Ontology Contraction: beyond Propositional Paradise Bernardo Cuenca Grau, Computer Science Department, University of Oxford Evgeny Kharlamov, Dmitriy Zheleznyakov KRDB research centre, Free University of Bozen-Bolzano AMW 2012, Ouro Preto, Brazil

/30 o Schema provide o standard vocabularies for data o classes (concepts) o properties (roles) o a way to structure data o means for machines to be able to understand data o Data is a collections of facts o Instantiations of classes o Instantiations of properties 2 Ontologies: schema + data

/30 o Ontology Based Data Access o provide unified query interface to heterogeneous data sources o e.g., Quest, OWLIM o Web Knowledge Bases o Wiki based Knowledge bases o e.g., Jago, DBpedia o Clinical sciences ontologies o provide standard vocabularies to communities o e.g., SNOMED CT, NCIt 3 Usage of Ontologies

/30 o Ontology Based Data Access o the schema may change o Web Knowledge Bases o Wiki changes all the time, and so does Wiki-based knowledge bases o Clinical sciences ontologies o from 2002 to 2008 SNOMED went from 278k to 311k concepts 4 Evolution of Ontologies (1)

/30 o At the high level ontologies are changed by o addition of information o usually referred as revision or update o deletion of information o usually referred as contraction o Evolution may affect both o schema level o data level 5 Evolution of Ontologies

/30 o Evolution of knowledge is a classical problem in KR o intensively studied for propositional logic o there are different semantics for evolution o many complexity results o very few results beyond propositional case o Two main types of approaches to evolution o Model-Based Approach (MBA) o Formula-Based Approach (FBA) o Principal of minimal change o a knowledge base should change as little as possible 6 Can Previous Works Help? Adam in the Garden of Eden

/30 7 MBA: Contraction Process new data processing Contraction operator : takes models of the original ontology, transform them so they do not entail axioms to be deleted

/30 8 MBA: Propositional Case (1) o choose models of M 2 less distanced from M 1 o distance is based on symmetric difference between models o I = {a, b} J = {b, c}  diff( I, J ) = {a, c} o lots of operators to compute the distance between sets of models : o Winslett’s operator o Satoh’s operator o…o… M1M1 M2M2 M3M3 [EG’92]

/30 9 MBA: Propositional Case (2) o Is M 3 axiomatizable in the propositional logic? o Yes! o The number of models is just exponential in the size of the original ontology M1M1 EXP number Adam in the Garden of Eden M3M3

/30 10 FBA: Contraction Process new data processing Contraction operator: takes a subset of the ontology deductive closure which does not entail axioms to be deleted

/30 11 FBA: Propositional Case o What subset to choose? o WIDTIO operator o Cross-product operator o…o… o Is evolved closure axiomatizable in the propositional logic? o Yes! o The size of closure is exponential in the size of the original ontology Adam and Eve in the Garden of Eden [EG’92]

/30 1.Languages for ontologies 2.Ontology evolution under MBA 3.Ontology evolution under FBA 4.Evolution under semantic constraints 5.Conclusion & directions 12 Outline

/30 o Languages that are natural for real-life ontologies o flexible to capture complex interaction o logic-based o propositional logic is not enough o fragments of FOL are needed o the situation becomes much more difficult 13 Languages for Ontologies The Fall of Adam and EveThe Expulsion from Paradise

/30 o Languages that are natural for real-life ontologies o flexible to capture complex interaction o logic-based o propositional logic is not enough o fragments of FOL are needed o the situation becomes much more difficult o Ontology Web Language: OWL 2 – W3C standard o OWL 2 (based on SROIQ ) o OWL 2 QL (based on DL-Lite ) o OWL 2 EL (based on EL, EL ++ ) o e.g. SNOMED 14 Languages for Ontologies

/30 15 Description Logics DL-Lite & EL Concepts DL-LiteEL SyntaxExample,The Universe, The Nothing A, R Koala, hasFather R–R– hasFather – = isFather C 1 ⊓ C 2 Koala ⊓ Gourmet ∃ R. ∃ R.C ∃ likes.FrenchFood Axioms DL-LiteEL SyntaxExample C 1 ⊑ C 2 Koala ⊑ Mammal C 1 ⊑ ¬C 2 – Koala ⊑ ¬Human (funct R)– (funct hasFather)

/30 1.Languages for ontologies 2.Ontology evolution under MBA 3.Ontology evolution under FBA 4.Evolution under semantic constraints 5.Conclusion & directions 16 Outline

/30 17 MBA: Contraction Process new data processing

/30 18 MBA: FOL Case o How to measure distance between models of a FOL theory? o There are two ways to generalize the propositional approach o propositional case: I = {a, b} J = {b, c}  diff p ( I, J ) = {a, c} o FOL case 1: I = {A(a), B(b)} J = {B(b), A(c)}  diff 1 ( I, J ) = {A(a), A(c)} o FOL case 2: I = {A(a), B(b)} J = {B(b), A(c)}  diff 2 ( I, J ) = {A} o Each of the propositional operators can be generalized in two ways M1M1 M2M2 M3M3

