The effect of algebraic structure on the computational complexity of matrix multiplication
The effect of algebraic structure on the computational complexity of matrix multiplication
Boolean matrix multiplication and transitive closure
SWAT '71 Proceedings of the 12th Annual Symposium on Switching and Automata Theory (swat 1971)
A 2.5 n-lower bound on the combinational complexity of Boolean functions
STOC '75 Proceedings of seventh annual ACM symposium on Theory of computing
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We are interested in combinational circuits synthesized from and-gates and or-gates. We first show that n3 distinct and-gate inputs are needed to form the product of two Boolean matrices, and hence O(n3) two-input and-gates are needed to compute the transitive closure of a Boolean matrix. While this result has the flavor of Kerr's (achievable)lower bound [Kerr 1970] of n3+-gates for computing the min/+ product of integer-valued matrices using only min-gates and +-gates, the problem turns out on closer inspection to be considerably more subtle, and in fact we have been able to come only to within a factor of two of the best known upper bound of n3and-gates. Secondly we use this result to study the effect on combinational circuts of not using not-gates inverters).