On the origin of events: Branching cells as stubborn sets

Abstract

In prime event structures with binary conflicts (pes-bc) a branching cell [1] is a subset of events closed under downward causality and immediate conflict relations. This means that no event outside the branching cell can be in conflict with or enable any event inside the branching cell. It bears a strong resemblance to stubborn sets, a partial order reduction method on transition systems. A stubborn set (at a given state) is a subset of actions such that no execution consisting entirely of actions outside the stubborn set can be in conflict with or enable actions that are inside the stubborn set. A rigorous study of the relationship between the two ideas, however, is not straightforward due to the facts that 1) stubborn sets utilise sophisticated causality and conflict relations that invalidate the stability and coherence of event structures [18], 2) without stability it becomes very difficult to define concepts like prefixes and branching cells, which prerequire a clear notion of causality, and 3) it is challenging to devise a technique for identifying 'proper' subsets of transitions as 'events' such that the induced event-based system captures exactly the causality and conflict information needed by stubborn sets. In this paper we give a solution to the problems by providing an unfolding of labelled transition systems into configuration structures, a more general structure supporting or-causality and finite conflict. We show that the branching cell definition can be extended to configuration structures and that each branching cell in the unfolding is a long-lived stubborn set, such that no matter how the system evolves, what remains of the branching cell is always a stubborn set. © 2011 Springer-Verlag.