Satisfiability and Containment of Recursive SHACL

TL;DR

This paper provides a comprehensive logical framework for analyzing the satisfiability and containment problems in SHACL, including recursive semantics, and identifies decidable fragments despite the language's overall undecidability.

Contribution

It introduces a new formal logic translation capturing SHACL semantics and explores the decidability and complexity of key decision problems across various SHACL fragments.

Findings

Satisfiability and containment are undecidable for the full language.

Decidable fragments exist with specific feature combinations.

A new logical framework models recursive SHACL semantics effectively.

Abstract

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 and 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.