Shapes from Examples: Foundations of Shape Learning in Recursive SHACL

2026-07-30

AI Summary

This study addresses the problem of automatically learning shape expressions that conform to recursive SHACL semantics from positive and negative example nodes, thereby enabling effective validation of knowledge graphs. Focusing on the ELI fragment of description logics, the work establishes—for the first time—the theoretical foundations of shape learning under multiple semantics of recursive SHACL, including well-founded, stable, and supported interpretations, and precisely delineates the boundaries of learnability. The core contributions include proving that both the existence of a fitting shape expression and the computation of a most specific fitting expression are solvable in exponential time, while also providing polynomial-time algorithms for several practically relevant special cases. Furthermore, the paper establishes tight upper bounds on the associated computational complexity.

Abstract

SHACL shapes enable data graph validation, making automatic shape learning essential for knowledge graph applications. We investigate the well-known fitting approach to this task: given sets P and N of positive and negative example nodes from an input graph, compute a shape expression C, possibly using shape names defined in a recursive shape catalogue, that validates at every node in P and none in N. We focus on the case where C is written in a core fragment of SHACL corresponding to the Description Logic ELI. For the catalogue, we consider the well-founded, stable, and supported semantics. We address fitting existence and most specific fitting computation, establish tight exponential-time upper bounds for both problems, and obtain polynomial bounds for relevant special cases.

Problem

• SHACL
• shape learning
• recursive shapes
• Description Logic ELI
• fitting problem

Innovation

• shape learning
• recursive SHACL
• Description Logic ELI
• fitting problem
• well-founded semantics