The Shapes Constraint Language (SHACL) is the recent W3C recommendation language for validating RDF data, by verifying certain shapes on graphs. Previous work has largely focused on the validation problem, while the standard decision problems of satisfiability and containment, crucial for design and optimisation purposes, have only been investigated for simplified versions of SHACL. Moreover, the SHACL specification does not define the semantics of recursively-defined constraints, which led to several alternative recursive semantics being proposed in the literature. The interaction between these different semantics and important decision problems has not been investigated yet. In this article we provide a comprehensive study of the different features of SHACL, by providing a translation to a new first-order language, called SCL, that precisely captures the semantics of SHACL. We also present MSCL, a second-order extension of SCL, which allows us to define, in a single formal logic framework, the main recursive semantics of SHACL. Within this language we also provide an effective treatment of filter constraints which are often neglected in the related literature. Using this logic we provide a detailed map of (un)decidability and complexity results for the satisfiability and containment decision problems for different SHACL fragments. Notably, we prove that both problems are undecidable for the full language, but we present decidable combinations of interesting features, even in the face of recursion.

形状约束语言(SHACL) 是最近的W3C 推荐语言，用于通过验证图形上的某些形状来验证 RDF 数据。以前的工作主要集中在验证问题上，而对设计和优化目的至关重要的可满足性和包含性的标准决策问题仅针对简化版本的 SHACL 进行了研究。此外，SHACL 规范没有定义递归定义的约束的语义，这导致了几种替代递归文献中提出的语义。这些不同语义和重要决策问题之间的相互作用尚未被研究。在本文中，我们通过提供一种新的一阶语言（称为 SCL）的翻译，对 SHACL 的不同特性进行了全面研究，该语言精确地捕捉了 SHACL 的语义。我们还介绍了 MSCL，它是 SCL 的二阶扩展，它允许我们在单个形式逻辑框架中定义 SHACL 的主要递归语义。在这种语言中，我们还提供了对相关文献中经常被忽略的过滤器约束的有效处理。使用此逻辑，我们为不同 SHACL 片段的可满足性和包含决策问题提供了（未）可判定性和复杂性结果的详细地图。尤其，