Infinite Communication Complexity

TL;DR

This paper introduces a formal framework for measuring communication complexity when dealing with infinite strings, connecting it to classical concepts and applying it to problems in tilings and sofic shifts.

Contribution

It proposes a formalism for infinite communication complexity, demonstrating its properties and relation to classical amortized complexity, and applies it to conjectures in tilings and multidimensional shifts.

Findings

Formalism for infinite communication complexity established

Shows equivalence with classical amortized communication complexity in relevant cases

Provides an application to conjectures in tilings and multidimensional sofic shifts

Abstract

Suppose that Alice and Bob are given each an infinite string, and they want to decide whether their two strings are in a given relation. How much communication do they need? How can communication be even defined and measured for infinite strings? In this article, we propose a formalism for a notion of infinite communication complexity, prove that it satisfies some natural properties and coincides, for relevant applications, with the classical notion of amortized communication complexity. More-over, an application is given for tackling some conjecture about tilings and multidimensional sofic shifts.