Fixpoint Semantics for Recursive SHACL

TL;DR

This paper develops a formal fixpoint semantics for recursive SHACL schemas by leveraging Approximation Fixpoint Theory, providing a solid theoretical foundation for future extensions of the W3C standard.

Contribution

It introduces a three-valued semantic operator for recursive SHACL schemas based on AFT, enabling formal analysis and diverse semantics.

Findings

Defines a three-valued semantic operator for recursive SHACL

Derives supported, stable, and well-founded semantics for recursive schemas

Provides a theoretical foundation for future SHACL standard extensions

Abstract

SHACL is a W3C-proposed language for expressing structural constraints on RDF graphs. The recommendation only specifies semantics for non-recursive SHACL; recently, some efforts have been made to allow recursive SHACL schemas. In this paper, we argue that for defining and studying semantics of recursive SHACL, lessons can be learned from years of research in non-monotonic reasoning. We show that from a SHACL schema, a three-valued semantic operator can directly be obtained. Building on Approximation Fixpoint Theory (AFT), this operator immediately induces a wide variety of semantics, including a supported, stable, and well-founded semantics, related in the expected ways. By building on AFT, a rich body of theoretical results becomes directly available for SHACL. As such, the main contribution of this short paper is providing theoretical foundations for the study of recursive SHACL, which can later enable an informed decision for an extension of the W3C recommendation.