Well-structured transition systems form a large class of infinite-state systems, for which safety verification is decidable thanks to a generic backward coverability algorithm. However, for several classes of systems, the generic upper bounds one can extract from the algorithm are far from optimal. In particular, in the case of vector addition systems (VAS) and several of their extensions, the known tight upper bounds were rather derived thanks to ad-hoc arguments based on Rackoff's small witness property.

We show how to derive the same bounds directly on the computations of the VAS instantiation of the generic backward coverability algorithm. This relies on a dual view of the algorithm using ideal decompositions of downwards-closed sets, which exhibits a key structural invariant in the VAS case. This reasoning offers a uniform setting for all well-structured transition systems, including branching ones, and we further apply it to several VAS extensions, deriving optimal upper bounds.

结构良好的过渡系统构成了一大类无限状态系统，对于这些系统，由于具有通用的后向可覆盖性算法，因此可以确定安全性。但是，对于几类系统，可以从算法中提取的通用上限远非最佳。特别是在向量加法系统（VAS）及其几个扩展的情况下，由于基于Rackoff的小型见证人属性的即席参数，已知的紧密上限相当容易得出。

我们展示了如何直接在通用后向可覆盖性算法的VAS实例的计算上得出相同的边界。这依赖于使用向下封闭集合的理想分解的算法的双重视图，这在VAS情况下显示出关键的结构不变性。这种推理为所有结构良好的过渡系统（包括分支系统）提供了统一的设置，并且我们将其进一步应用于多个VAS扩展，从而得出最佳上限。