Compositional model checking approaches attempt to limit state space explosion by iteratively combining the behaviour of the components in a concurrent system and reducing the result modulo an appropriate equivalence relation. In this article, we consider Labelled Transition Systems (LTSs), in which transitions are labelled by actions, to describe component behaviour, and LTS networks to combine the behaviour of all components in a system.

For an equivalence relation to be useful for the compositional model checking of LTS networks, it should be a congruence for the parallel composition operator that is used to combine component behaviour. Such an operator may define synchronisations between the actions of component transitions.

An equivalence relation preserving both safety and liveness properties is divergence-preserving branching bisimilarity (DPBB). It has long been generally assumed that DPBB is a congruence for parallel composition. Fokkink, Van Glabbeek and Luttik recently proposed a congruence format that implies that this is the case. In parallel, we were the first to prove, by means of the Coq proof assistant, that DPBB is a congruence for the parallel composition of two LTS networks with synchronisation on transition labels. In the current article, we also consider an instance of our parallel composition operator that is both associative and commutative, which are two essential properties for the compositional construction of state spaces.

Furthermore, we show that DPBB is a congruence for LTS networks in which many LTSs are composed in parallel at once with support for multi-party synchronisation. Additionally, we discuss how to safely decompose an existing LTS network into components such that their recomposition is equivalent to the original LTS network.

Finally, to demonstrate the effectiveness of compositional model checking with intermediate DPBB reductions, we discuss the results we obtained after having conducted a number of experiments.

组成模型检查方法试图通过迭代合并并发系统中组件的行为并以适当的等价关系为模来减少结果，从而限制状态空间爆炸。在本文中，我们考虑标记过渡系统（LTS），其中过渡以动作标记，以描述组件的行为，而LTS网络则结合了系统中所有组件的行为。

为了使等价关系对LTS网络的组成模型检查有用，用于组合组成行为的并行组成算子应该是同等的。这样的操作员可以定义组件转换动作之间的同步。

既保持安全性又保持活力的等价关系是保持分歧的分支双相似性（DPBB）。人们长期以来一直认为DPBB是平行合成的全等。Fokkink，Van Glabbeek和Luttik最近提出了一种全等格式，这表明情况确实如此。同时，我们是第一个通过Coq证明助手证明DPBB是两个LTS网络的并行组成的全等，并且在过渡标签上具有同步。在当前文章中，我们还将考虑并行组合运算符的一个实例，该实例既具有关联性又具有交换性，这是状态空间的组合构造的两个基本属性。

此外，我们表明DPBB是LTS网络的一个统筹，其中许多LTS一次并行组成，并支持多方同步。此外，我们讨论了如何安全地将现有LTS网络分解为组件，使其重组等效于原始LTS网络。

最后，为了证明成分模型检查具有中等DPBB降低的有效性，我们讨论了进行大量实验后获得的结果。