Static Analysis of Recursive SHACL

2026-05-04

AI Summary

This study addresses the entailment problem for recursive SHACL documents under the well-founded semantics: given two SHACL documents, it determines whether every RDF graph accepted by the first is also accepted by the second. To this end, the work establishes, for the first time, a theoretical correspondence between the well-founded semantics of SHACL and hybrid μ-calculus, translating shape validation into fixed-point modal logic formulas. By integrating automata-theoretic techniques with the description logic ALCIO, the authors devise a worst-case optimal decision procedure. The main contribution is proving that this entailment problem is decidable in single-exponential time under the well-founded semantics, whereas it becomes undecidable under alternative semantics, thereby providing the first decidable and efficient method for static analysis of SHACL.

Abstract

SHACL (Shapes Constraint Language) expresses constraints on RDF data by means of so-called shapes. Its central service is validation: verifying whether a data graph complies with a SHACL document. But so far, there are no static analysis services to compare documents. In this paper, we study the following problem: decide whether all graphs that validate one SHACL document also validate another. Unlike previous works that have considered the implication of shape expressions only, we consider documents comprising (recursive) shape definitions and targets. We show that implication (a.k.a. containment) is undecidable under the supported and the stable model semantics, even for the fragment that uses the description logic ALCIO for shape expressions. Under the well-founded semantics, in surprising contrast, it is decidable in single exponential time. Our key technical contribution is a translation of SHACL under the well-founded semantics into the full hybrid mu-calculus, revealing a novel link between well-founded models and a fixed point modal logic, and a worst-case optimal automata-based decision procedure.

Problem

• SHACL
• containment
• implication
• static analysis
• recursive shapes

Innovation

• SHACL
• well-founded semantics
• hybrid mu-calculus
• containment
• static analysis