Fair Termination for Parameterized Probabilistic Concurrent Systems (Technical Report)

TL;DR

This paper introduces a novel automatic verification method for probabilistic parameterized systems that ensures termination under realistic fairness assumptions, applicable to distributed algorithms and biological models.

Contribution

It provides the first systematic, regularity-preserving encoding of finitary fairness in regular model checking for probabilistic systems, enabling automatic termination proofs.

Findings

Verified termination for Herman's protocol, Moran process, and cell cycle switch.

First fully-automatic method for these probabilistic, fair termination problems.

Demonstrated applicability to distributed algorithms and biological models.

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.