Cluster Computing and the Power of Edge Recognition

TL;DR

This paper investigates the robustness of the cluster class CL#P, introduces a more natural variant CLU#P, and demonstrates their equivalence, leading to new insights into the class's closure properties and edge recognition complexity.

Contribution

It defines a new cluster class CLU#P, proves its equivalence to CL#P, and explores robustness and closure properties using edge recognition and unique discovery techniques.

Findings

CL#P equals CLU#P, showing naturalness is costless.

New robustness results for CL#P.

Expanded understanding of closure properties of CL#P.

Abstract

We study the robustness--the invariance under definition changes--of the cluster class CL#P [HHKW05]. This class contains each #P function that is computed by a balanced Turing machine whose accepting paths always form a cluster with respect to some length-respecting total order with efficient adjacency checks. The definition of CL#P is heavily influenced by the defining paper's focus on (global) orders. In contrast, we define a cluster class, CLU#P, to capture what seems to us a more natural model of cluster computing. We prove that the naturalness is costless: CL#P = CLU#P. Then we exploit the more natural, flexible features of CLU#P to prove new robustness results for CL#P and to expand what is known about the closure properties of CL#P. The complexity of recognizing edges--of an ordered collection of computation paths or of a cluster of accepting computation paths--is central to this study. Most particularly, our proofs exploit the power of unique discovery of edges--the ability of nondeterministic functions to, in certain settings, discover on exactly one (in some cases, on at most one) computation path a critical piece of information regarding edges of orderings or clusters.