Abstract
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 paper provides an extensive comparison between these two methods, and answers the question how they relate in terms of reduction power when preserving branching time properties. We prove that, while both preserve the same 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.The main challenge for the comparison is that confluence reduction was defined in an action-based setting, whereas ample set reduction is often defined in a state-based setting. We therefore redefine confluence reduction in the state-based setting of Markov decision processes, and provide a nontrivial proof of its correctness. Additionally, we pinpoint precisely in what way confluence reduction is more general, and provide conditions under which the two notions 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 discuss 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 to ample sets.
Highlights
Probabilistic model checking has proved to be an effective way for improving the quality of communication protocols and encryption techniques, for studying biological systems, and measuring the performance of networks
We show that when preserving branching time behaviour, confluence reduction is strictly more powerful than ample set reduction, by proving that every nontrivial ample set can be mimicked by a confluent set, while providing examples where confluent transitions do not qualify as ample sets
Note that, compared to the maximal ample set reduction that could be obtained for this Markov decision processes (MDPs), we reduced on two more occasions in the MDP
Summary
Probabilistic model checking has proved to be an effective way for improving the quality of communication protocols and encryption techniques, for studying biological systems, and measuring the performance of networks. Two powerful techniques of this kind were generalised from non-probabilistic model checking to the probabilistic setting: partial order reduction [1, 2, 3] and confluence reduction [4, 5] Both use a notion of independence between transitions of a system, either explicitly or implicitly, and try to reduce the state space by eliminating redundant paths through the system (and often states). In [17, 18] a probabilistic variant was introduced that, just like the ample set reduction of [13], preserves branching properties It was defined as a reduction technique for action-based probabilistic automata [19], but as we will show in this paper, it can be used in the context of MDPs. Ample sets and confluent transitions are defined and detected quite differently: ample sets are defined by first giving an independence relation for the action labels, whereas confluence is a property of a set of (invisible) transitions in the final state space.
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