A comparison of confluence and ample sets in probabilistic and non-probabilistic branching time

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.

Confluence reduction for Markov automata

“…Hence, it would not even preserve Markovian divergence-insensitive branching bisimulation. We now improve on the definition to resolve this issue, introducing τ -loops in the reduced system for states having confluent divergence in the original system (inspired by the way [17] deals with divergences). This not only makes the theory work for MAs, it even yields preservation of divergence-sensitive branching bisimulation, and hence of minimal reachability probabilities.…”

“…Also, several types of partial order reduction (POR) have been defined, both for nonprobabilistic [26,21,16] and probabilistic systems [11,4,3]. These techniques are based on ideas similar to confluence, and have been compared to confluence recently, both in a theoretical [17] and in a practical manner [18]. The results showed that branching-time POR is strictly subsumed by confluence, and that the additional advantages of confluence can be employed nicely in the context of statistical model checking.…”

On-the-Fly Confluence Detection for Statistical Model Checking

“…POR was first generalised to the probabilistic domain preserving linear time properties [2,6], with a later extension to preserve branching time properties [1]. Confluence reduction was generalised in [8,14], preserving branching time properties.…”

Confluence Reduction for Markov Automata

“…Also, several types of partial order reduction (POR) have been defined, both for nonprobabilistic [26,21,16] and probabilistic systems [11,4,3]. These techniques are based on ideas similar to confluence, and have been compared to confluence recently, both in a theoretical [17] and in a practical manner [18]. The results showed that branching-time POR is strictly subsumed by confluence, and that the additional advantages of confluence can be employed nicely in the context of statistical model checking.…”

“…It is key to the way we detect confluence on MAPA specifications. -We now do preserve divergences and hence minimal reachability probabilities, incorporating a technique used earlier in [17].…”