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. It has been recently proved that there are correctness properties that require a large number of opinions to be monitored, an opinion being of the form true, false, perhaps, probably true, probably no, etc. The main outcome of this paper is to show that 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 require such 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. We also provide an algorithm to monitor k-set-agreement with min{2k, n} + 1 opinions, showing that the lower bound is tight.