An algebraic approach to communication complexity

Abstract

In this work, we define the communication complexity of a monoid M as the maximum complexity of any language recognized by M, and show that monoid classes defined by, communication complexity classes form varieties. Then we prove that a group G has constant communication complexity for k players if and only if G is a nilpotent group of class at most ( k - 1), and has linear complexity otherwise. When M is aperiodic, we show that its 2-party communication complexity is constant if M is commutative, logarithmic if M is not commutative but is in the variety DA, and is linear otherwise. Moreover, we show that if M is in DA, there exists a k such that M has constant k-party communication complexity. We conjecture that this condition is also necessary and prove lower bounds in that direction.These results lead us to state a conjecture which would generalize Szegedy's theorem to O(1) players. They also suggest several interesting possibilities to further uncover algebraic characterizations of communication complexity classes. (Abstract shortened by UMI.