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PN-OWL: A Two Stage Algorithm to Learn Fuzzy Concept Inclusions from OWL Ontologies

Franco Alberto Cardillo, Franca Debole, Umberto Straccia

TL;DR

PN-OWL tackles learning fuzzy $EL({\mathbf{D}})$ concept inclusions from OWL ontologies by introducing a two-stage PN-rule-inspired framework. The P-stage seeks broad positive coverage (high recall) while the N-stage prunes false positives to improve precision, and the two rule sets are aggregated into a final fuzzy inclusion that can be encoded in and reasoned with Fuzzy OWL 2. The method supports fuzzy datatypes and automatically constructs fuzzy concepts, enabling graded classification via a Fuzzy OWL 2 reasoner. Across multiple ontologies, PN-OWL generally outperforms a fuzzy Foil baseline, highlighting the benefit of the N-stage in reducing false positives. The work points to future directions in datatype coverage, efficiency, alternative stage learners, and integration with sub-symbolic methods, with potential extensions to thresholded concepts for finer-grained rules.

Abstract

OWL ontologies are a quite popular way to describe structured knowledge in terms of classes, relations among classes and class instances. In this paper, given a target class T of an OWL ontology, with a focus on ontologies with real- and boolean-valued data properties, we address the problem of learning graded fuzzy concept inclusion axioms with the aim of describing enough conditions for being an individual classified as instance of the class T. To do so, we present PN-OWL that is a two-stage learning algorithm made of a P-stage and an N-stage. Roughly, in the P-stage the algorithm tries to cover as many positive examples as possible (increase recall), without compromising too much precision, while in the N-stage, the algorithm tries to rule out as many false positives, covered by the P-stage, as possible. PN-OWL then aggregates the fuzzy inclusion axioms learnt at the P-stage and the N-stage by combining them via aggregation functions to allow for a final decision whether an individual is instance of T or not. We also illustrate its effectiveness by means of an experimentation. An interesting feature is that fuzzy datatypes are built automatically, the learnt fuzzy concept inclusions can be represented directly into Fuzzy OWL 2 and, thus, any Fuzzy OWL 2 reasoner can then be used to automatically determine/classify (and to which degree) whether an individual belongs to the target class T or not.

PN-OWL: A Two Stage Algorithm to Learn Fuzzy Concept Inclusions from OWL Ontologies

TL;DR

PN-OWL tackles learning fuzzy concept inclusions from OWL ontologies by introducing a two-stage PN-rule-inspired framework. The P-stage seeks broad positive coverage (high recall) while the N-stage prunes false positives to improve precision, and the two rule sets are aggregated into a final fuzzy inclusion that can be encoded in and reasoned with Fuzzy OWL 2. The method supports fuzzy datatypes and automatically constructs fuzzy concepts, enabling graded classification via a Fuzzy OWL 2 reasoner. Across multiple ontologies, PN-OWL generally outperforms a fuzzy Foil baseline, highlighting the benefit of the N-stage in reducing false positives. The work points to future directions in datatype coverage, efficiency, alternative stage learners, and integration with sub-symbolic methods, with potential extensions to thresholded concepts for finer-grained rules.

Abstract

OWL ontologies are a quite popular way to describe structured knowledge in terms of classes, relations among classes and class instances. In this paper, given a target class T of an OWL ontology, with a focus on ontologies with real- and boolean-valued data properties, we address the problem of learning graded fuzzy concept inclusion axioms with the aim of describing enough conditions for being an individual classified as instance of the class T. To do so, we present PN-OWL that is a two-stage learning algorithm made of a P-stage and an N-stage. Roughly, in the P-stage the algorithm tries to cover as many positive examples as possible (increase recall), without compromising too much precision, while in the N-stage, the algorithm tries to rule out as many false positives, covered by the P-stage, as possible. PN-OWL then aggregates the fuzzy inclusion axioms learnt at the P-stage and the N-stage by combining them via aggregation functions to allow for a final decision whether an individual is instance of T or not. We also illustrate its effectiveness by means of an experimentation. An interesting feature is that fuzzy datatypes are built automatically, the learnt fuzzy concept inclusions can be represented directly into Fuzzy OWL 2 and, thus, any Fuzzy OWL 2 reasoner can then be used to automatically determine/classify (and to which degree) whether an individual belongs to the target class T or not.
Paper Structure (13 sections, 15 equations, 4 figures, 5 tables, 3 algorithms)

This paper contains 13 sections, 15 equations, 4 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Mathematical Fuzzy Logic.
  4. Fuzzy Description Logics basics.
  5. PN-OWL
  6. The Learning Problem
  7. Conceptual Illustration of the Learning Method.
  8. The Learning Algorithm PN-OWL
  9. The Stage Learner pnFoil-$\mathcal{DL}$
  10. Evaluation
  11. Related Work
  12. Conclusions & Future Work
  13. Brief Description of the Datasets

Figures (4)

  • Figure 1: Excerpt of the mammographic ontology.
  • Figure 2: (a) Trapezoidal function $\mathit{trz}(a,b,c,d)$, (b) triangular function $\mathit{tri}(a,b,c)$, (c) left shoulder function $\mathit{ls}(a,b)$, and (d) right shoulder function $\mathit{rs}(a,b)$.
  • Figure 3: Uniform fuzzy sets over salaries: trapezoidal (left) or triangular (right).
  • Figure 4: How PN-OWL works. (a) Original training set; (b) Coverage $Cov_{\theta_P}(h_P)$ w.r.t. learnt hypothesis $h_P$ after the P-stage; (c) Starting dataset for N-stage: the new target class are the false positives $FP$ of the P-stage, while the negative individuals are the initial positives; (d) Coverage $Cov_{\theta_N}(h_N)$ w.r.t. learnt hypothesis $h_N$ after the N-stage; (e) Final scenario.

Theorems & Definitions (8)

  • Example 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Example 4.1