One approach to verify a property expressed as a modal \(\mu \)-calculus formula on a system with several concurrent processes is to build the underlying state space compositionally (i.e., by minimizing and recomposing the state spaces of individual processes in a hierarchical way, keeping visible only the relevant actions occurring in the formula), and check the formula on the resulting state space. It was shown previously that, when checking the formulas of the \(L_{\mu }^{ dbr }\) fragment of the \(\mu \)-calculus (consisting of weak modalities only), individual processes can be minimized modulo divergence-preserving branching (divbranching for short) bisimulation. In this paper, we refine this approach to handle formulas containing both strong and weak modalities, so as to enable a combined use of strong or divbranching bisimulation minimization on concurrent processes depending whether they contain or not the actions occurring in the strong modalities of the formula. We extend \(L_{\mu }^{ dbr }\) with strong modalities and show that the combined minimization approach preserves the truth value of formulas of the extended fragment. We implemented this approach on top of the CADP verification toolbox and demonstrated how it improves the capabilities of compositional verification on realistic examples of concurrent systems. In particular, we applied our approach to the verification problems of the RERS 2019 challenge and observed drastic reductions of the state space compared to the approach in which only strong bisimulation minimization is used, on formulas not preserved by divbranching bisimulation.

验证在具有多个并发进程的系统上以模态\（\ mu \）-演算公式表示的属性的一种方法是按组成方式构建基础状态空间（即，通过最小化和重组分层结构中各个进程的状态空间）方式，仅使公式中发生的相关动作可见），并在结果状态空间上检查公式。先前已显示，当检查\（\ mu \）的\（L _ {\ mu} ^ {dbr} \）片段的公式时，-演算（仅由弱模态组成），可以将单个过程最小化以模散度保留的分支（简称分叉）双仿真。在本文中，我们改进了这种方法来处理同时包含强和弱模态的公式，以便能够在并发过程中组合使用强或分叉双模拟最小化，这取决于它们是否包含在公式的强模态中发生的动作。我们扩展\（L _ {\ mu} ^ {dbr} \）具有强大的模态，并且表明组合最小化方法保留了扩展片段公式的真实值。我们在CADP验证工具箱的顶部实施了此方法，并演示了它如何在并发系统的实际示例中提高成分验证的功能。特别是，我们将方法应用于RERS 2019挑战的验证问题，并且与仅使用强双模拟最小化的方法相比，在未通过分叉双模拟保留的公式上，观察到状态空间的急剧减少。