Multilevel modeling extends traditional modeling techniques with a potentially unlimited number of abstraction levels. Multilevel models can be formally represented by multilevel typed graphs whose manipulation and transformation are carried out by multilevel typed graph transformation rules. These rules are cospans of three graphs and two inclusion graph homomorphisms where the three graphs are multilevel typed over a common typing chain. In this paper, we show that typed graph transformations can be appropriately generalized to multilevel typed graph transformations improving preciseness, flexibility and reusability of transformation rules. We identify type compatibility conditions, for rules and their matches, formulated as equations and inequations, respectively, between composed partial typing morphisms. These conditions are crucial presuppositions for the application of a rule for a match---based on a pushout and a final pullback complement construction for the underlying graphs in the category Graph---to always provide a well-defined canonical result in the multilevel typed setting. Moreover, to formalize and analyze multilevel typing as well as to prove the necessary results, in a systematic way, we introduce the category Chain of typing chains and typing chain morphisms.

中文翻译：

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多级类型图转换

多级建模扩展了传统建模技术，具有潜在无限数量的抽象级别。多级模型可以由多级类型图形式化表示，其操作和转换由多级类型图转换规则执行。这些规则是三个图和两个包含图同态的 cospan，其中三个图在公共类型链上是多级类型的。在本文中，我们表明类型图转换可以适当地推广到多级类型图转换，从而提高转换规则的精确性、灵活性和可重用性。我们为规则及其匹配确定类型兼容性条件，分别表述为组合部分类型态射之间的方程和不等式。这些条件是应用匹配规则的关键前提——基于类别 Graph 中底层图的推出和最终回撤补充构造——始终在多级中提供明确定义的规范结果键入设置。此外，为了形式化和分析多级类型以及证明必要的结果，我们系统地引入了类型链的范畴链和类型链态射。