"Declutter your Justifications" @ WoDOOM 2012, Galway, Ireland

My presentation on "Declutter your Justifications" from the First International Workshop on Debugging Ontologies and Ontology Mappings, October 8, 2012 in Galway, Ireland (an EKAW 2012 workshop)

Transcript

1. Samantha Bail, Bijan Parsia, Uli Sattler The University of Manchester Determining Similarity Between OWL Explanations Declutter Your Justi cations WoDOOM, 8th October 2012
2. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Background: OWL & Justi cations 2 • Justi cations pinpoint the causes for an entailment ‣ Restrict attention to the relevant axioms ‣ Focus on a potentially smaller set of axioms • Best understood and most promising explanation type ‣ for ontology debugging (understanding & xing errors) ‣ for ontology comprehension De nition J is a justification for O |= η if J ⊆ O, J |= η and for all J ￿ ⊂ J it holds that J ￿ ￿ η.
3. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Motivation: Multiple Justi cations • Entailments can have multiple justi cations • potentially exponential in the size of the ontology • We may also want to repair multiple entailments • Large numbers of justi cations are an unordered, unmanageable mess • But: Justi cations are often similar (if we look closely) 3
4. • Pizza ontology ( ): 100 classes, 672 axioms • >100 justi cations for an entailment SHOIN
5. • Pizza ontology ( ): 100 classes, 672 axioms • >100 justi cations for an entailment SHOIN
6. • Pizza ontology ( ): 100 classes, 672 axioms • >100 justi cations for an entailment SHOIN
7. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Motivation: Multiple Justi cations • Entailments can have multiple justi cations • potentially exponential in the size of the ontology • We may also want to repair multiple entailments • Large numbers of justi cations are an unordered, unmanageable mess • But: Justi cations are often similar (if we look closely) 5
8. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Motivation: Multiple Justi cations • Entailments can have multiple justi cations • potentially exponential in the size of the ontology • We may also want to repair multiple entailments • Large numbers of justi cations are an unordered, unmanageable mess • But: Justi cations are often similar (if we look closely) • The logical diversity is not as great as it may seem • ... how do we determine similarity between justi cations? 6
9. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Structural Equivalence • Structural equivalence [1] of OWL axioms is well de ned: ‣ We have an equivalence relation! ‣ ... but only a boring one 7 Example 1) InterestingPizza ≡ Pizza ￿ ￿ 3 hasTopping 2) ￿ 3 hasTopping ￿ Pizza ≡ InterestingPizza [1] http://www.w3.org/TR/owl2-syntax/
10. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations • Isomorphism [2] between justi cations is well de ned • It describes an equivalence relation 8 [2] Matthew Horridge, Samantha Bail, Bijan Parsia, and Ulrike Sattler. The cognitive complexity of OWL justi cations. In Proceedings of ISWC-11, 2011. Justi cation Isomorphism Example J1 = {A ￿ B ￿ ∃r.C, B ￿ ∃r.C ￿ D} |= A ￿ D J2 = {E ￿ B ￿ ∃s.F, B ￿ ∃s.F ￿ D} |= E ￿ D φ = {A ￿→ E, C ￿→ F, r ￿→ s}
11. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations We Want More! • ... but these do not cover all possible “similarities”: 9 J1 = {A ￿ B ￿ C, B ￿ C ￿ D} |= A ￿ D J2 = {A ￿ ∃r.C, ∃r.C ￿ D} |= A ￿ D Example: Subexpressions
12. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations We Want More! • ... but these do not cover all possible “similarities”: 10 J1 = {A ￿ B ￿ C, B ￿ C ￿ D} |= A ￿ D J2 = {A ￿ ∃r.C, ∃r.C ￿ D} |= A ￿ D X1 Example: Subexpressions
13. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations We Want More! • ... but these do not cover all possible “similarities”: 11 J1 = {A ￿ B ￿ C, B ￿ C ￿ D} |= A ￿ D J2 = {A ￿ ∃r.C, ∃r.C ￿ D} |= A ￿ D X1 X1 Example: Subexpressions
14. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations We Want More! • ... but these do not cover all possible “similarities”: 12 J1 = {A ￿ B ￿ C, B ￿ C ￿ D} |= A ￿ D J2 = {A ￿ ∃r.C, ∃r.C ￿ D} |= A ￿ D X1 X1 X2 Example: Subexpressions
15. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Example: Subexpressions Declutter Your Justi cations We Want More! • ... but these do not cover all possible “similarities”: 13 J1 = {A ￿ B ￿ C, B ￿ C ￿ D} |= A ￿ D J2 = {A ￿ ∃r.C, ∃r.C ￿ D} |= A ￿ D X1 X1 X2 X2
16. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations We Want More! • ... but these do not cover all possible “similarities”: 14 J1 = {A ￿ B ￿ C, B ￿ C ￿ D} |= A ￿ D J2 = {A ￿ ∃r.C, ∃r.C ￿ D} |= A ￿ D X1 X1 X2 X2 Example: Different number of axioms J1 = {A ￿ B, B ￿ C} |= A ￿ C J2 = {A ￿ B, B ￿ C, C ￿ D} |= A ￿ D Example: Subexpressions
17. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations We Want More! • ... but these do not cover all possible “similarities”: 15 J1 = {A ￿ B ￿ C, B ￿ C ￿ D} |= A ￿ D J2 = {A ￿ ∃r.C, ∃r.C ￿ D} |= A ￿ D X1 X1 X2 X2 Example: Different number of axioms J1 = {A ￿ B, B ￿ C} |= A ￿ C J2 = {A ￿ B, B ￿ C, C ￿ D} |= A ￿ D Example: Subexpressions ...
18. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Subexpression-Isomorphism 16 De nition: S-Isomorphism Two justifications (J1, η1), (J2, η2) are s-isomorphic ((J1, η1) ≈s (J2, η2)) if there exists a justification (J , η) and two injective substitutions φ1, φ2, such that φ1(J ) = J1, φ2(J ) = J2, φ1(η) = η1, and φ2(η) = η2. • S-isomorphism is re exive, symmetric, and transitive ‣ It is an equivalence relation ‣ It partitions a set of justi cations
19. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Lemma-Isomorphism 17 • Lemma: “intermediate proof step” • Entailment of a subset of a justi cation [3] [3] Matthew Horridge, Bijan Parsia, and Ulrike Sattler. Lemmas for justi cations in OWL. In Proceedings of DL 2009, 2009.
20. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Lemma-Isomorphism 18 Example: Lemmatisation • Lemma: “intermediate proof step” • Entailment of a subset of a justi cation [3] Justi cation [3] Matthew Horridge, Bijan Parsia, and Ulrike Sattler. Lemmas for justi cations in OWL. In Proceedings of DL 2009, 2009. {A ￿ ∃r.B, B ￿ C C ￿ D ∃r.D ￿ E}
21. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Lemma-Isomorphism 19 Example: Lemmatisation B ￿ D } • Lemma: “intermediate proof step” • Entailment of a subset of a justi cation [3] Justi cation Lemma [3] Matthew Horridge, Bijan Parsia, and Ulrike Sattler. Lemmas for justi cations in OWL. In Proceedings of DL 2009, 2009. {A ￿ ∃r.B, B ￿ C C ￿ D ∃r.D ￿ E}
22. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Lemma-Isomorphism 20 Example: Lemmatisation B ￿ D } • Lemma: “intermediate proof step” • Entailment of a subset of a justi cation [3] Justi cation Lemma Lemmatisation [3] Matthew Horridge, Bijan Parsia, and Ulrike Sattler. Lemmas for justi cations in OWL. In Proceedings of DL 2009, 2009. {A ￿ ∃r.B, B ￿ C C ￿ D ∃r.D ￿ E} {A ￿ ∃r.B, B ￿ D ∃r.D ￿ E}
23. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Lemma-Isomorphism 21 ‣ L-isomorphism with arbitrary lemmas is not transitive! ‣ Arbitrary lemmatisations may be entirely different from the original justi cation ‣ e.g. lemmatisation = entailment De nition: L-Isomorphism Two justifications (J1, η1), (J2, η2) are ￿-isomorphic ((J1, η) ≈￿ (J2, η)) if there exist lemmatisations J Λ1 1 , J Λ2 2 which are s-isomorphic: J Λ1 1 ≈s J Λ2 2 .
24. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Lemma Restriction • We need to restrict the selection of lemmas: ‣ Allow only summarising lemmas ‣ Allow only obvious steps to be substituted • Frequent pattern: Atomic subsumption chains ‣ ... they seem like a good start! ‣ We substitute atomic subsumption chains in a justi cation with their entailment 22
25. Samantha Bail, Bijan Parsia, Uli Sattler • 83 ontologies from BioPortal ‣ 85 to 70,015 axioms (median: 962) ‣ 23 to 33,913 classes (median: 552) ‣ Expressivity: to • Computed 6,744 justi cations in total bails@cs.man.ac.uk Declutter Your Justi cations BioPortal Survey 23 AL SROIQ(D)
26. Samantha Bail, Bijan Parsia, Uli Sattler • 83 ontologies from BioPortal ‣ 85 to 70,015 axioms (median: 962) ‣ 23 to 33,913 classes (median: 552) ‣ Expressivity: to • Computed 6,744 justi cations in total • Large reductions visible across corpus: • 7.7 -11 justi cations per template • all to iso: 90.9% • iso to s-iso: 25.7% • s-iso to l-iso: 15.8% bails@cs.man.ac.uk Declutter Your Justi cations BioPortal Survey 24 0 1,750 3,500 5,250 7,000 all iso s-iso l-iso 384 456 614 6744 AL SROIQ(D)
27. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations 25 musculoskeletal_bleeding SubClassOf disease_o musculoskeletal_bleeding SubClassOf disease_o musculoskeletal_bleeding SubClassOf disease_o aspirin SubClassOf disease_or_disorder aspirin SubClassOf disease_or_disorder aspirin SubClassOf disease_or_disorder aspirin SubClassOf disease_or_disorder Selected Instantiations musculoskeletal_bleeding SubClassOf disease_o musculoskeletal_bleeding SubClassOf disease_o aspirin SubClassOf disease_or_disorder Entailments template 1 (2 axioms) Strict iso templates L-iso templates S-iso templates Instantiations of selected templates A potential application...
28. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Another potential application... 26 J1 = {A ￿ B ￿ C, B ￿ C ￿ D} |= A ￿ D J2 = {A ￿ ∃r.C, ∃r.C ￿ D} |= A ￿ D X1 X1 X2 X2 J1 = {A ￿ B, B ￿ C} |= A ￿ C J2 = {A ￿ B, B ￿ C, C ￿ D} |= A ￿ D ...
29. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Another potential application... 26 J1 = {A ￿ B ￿ C, B ￿ C ￿ D} |= A ￿ D J2 = {A ￿ ∃r.C, ∃r.C ￿ D} |= A ￿ D X1 X1 X2 X2 J1 = {A ￿ B, B ￿ C} |= A ￿ C J2 = {A ￿ B, B ￿ C, C ￿ D} |= A ￿ D ... OWL developer: “That would be tremendously helpful”
30. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Summary and Future Work • We have introduced new equivalence relations: ‣ Subexpression-isomorphism ‣ Lemma-isomorphism • We implemented isomorphism detection • We surveyed a set of BioPortal ontologies • Future work: • Exploit equivalence relations in OWL applications • Explore further obvious steps for lemmatisations 27
32. Samantha Bail, Bijan Parsia, Uli Sattler bails@cs.man.ac.uk Declutter Your Justi cations Template Frequency 29