
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
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
Depth reduction for circuits of unbounded fan-in
Information and Computation
On the computational power of depth 2 circuits with threshold and modulo gates
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
A weight-size trade-off for circuits with MOD m gates
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
A note on the power of majority gates and modular gates
Information Processing Letters
When do extra majority gates help?: polylog(N) majority gates are equivalent to one
Computational Complexity - Special issue on circuit complexity
Computational Complexity - Special issue on circuit complexity
Representing Boolean functions as polynomials modulo composite numbers
Computational Complexity - Special issue on circuit complexity
A note on a theorem of Barrington, Straubing and The´rien
Information Processing Letters
Threshold circuits of small majority-depth
Threshold circuits of small majority-depth
Separating the polynomial-time hierarchy by oracles
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
A note on the power of threshold circuits
SFCS '89 Proceedings of the 30th Annual Symposium on Foundations of Computer Science
SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
The correlation between parity and quadratic polynomials mod 3
Journal of Computer and System Sciences - Special issue on computational complexity 2002
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We investigate the complexity of depth-3 threshold circuits with majority gates at the output, possibly negated AND gates at level two, and MODm gates at level one. We show that the fan-in of the AND gates can be reduced to O(log n) in the case where m is unbounded, and to a constant in the case where m is constant. We then use these upper bounds to derive exponential lower bounds for this class of circuits. In the unbounded m case, this yields a new proof of a lower bound of Grolmusz; in the constant m case, our result sharpens his lower bound. In addition, we prove an exponential lower bound if OR gates are also permitted on level two and m is a constant prime power.