# Fixpoint Semantics for Recursive SHACL

arXiv: https://arxiv.org/abs/2109.08285 · HTML: https://mlsanity.com/paper/2109.08285/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 31 references that are in the map.
- First posted on arXiv 17 Sep 2021.

## Builds on (most cited first, 3 of 3)

- [Predicate Logic as a Modelling Language: The IDP System](https://mlsanity.com/paper/1401.6312/) (2014)
- [SHACL: A Description Logic in Disguise](https://mlsanity.com/paper/2108.06096/) (2021)
- [Approximation Fixpoint Theory and the Well-Founded Semantics of Higher-Order Logic Programs](https://mlsanity.com/paper/1804.08335/) (2018)

## Built on by (most cited first, 6 of 6)

- [Non-Deterministic Approximation Fixpoint Theory and Its Application in Disjunctive Logic Programming](https://mlsanity.com/paper/2211.17262/) (2022)
- [Common Foundations for SHACL, ShEx, and PG-Schema](https://mlsanity.com/paper/2502.01295/) (2025)
- [Shape Fragments](https://mlsanity.com/paper/2112.11796/) (2021)
- [A Knowledge Engineering Primer](https://mlsanity.com/paper/2305.17196/) (2023)
- [The Stable Model Semantics for Higher-Order Logic Programming](https://mlsanity.com/paper/2408.10563/) (2024)
- [Shapes from Examples: Foundations of Shape Learning in Recursive SHACL](https://mlsanity.com/paper/2607.27934/) (2026)

## Abstract (arXiv)

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.