Common Foundations for Recursive Shape Languages

2026-04-22

AI Summary

This work addresses the semantic divergence between ShEx and SHACL, which adopt greatest fixed-point (GFP) and least fixed-point (LFP) semantics for recursion, respectively, thereby hindering interoperability. The paper proposes a unified formal semantic framework to systematically compare LFP, GFP, and supported model semantics (SMS), revealing a duality between these approaches over stratified fragments. It identifies large-scale syntactic fragments of ShEx and SHACL that are semantically equivalent and provides a complete characterization of data and combined complexity under all three semantics. The results demonstrate that ShEx and SHACL are semantically compatible in mainstream use cases, while SMS incurs higher computational costs under standard complexity assumptions, offering a theoretical foundation for future standardization efforts and tool development.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Data Mining & Knowledge Management: Semantic Web
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

As schema languages for RDF data become more mature, we are seeing efforts to extend them with recursive semantics, applying diverse ideas from logic programming and description logics. While ShEx has an official recursive semantics based on greatest fixpoints (GFP), the discussion for SHACL is ongoing and seems to be converging towards least fixpoints (LFP). A practical study we perform shows that, indeed, ShEx validators implement GFP, whereas SHACL validators are more heterogeneous. This situation creates tension between ShEx and SHACL, as their semantic commitments appear to diverge, potentially undermining interoperability and predictability. We aim to clarify this design space by comparing the main semantic options in a principled yet accessible way, hoping to engage both theoreticians and practitioners, especially those involved in developing tools and standards. We present a unifying formal semantics that treats LFP, GFP, and supported model semantics (SMS), clarifying their relationships and highlighting a duality between LFP and GFP on stratified fragments. Next, we investigate to which extent the directions taken by SHACL and ShEx are compatible. We show that, although ShEx and SHACL seem to be going in different directions, they include large fragments with identical expressive power. Moreover, there is a strong correspondence between these fragments through the aforementioned principle of duality. Finally, we present a complete picture of the data and combined complexity of ShEx and SHACL validation under LFP, GFP, and SMS, showing that SMS comes at a higher computational cost under standard complexity-theoretic assumptions.

Problem

• recursive semantics
• ShEx
• SHACL
• greatest fixpoint
• least fixpoint

Innovation

• recursive shape languages
• least fixpoint
• greatest fixpoint
• supported model semantics
• semantic duality