Why Confluence is More Powerful than Ample Sets in Probabilistic and Non-Probabilistic Branching Time

Confluence reduction and partial order reduction by means of ample sets are two different techniques for state space reduction in both traditional and probabilistic model checking. This presentation provides an extensive comparison between these two methods, answering the long-standing question of how they relate. We show that, while both preserve branching time properties, confluence reduction is strictly more powerful than partial order reduction: every reduction that can be obtained with partial order reduction can also be obtained with confluence reduction, but the converse is not true. A core problem in the comparison is that confluence reduction was de﬿ned in an action-based setting, whereas partial order reduction was de﬿ned in a state-based setting. We therefore rede﬿ne confluence reduction in the state-based setting of Markov decision processes, and discuss a nontrivial proof of its correctness. Additionally, we pinpoint precisely in what way confluence reduction is more general, and provide a restricted variant of confluence and relaxed variant of partial order reduction that exactly coincide. The results we present also hold for non-probabilistic models, as they can just as well be applied in a context where all transitions are non-probabilistic. To show the practical applicability of our results, we adapt a state space generation technique based on representative states, already known in combination with confluence reduction, so that it can also be applied with partial order reduction.