The layer complexity of Arthur-Merlin-like communication

TL;DR

This paper introduces a new class called SLAM within the Arthur-Merlin communication complexity framework, explores its properties, and discusses the challenges in proving super-logarithmic lower bounds for AM complexity.

Contribution

The paper defines the SLAM class, shows its relation to existing classes, and analyzes the difficulty of establishing super-logarithmic lower bounds in AM communication complexity.

Findings

SLAM is strictly included in AM and includes known subclasses.

SLAM is subject to discrepancy bounds, limiting protocol efficiency.

Proving super-logarithmic lower bounds for AM remains a fundamental challenge.

Abstract

In communication complexity the Arthur-Merlin (AM) model is the most natural one that allows both randomness and non-determinism. Presently we do not have any super-logarithmic lower bound for the AM-complexity of an explicit function. Obtaining such a bound is a fundamental challenge to our understanding of communication phenomena. In this article we explore the gap between the known techniques and the complexity class AM. In the first part we define a new natural class, Small-advantage Layered Arthur-Merlin (SLAM), that has the following properties: - SLAM is (strictly) included in AM and includes all previously known subclasses of AM with non-trivial lower bounds. - SLAM is qualitatively stronger than the union of those classes. - SLAM is a subject to the discrepancy bound: in particular, the inner product function does not have an efficient SLAM-protocol. Structurally this can be summarised as SBP $\cup$ UAM $\subset$ SLAM $\subseteq$ AM $\cap$ PP. In the second part we ask why proving a lower bound of $\omega(\sqrt n)$ on the MA-complexity of an explicit function seems to be difficult. Both of these results are related to the notion of layer complexity, which is, informally, the number of "layers of non-determinism" used by a protocol.