Probabilistic Bisimulation for Parameterized Systems

Abstract

Probabilistic bisimulation is a fundamental notion of process equivalence for probabilistic systems. It has important applications, including the formalisation of the anonymity property of several communication protocols. While there is a large body of work on verifying probabilistic bisimulation for finite systems, the problem is in general undecidable for parameterized systems, i.e., for infinite families of finite systems with an arbitrary number n of processes. In this paper we provide a general framework for reasoning about probabilistic bisimulation for parameterized systems. Our approach is in the spirit of software verification, wherein we encode proof rules for probabilistic bisimulation and use a decidable first-order theory to specify systems and candidate bisimulation relations, which can then be checked automatically against the proof rules. We work in the framework of regular model checking, and specify an infinitestate system as a regular relation described by a first-order formula over a universal automatic structure, i.e., a logical theory over the string domain. For probabilistic systems, we show how probability values (as well as the required operations) can be encoded naturally in the logic. Our main result is that one can specify the verification condition of whether a given regular binary relation is a probabilistic bisimulation as a regular relation. Since the first-order theory of the universal automatic structure is decidable, we obtain an effective method for verifying probabilistic bisimulation for infinite-state systems, given a regular relation as a candidate proof. As a case study, we show that our framework is sufficiently expressive for proving the anonymity property of the parameterized dining cryptographers protocol and the parameterized grades protocol. Both of these protocols hitherto could not be verified by existing automatic methods.

Quantitative Verification of Stochastic Regular Expressions

“…This transformation generates PRA in the size of SREs; however, the obtained automata is still not minimized. There, some techniques can be applied using probabilistic bisimulation [86], or state equivalence techniques [50] to minimize the PRA.…”

Regular Model Checking Revisited (Technical Report)

“…Numerous results suggest that the answer is that it is very expressive. On the practical side, many benchmarks (especially from paramterized systems) have indicated this to be the case, e.g., see [36,17,34,30,37,3,44,38,24,33]. On the theoretical side, this framework is in fact complete for important properties like safety and liveness for many classes of infinite-state systems that can be captured by regular model checking, including pushdown systems, reversal-bounded counter systems, two-dimensional vector addition systems, communication-free Petri nets, and tree-rewrite systems (for the extension to trees), among others, e.g., see [41,42,7,32,23,35].…”

“…Encoding bisimulation and probabilistic bisimulation for parameterized systems is a bit trickier since we will need infinitely many action labels (i.e. to distinguish the action of the ith process), but this can also be encoded in our framework; see the first-order proof rules over automatic structures in the recent paper [24].…”

“…liveness, even for probabilistic distributed protocols [34,30]), as well as new correctness properties (e.g. safety games [37] and probabilistic bisimulation with applications to proving anonymity [24]). Despite these recent successes, these techniques are rather ad-hoc, and often difficult to adapt to new correctness properties.…”

“…Finally, we show that restricting to regular proofs (i.e. proofs that can be expressed by finite automata) is sufficient in practice, and allows us to have powerful verification algorithms that unify the recent successful automata synthesis algorithms [36,34,30,24] for safety, liveness, reachability games, and other interesting correctness properties.…”