Abstract
The “log rank” conjecture consists of the question how exactly the deterministic communication complexity of a problem can be determined in terms of algebraic invariants of the communication matrix of this problem. In the following, we answer this question in the context of modular communication complexity. We show that the modular communication complexity can be characterised precisely in terms of the logarithm of a certain rigidity function of the communication matrix. Thus, we are able to determine precisely the modular communication complexity of several problems, such as, e.g., set disjointness, comparability, and undirected graph connectivity. From the obtained bounds for the modular communication complexity, we can conclude exponential lower bounds on the size of depth two circuits having arbitary symmetric gates at the bottom level and a MODm-gate at the top.
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