Checking Qualitative Liveness Properties of Replicated Systems with Stochastic Scheduling

Abstract

We present a sound and complete method for the verification of qualitative liveness properties of replicated systems under stochastic scheduling. These are systems consisting of a finite-state program, executed by an unknown number of indistinguishable agents, where the next agent to make a move is determined by the result of a random experiment. We show that if a property of such a system holds, then there is always a witness in the shape of a Presburger stage graph : a finite graph whose nodes are Presburger-definable sets of configurations. Due to the high complexity of the verification problem (non-elementary), we introduce an incomplete procedure for the construction of Presburger stage graphs, and implement it on top of an SMT solver. The procedure makes extensive use of the theory of well-quasi-orders, and of the structural theory of Petri nets and vector addition systems. We apply our results to a set of benchmarks, in particular to a large collection of population protocols, a model of distributed computation extensively studied by the distributed computing community.

Population Protocols: Beyond Runtime Analysis

“…This time, thanks to new progress by Leroux on the theory of Petri nets, we proved that the correctness problem is decidable, although as hard as the reachability problem for Petri nets [31]. This was the starting point of a research program devoted to the theory and practice of verifying population protocols, which reached an important milestone in 2020 with the release of Peregrine 2.0, a verifier based on new theoretical results [15,32]. The first part of this note surveys this research, adding all the work carried out since 2017 to a brief previous survey [28].…”

“…They emerged when we started to investigate how to automatically compute an upper bound on the expected runtime of a protocol. This work, published in [19], introduced stage graphs, a notion that, after many rewrites, finally led to the stage graph proof methodology of [15], which we describe now.…”

The complexity of verifying population protocols

“…The TOWER-hard lower bound is also proved in [32] by reduction to the reachability problem for Petri nets, which is shown to be TOWER-hard in [25]. Practical verification algorithms for PP have been given in [14,15,17]. The complexity of other verification problems beyond correctness is studied in [31].…”

Computing Parameterized Invariants of Parameterized Petri Nets

“…We present a technique that automatically computes a finite set of parameterized invariants, readable by humans. This is achieved by lifting a CEGAR (counterexampleguided abstraction refinement) loop, introduced in [28] and further developed in [27,29,12], to the parameterized case. Each iteration of the loop of [28,27] first computes a counterexample, i.e., a marking that violates the desired safety property but satisfies all invariants computed so far, and then computes a Pcomponent, siphon, or trap showing that the marking is not reachable.…”