Let M be a finite monoid: define C(k)(M) to be the maximum number of bits that need to be exchanged in the k-party communication game to decide membership in any language recognized by M. We prove the following: a) If M is a group then, for any k, C(k)(M) = O(1) if M is nilpotent of class k - 1 and C(k)(M) = Θ(n) otherwise. b) If M is aperiodic, then C(2)(M) = O(1) if M is commutative, C(2)(AT) = Θ(log n) if M belongs to the variety DA but is not commutative and C (2)(M) = Θ(n) otherwise. We also show that when M is in DA, C(k)(M) = O(1) for some k and conjecture that this algebraic condition is also necessary.
CITATION STYLE
Raymond, J. F., Tesson, P., & Thérien, D. (1998). An algebraic approach to communication complexity. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 1443 LNCS, pp. 29–40). Springer Verlag. https://doi.org/10.1007/bfb0055038
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