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On the Power and Limitations of Examples for Description Logic Concepts

Balder ten Cate, Raoul Koudijs, Ana Ozaki

TL;DR

The paper systematically analyzes when finite characterisations exist for description-logic concepts, across a spectrum of languages from ${\mathcal{EL}}$ to ${\mathcal{ALCQI}}$ and with/without DL-Lite ontologies. It introduces a robust framework based on labeled finite interpretations and develops both constructive techniques (frontier methods, canonical models) and hardness results to classify fragments into those with and without finite characterisations, including polynomial-time computable cases. Notably, it provides an elementary doubly-exponential construction for the fragment ${\mathcal L}(\geq,\sqcap,\top)$ via frontier methods, and proves negative results for several other fragments, highlighting inherent limits on the power of labeled examples for exact characterisation. The work connects these characterisation results to exact learnability with membership queries, delineating when efficient learning is possible and outlining open questions for remaining fragments and ontology settings, with practical implications for debugging, teaching, and concept refinement in Description Logics.

Abstract

Labeled examples (i.e., positive and negative examples) are an attractive medium for communicating complex concepts. They are useful for deriving concept expressions (such as in concept learning, interactive concept specification, and concept refinement) as well as for illustrating concept expressions to a user or domain expert. We investigate the power of labeled examples for describing description-logic concepts. Specifically, we systematically study the existence and efficient computability of finite characterisations, i.e. finite sets of labeled examples that uniquely characterize a single concept, for a wide variety of description logics between EL and ALCQI, both without an ontology and in the presence of a DL-Lite ontology. Finite characterisations are relevant for debugging purposes, and their existence is a necessary condition for exact learnability with membership queries.

On the Power and Limitations of Examples for Description Logic Concepts

TL;DR

The paper systematically analyzes when finite characterisations exist for description-logic concepts, across a spectrum of languages from to and with/without DL-Lite ontologies. It introduces a robust framework based on labeled finite interpretations and develops both constructive techniques (frontier methods, canonical models) and hardness results to classify fragments into those with and without finite characterisations, including polynomial-time computable cases. Notably, it provides an elementary doubly-exponential construction for the fragment via frontier methods, and proves negative results for several other fragments, highlighting inherent limits on the power of labeled examples for exact characterisation. The work connects these characterisation results to exact learnability with membership queries, delineating when efficient learning is possible and outlining open questions for remaining fragments and ontology settings, with practical implications for debugging, teaching, and concept refinement in Description Logics.

Abstract

Labeled examples (i.e., positive and negative examples) are an attractive medium for communicating complex concepts. They are useful for deriving concept expressions (such as in concept learning, interactive concept specification, and concept refinement) as well as for illustrating concept expressions to a user or domain expert. We investigate the power of labeled examples for describing description-logic concepts. Specifically, we systematically study the existence and efficient computability of finite characterisations, i.e. finite sets of labeled examples that uniquely characterize a single concept, for a wide variety of description logics between EL and ALCQI, both without an ontology and in the presence of a DL-Lite ontology. Finite characterisations are relevant for debugging purposes, and their existence is a necessary condition for exact learnability with membership queries.

Paper Structure

This paper contains 25 sections, 61 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Related Work
  4. Basic Definitions and Framework
  5. Concept Languages
  6. Ontology Language
  7. Semantics
  8. Size of Concepts and Examples
  9. Labeled Examples and Finite characterisations
  10. Finite Characterisations without Ontology
  11. Finite Characterisations for L(forall,exists,geq,and,top)
  12. An Elementary Construction of Finite Characterisations for L(exists,geq,and,top)
  13. Negative Results
  14. Finite Characterisations With Ontologies
  15. L(exists,and,top,bot) Admits Finite Characterisations Under DL-Lite Ontologies
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\{\exists,\sqcap\}\subseteq \mathbf{O}\xspace\subseteq\{\forall,\exists,\geq,-,\sqcap,\sqcup,\top,\bot,\neg\}$.

Figures (2)

  • Figure 1: Summary of Thm. \ref{['thm:mainone']} and Thm. \ref{['thm:maintwo']}.
  • Figure 2: A frontier in the concept subsumption hierarchy

Theorems & Definitions (129)

  • Example 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • ...and 119 more