The Layer Complexity of Arthur-Merlin-like Communication
2021
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 work 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 have the following properties:
- SLAM is (strictly) included in AM and includes all previously known sub-AM classes 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. We believe that further investigation of this concept may lead to better understanding of the communication model AM and to non-trivial lower bounds against it.
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