The monotone circuit complexity of Boolean functions
Combinatorica
On the complexity of cutting-plane proofs
Discrete Applied Mathematics
On the method of approximations
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Circuits and local computation
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Characterizing non-deterministic circuit size
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Top-down lower bounds for depth-three circuits
Computational Complexity
Computing threshold functions by depth-3 threshold circuits with smaller thresholds of their gates
Information Processing Letters
Lower bounds for cutting planes proofs with small coefficients
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
A Note on the Bottleneck Counting Argument
CCC '97 Proceedings of the 12th Annual IEEE Conference on Computational Complexity
Counting bottlenecks to show monotone P ? NP
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
On monotone formulae with restricted depth
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Potential of the approximation method
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Higher lower bounds on monotone size
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Tight bounds for monotone switching networks via fourier analysis
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
| Hi-index | 0.01 |
Our main result is a combinatorial lower bounds criterion for a monotone circuits over the reals. Gates are any unbounded fanin non-decreasing real-valued functions. We require only that they are local (in a natural sense). Unbounded fanin AND and OR gates, as well as any threshold gate with small enough threshold value, are examples of local gates. The proof is relatively simple and direct, and combines the bottlenecks counting approach of Haken with the idea of finite limit due to Sipser. Apparently this is the first combinatorial lower bounds criterion for monotone computations. It is symmetric and yields (in a uniform and easy way) exponential lower bounds.