For a given labelled transition system (LTS), synthesis is the task to find an unlabelled Petri net with an isomorphic reachability graph. Even when just demanding an embedding into a reachability graph instead of an isomorphism, a solution is not guaranteed. In such a case, label splitting is an option, i.e. relabelling edges of the LTS such that differently labelled edges remain different. With an appropriate label splitting, we can always obtain a solution for the synthesis or embedding problem. Using the label splitting, we can construct a labelled Petri net with the intended bahaviour (e.g. embedding the given LTS in its reachability graph). As the labelled Petri net can have a large number of transitions, an optimisation may be desired, limiting the number of labels produced by the label splitting. We show that such a limitation will turn the problem from being solvable in polynomial time into an NP-complete problem.

中文翻译：

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将 LTS 嵌入任意 Petri 网可达图的最佳标签拆分是 NP 完全的

对于给定的标记转换系统 (LTS)，合成是找到具有同构可达图的未标记 Petri 网的任务。即使只是要求嵌入可达图而不是同构，也不能保证解决方案。在这种情况下，标签拆分是一种选择，即重新标记 LTS 的边缘，以便不同标记的边缘保持不同。通过适当的标签拆分，我们总能获得合成或嵌入问题的解决方案。使用标签拆分，我们可以构建一个带有预期行为的标记 Petri 网（例如，将给定的 LTS 嵌入其可达性图中）。由于带标签的 Petri 网可以有大量的转换，因此可能需要进行优化，限制标签拆分产生的标签数量。