The complexity of Boolean functions
The complexity of Boolean functions
Lower bounds for non-commutative computation
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
On submodular complexity measures
Poceedings of the London Mathematical Society symposium on Boolean function complexity
Lattices, mobius functions and communications complexity
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Theoretical Computer Science
An upper bound for nonnegative rank
Journal of Combinatorial Theory Series A
| Hi-index | 0.89 |
For real numbers a"1,...,a"n, let Q(a"1,...,a"n) be the nxn matrix whose i,j-th entry is (a"i-a"j)^2. We show that Q(1,...,n) has nonnegative rank at most 2log"2n+2. This refutes a conjecture from Beasley and Laffey (2009) [1] (and contradicts a ''theorem'' from Lin and Chu (2010) [5]). We give other examples of sequences a"1,...,a"n for which Q(a"1,...,a"n) has logarithmic nonnegative rank, and pose the problem whether this is always the case. We also discuss examples of matrices based on hamming distances between inputs of a Boolean function, and note that a lower bound on their nonnegative rank implies lower bounds on Boolean formula size.