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
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
Annual review of computer science: vol. 3, 1988
A lower bound for matrix multiplication
SIAM Journal on Computing
Size--Depth Tradeoffs for Threshold Circuits
SIAM Journal on Computing
An exponential lower bound for depth 3 arithmetic circuits
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Depth-3 Arithmetic Formulae over Fields of Characteristic Zero
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Nearly tight bounds on the learnability of evolution
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Superconcentrators, generalizers and generalized connectors with limited depth
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
On the complexity of matrix product
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Affine projections of symmetric polynomials
Journal of Computer and System Sciences - Complexity 2001
Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Polynomial Identity Testing for Depth 3 Circuits
Computational Complexity
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We prove super-linear lower bounds for the number of edges in constant depth circuits with n inputs and up to n outputs. Our lower bounds are proved for all types of constant depth circuits, e.g., constant depth arithmetic circuits and constant depth Boolean circuits with arbitrary gates. The bounds apply for several explicit functions, and, most importantly, for matrix product. In particular, we obtain the following results:We show that the number of edges in any constant depth arithmetic circuit for matrix product (over any field is super-linear in m^2 (where m \times m is the size of each matrix). That is, the lower bound is super-linear in the number of input variables. Moreover, if the circuit is bilinear the result applies also for the case where the circuit gets for free any product of two linear functions.We show that the number of edges in any constant depth arithmetic circuit for the trace of the product of 3 matrices (over fields with characteristic~0) is super-linear in m^2. (Note that the trace is a single-output function).We give explicit examples for n Boolean functions f_1,\dots,f_ , such that any constant depth for f_1,...,f_n has a super-linear number of edges. The lower bound is proved also for circuits with arbitrary gates over any finite field. The bound applies for matrix product over finite fields as well as for several other explicit functions.