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FALCON: Scalable Reasoning over Inconsistent ALC Ontologies

Tilman Hinnerichs, Zhenwei Tang, Xi Peng, Xiangliang Zhang, Robert Hoehndorf

TL;DR

This work proposes FALCON, a Fuzzy Ontology Neural reasoner, a Fuzzy Ontology Neural reasoner to approximate reasoning over ALC ontologies, and provides an approximate technique for the model generation step in classical ALC reasoners.

Abstract

Ontologies are one of the richest sources of knowledge. Real-world ontologies often contain thousands of axioms and are often human-made. Hence, they may contain inconsistency and incomplete information which may impair classical reasoners to compute entailments that are considered as useful. To overcome these two challenges, we propose FALCON, a Fuzzy Ontology Neural reasoner to approximate reasoning over ALC ontologies. We provide an approximate technique for the model generation step in classical ALC reasoners. Our approximation is not guaranteed to construct exact logical models, but can approximate arbitrary models, which is notably faster for some large ontologies. Moreover, by sampling multiple approximate logical models, our technique supports approximate entailment also over inconsistent ontologies. Theoretical results show that more models generated lead to closer, i.e., faithful approximation of entailment over ALC entailments. Experimental results show that FALCON enables approximate reasoning and reasoning in the presence of inconsistency. Our experiments further demonstrate how ontologies can improve knowledge base completion in biomedicine by incorporating knowledge expressed in ALC.

FALCON: Scalable Reasoning over Inconsistent ALC Ontologies

TL;DR

This work proposes FALCON, a Fuzzy Ontology Neural reasoner, a Fuzzy Ontology Neural reasoner to approximate reasoning over ALC ontologies, and provides an approximate technique for the model generation step in classical ALC reasoners.

Abstract

Ontologies are one of the richest sources of knowledge. Real-world ontologies often contain thousands of axioms and are often human-made. Hence, they may contain inconsistency and incomplete information which may impair classical reasoners to compute entailments that are considered as useful. To overcome these two challenges, we propose FALCON, a Fuzzy Ontology Neural reasoner to approximate reasoning over ALC ontologies. We provide an approximate technique for the model generation step in classical ALC reasoners. Our approximation is not guaranteed to construct exact logical models, but can approximate arbitrary models, which is notably faster for some large ontologies. Moreover, by sampling multiple approximate logical models, our technique supports approximate entailment also over inconsistent ontologies. Theoretical results show that more models generated lead to closer, i.e., faithful approximation of entailment over ALC entailments. Experimental results show that FALCON enables approximate reasoning and reasoning in the presence of inconsistency. Our experiments further demonstrate how ontologies can improve knowledge base completion in biomedicine by incorporating knowledge expressed in ALC.
Paper Structure (30 sections, 5 theorems, 17 equations, 2 figures, 6 tables)

This paper contains 30 sections, 5 theorems, 17 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Description Logic $\mathcal{ALC}$
  4. Methods
  5. Generating model structures
  6. Constructing the Universe
  7. Interpreting Concept Symbols
  8. Relation Symbols
  9. Concept Descriptions
  10. Optimization
  11. Faithful semantic entailment
  12. Approximate Reasoning
  13. Theoretical results
  14. Experiments
  15. Experimental settings
  16. ...and 15 more sections

Key Result

Theorem 1

Let $\mathcal{T}$ be a TBox and $\mathcal{A}$ an ABox in $\mathcal{ALC}$ over signature $\Sigma = (\mathbf{C}, \mathbf{R}, \mathbf{I} = \mathbf{I_n} \cup \mathbf{I_{\mathbb{R}^n}})$. If $\mathcal{L}=0$ and the $t$-norm $\theta$ satisfies $\theta(x,y)=0$ if and only if $x=0$ or $y=0$, then the interp

Figures (2)

  • Figure 1: Learned degree of memberships of individuals in concepts in the Family. Dark and light grids denote the learned degrees of membership are (very close to and numerically indistinguishable from) 1 and 0, respectively. Anon $i$ denotes the randomly sampled anonymous individuals.
  • Figure 1: Experimental results of multi-model reasoning on Pizza. Avg denotes the average results (AUC) of independent single models. Multi denotes semantic entailment over multiple models described in Section \ref{['multiple_model_entailment']}.

Theorems & Definitions (16)

  • Example 1
  • Example 2
  • Definition 1: Concept satisfiability
  • Definition 2: Approximate degree of satisfiability
  • Definition 3: Approximate truth value of subsumption
  • Definition 4: Approximate concept instantiation
  • Definition 5: Approximate degree of consistency
  • Definition 6: $(M, \alpha)$-approximate entailment
  • Theorem 1: Faithfulness
  • Definition 7: Approximate model
  • ...and 6 more