Fair Termination for Parameterized Probabilistic Concurrent Systems

Abstract

Abstract. We consider the problem of automatically verifying that a parameterized family of probabilistic concurrent systems terminates with probability one for all instances against adversarial schedulers. A parameterized family defines an infinite-state system: for each number n, the family consists of an instance with n finite-state processes. In contrast to safety, the parameterized verification of liveness is currently still considered extremely challenging especially in the presence of probabilities in the model. One major challenge is to provide a sufficiently powerful symbolic framework. One well-known symbolic framework for the parameterized verification of non-probabilistic concurrent systems is regular model checking. Although the framework was recently extended to probabilistic systems, incorporating fairness in the framework -often crucial for verifying termination -has been especially difficult due to the presence of an infinite number of fairness constraints (one for each process). Our main contribution is a systematic, regularity-preserving, encoding of finitary fairness (a realistic notion of fairness proposed by Alur & Henzinger) in the framework of regular model checking for probabilistic parameterized systems. Our encoding reduces termination with finitary fairness to verifying parameterized termination without fairness over probabilistic systems in regular model checking (for which a verification framework already exists). We show that our algorithm could verify termination for many interesting examples from distributed algorithms (Herman's protocol) and evolutionary biology (Moran process, cell cycle switch), which do not hold under the standard notion of fairness. To the best of our knowledge, our algorithm is the first fully-automatic method that can prove termination for these examples.

Learning to prove safety over parameterised concurrent systems

“…Despite our results that SAT-based learning seems to be less efficient than L * learning for synthesising regular inductive invariants in regular model checking, SAT-based learning is more general and more easily applicable when verifying other properties, e.g., liveness [53], fair termination [49], and safety games [57]. View abstraction [5] is a novel technique for parameterised verification.…”

Checking Qualitative Liveness Properties of Replicated Systems with Stochastic Scheduling

“…3], [ 31 , threshold-n ]) for the family of predicates for some constant ; two families for threshold predicates of the form [ 8 , 20 ]; and one family for remainder protocols of the form [ 22 ]. Further, we check approximate majority protocols ([ 27 , 56 ], [ 51 , coin game ]). As these protocols only compute the predicate with large probability but not almost surely, we only verify that they always converge to a stable consensus.…”

The complexity of verifying population protocols

“…Correctness of population protocols is however a liveness property under stochastic schedulers, which choose the agents at random. This distinguishes our work from recent contributions to parameterized verification [38,41].…”