Understanding conflicts between transformation steps and rules is an important topic in algebraic graph transformation. A conflict occurs when two transformation steps are not parallel independent, that is, when after applying one of them the other can no longer occur. A static analysis technique called Critical Pair Analysis allows the detection of all potential conflicts between pairs of rules, by enumerating Critical Pairs. Since these are often too numerous for even simple rules, finding appropriate subsets of critical pairs is the topic of ongoing research. We contribute to this thread by proposing a new characterization for root causes of conflicts, called “conflict essences”, exploiting a recently proposed characterization of parallel independence. Furthermore we show that conflict essences are at least as precise as the “conflict reasons” previously proposed, and uniquely determine the so-called “initial conflicts”, an appropriate subset of critical pairs, under relatively mild assumptions on the underlying category. Finally, we show that several $\mathcal{M}$-adhesive categories of interest have the necessary properties for our results to hold, including typed, attributed and symbolic graphs. While our results are formulated for conflicts, they are directly applicable to dependencies in $\mathcal{M}$-adhesive transformation systems.

理解变换步骤和规则之间的冲突是代数图变换的重要主题。当两个转换步骤不是并行独立的时，即在应用其中一个转换步骤之后，另一个不再发生时，就会发生冲突。称为关键对分析的静态分析技术允许通过枚举关键对来检测规则对之间的所有潜在冲突。由于即使对于简单的规则而言，这些数目通常也太多了，因此寻找关键对的适当子集是正在进行的研究的主题。我们通过为冲突的根本原因提出一种新的表征（称为“冲突本质”），并利用最近提出的并行独立性的表征为这一思路做出了贡献。此外，我们证明了冲突本质至少与先前提出的“冲突原因”一样精确，并且在基本类别的相对温和假设下，唯一确定了所谓的“初始冲突”，即关键对的适当子集。最后，我们表明$\mathcal{中号}$-感兴趣的胶粘剂类别具有必要的属性，可以保持我们的结果，包括类型化，属性图和符号图。虽然我们的结果是针对冲突制定的，但它们直接适用于$\mathcal{中号}$胶粘剂转化系统。