Measuring communication complexity using instance complexity with oracles

TL;DR

This paper links non-deterministic communication complexity to instance complexity, using algorithmic entropy and oracles to measure communication requirements in computational problems.

Contribution

It introduces a novel connection between communication complexity and instance complexity, providing a new perspective for analyzing information exchange in protocols.

Findings

Establishes a formal relationship between non-deterministic communication complexity and instance complexity.

Uses algorithmic entropy to quantify information in communication protocols.

Provides a framework for analyzing witnesses in non-deterministic protocols.

Abstract

We establish a connection between non-deterministic communication complexity and instance complexity, a measure of information based on algorithmic entropy. Let $\overline{x}$, $\overline{y}$ and $Y_1(\overline{x})$ be respectively the input known by Alice, the input known by Bob, and the set of all values of $y$ such that $f(\overline{x},y)=1$; a string is a witness of the non-deterministic communication protocol iff it is a program $p$ that "corresponds exactly" to the instance complexity $\ic^{f,t}(\overline{y}:Y_1(\overline{x}))$.