The Opinion Number of Set-Agreement

Abstract

This paper carries on the effort to bridging runtime verification with distributed computability, studying necessary conditions for monitoring failure prone asynchronous distributed systems. In a previous paper (to appear in RV 2014), it has been proved that there are correctness properties that require a large number of opinions (e.g., true, false, perhaps, probably true, probably no, etc.) to be monitored. The main outcome of this paper is that the need for this large number of opinions is not an artifact induced by the existence of artificial constructions. Instead, monitoring an important class of properties, requiring processes to produce at most k different values does requires a large number of opinions. Specifically, our main result is a proof that it is impossible to monitor k-set-agreement in an n-process system with fewer than min{2k, n} + 1 opinions. The lower bound is shown to be tight.

Minimizing the Number of Opinions for Fault-Tolerant Distributed Decision Using Well-Quasi Orderings

“…The results where later on extended in [15] to the set agreement task and in [14] proving nearly tight bounds on the number of opinions required to check any distributed language. In [12,16] the context of local distributed network computing [23] is considered.…”

“…Recent work in this area includes but is not limited to, deciding locally whether the nodes of a network are properly colored , checking the results obtained from the execution of a distributed program [13,15], designing time lower bounds on the hardness of distributed approximation [7], estimating the complexity of logics required for distributed run-time verification [14], and elaborating a distributed computing complexity theory [12,16].…”

“…In fact, it is known that up to n different opinions may be necessary to decide some languages [14]. For example, for the k-set agreement language, specifying that at most k leaders are elected, min{2k, n} + 1 opinions are necessary and sufficient in a system where n asynchronous processes may crash [15]. In general, one aims at minimizing the number of possible opinions.…”

Survey of Distributed Decision

“…For non projection-closed languages, [43] investigated more general interpretation of the individual opinions produced by the processes, beyond the logical conjunction of boolean opinons. In [41], it is proved that k-set agreement requires that the processes must be allowed to produce essentially k different opinions to be wait-free decided. The class Σ WFD 1 has been investigated in [44,45], with applications to the space complexity of failure detectors.…”