On the Non-deterministic Communication Complexity of Regular Languages

Abstract

The notion of communication complexity was introduced by Yao in his seminal paper [Yao79]. In [BFS86], Babai Frankl and Simon developed a rich structure of communication complexity classes to understand the relationships between various models of communication complexity. This made it apparent that communication complexity was a self-contained mini-world within complexity theory. In this thesis, we study the place of regular languages within this mini-world. In particular, we are interested in the nondeterministic communication complexity of regular languages.We show that a regular language has either O(1) or Ω(log n) non-deterministic complexity. We obtain several linear lower bound results which cover a wide range of regular languages having linear non-deterministic complexity.These lower bound results also imply a result in semigroup theory: we obtain sufficient conditions for not being in the positive variety P ol(Com).To obtain our results, we use algebraic techniques. In the study of regular languages, the algebraic point of view pioneered by Eilenberg ([Eil74]) has led to many interesting results. Viewing a semigroup as a computational device that recognizes languages has proven to be prolific from both semigroup theory and formal languages perspectives. In this thesis, we provide further instances of such mutualism. ii