Non-Deterministic Communication Complexity of Regular Languages

TL;DR

This thesis investigates the non-deterministic communication complexity of regular languages, revealing they are either very simple or have linear complexity, using algebraic techniques to establish new bounds and theoretical insights.

Contribution

It establishes a dichotomy in the non-deterministic communication complexity of regular languages and provides algebraic conditions for complexity bounds, advancing understanding in formal language theory.

Findings

Regular languages have either O(1) or Omega(log n) non-deterministic complexity.

Several regular languages have proven linear non-deterministic complexity.

Algebraic techniques yield new lower bounds and conditions related to semigroup theory.

Abstract

In this thesis, we study the place of regular languages within the communication complexity setting. In particular, we are interested in the non-deterministic communication complexity of regular languages. We show that a regular language has either O(1) or Omega(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 Pol(Com). To obtain our results, we use algebraic techniques. In the study of regular languages, the algebraic point of view pioneered by Eilenberg (\cite{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.