/30 19 MBA: DL-Lite & EL Cases o Theorem : In general, M 3 is not axiomatizable in DL-Lite, nor in EL o the number of models is continuum o evolved models are “too many” & “too irregular” to capture them M1M1 infinite number M3M3 The Flood

/30 20 MBA: Can We Do Anything? o Can we overcome the inexpressibility by allowing fewer models in the result? o E.g., take those models where there are less changes in roles I = {A(a), B(b), R(a,b)} J = {A(a), B(b)} K = {R(a,b)} [QD’09] ab R AB ab AB ab R o J or K is closer to I ? It is K, since it does not differ from I on roles o Conjecture : In general, for OWL 2 EL + functionality + inverses, the result of evolution is not FOL expressible

/30 21 MBA: Can We Do Anything? o Can we overcome the inexpressibility by allowing fewer models in the result? o E.g., take those models where there are less changes in roles I = {A(a), B(b), R(a,b)} J = {A(a), B(b)} K = {R(a,b)} [QD’09] ab R AB ab AB ab R o J or K is closer to I ? It is K, since it does not differ from I on roles o Conjecture : In general, for OWL 2 EL + functionality + inverses, the result of evolution is not FOL expressible need to distinguish even cycles of an arbitrary size impossible in FOL (locality property of FOL) Gehenna

/30 1.Languages for ontologies 2.Ontology evolution under MBA 3.Ontology evolution under FBA 4.Evolution under semantic constraints 5.Conclusion & directions 22 Outline

/30 23 FBA: Evolution Process new info processing Contraction operator: takes a maximal subset (w.r.t. set inclusion)of the ontology deductive closure which does not entail axioms to be deleted

/30 24 FBA: DL-Lite Case o What subset to choose? o WIDTIO operator o Cross-product operator o…o… o Theorem : DL-Lite is closed under FBA o closure is finite [EG’92]

/30 25 FBA: EL Case o What subset to choose? o WIDTIO operator o Cross-product operator o…o… Theorem : : In general, EL is not closed under FBA o too many (infinite number of) formulas to preserve o not always possible [EG’92] Tower of Bable

/30 1.Languages for ontologies 2.Ontology evolution under MBA 3.Ontology evolution under FBA 4.Evolution under semantic constraints 5.Conclusion & directions 26 Outline

/30 o Our view of principle of minimal change o maximize preservation of ontology structure o maximize preservation of ontology entailments o Preservation language (LP) tells us which class of entailments should be maximized 27 Our Proposal in a Nutshell [GJRKZ’12]

/30 28 Evolution under Semantic Constraints SA FBA

/30 29 Contraction Process [GJRKZ’12] processing new info Choosing relevant LP allows to o achieve expressibility (for any language) o reduce computational hardness

/30 1.Languages for ontologies 2.Ontology evolution under MBA 3.Ontology evolution under FBA 4.Evolution under semantic constraints 5.Conclusion & directions 30 Outline

/30 Expressibility & exponentiality Inexpressibility even in simple settings FOL case: classical wayFOL case: evolution under SC 31 Conclusion & Directions Propositional case:FOL case: classical way … sometimes FOL inexpressibility Handling inexpressibility by tuning LP. Practical and logically sound. Evolution under SC:

/30 o [HKR’08] Hartung, M.; Kirsten, T.; and Rahm, E Analyzing the evolution of life science ontologies and mappings. In Proc. of DILS, 11–27. o [SM] Spackman K. SNOMED RT and SNOMEDCT. Promise of an international clinical terminology. MD Comput Nov;17(6):29. o [SM-1] o [SM-2] rent-en-US_INT_ pdf o [FMA] o [NCI] o [HS’05]Haase, P., Stojanovic, L.: Consistent evolution of OWL ontologies. In: ESWC. (2005) o [KPSCG’06] Kalyanpur, A., Parsia, B., Sirin, E., Grau, B.C.: Repairing unsatisfiable concepts in OWL ontologies. In: ESWC. (2006) 170– References

/30 o [JRCGHB’11] Jimenez-Ruiz, E., Cuenca Grau, B., Horrocks, I., Berlanga, R.: Supporting concurrent ontology development: Framework, algorithms and tool. DKE. 70:1 (2011) o [CKNZ’10] Calvanese D., Kharlamov E., Nutt W., Zheleznyakov D Evolution of DL-Lite Knowledge Bases. In Proc. of ISWC, o [CJKZ’12] Cuenca Grau B., Jiménez-Ruiz E., Kharlamov E., Zheleznyakov D Ontology evolution under semantic constraints. In Proc. of KR. o [MSH’09] Motik B., Shearer R., Horrocks I Hyper-tableau reasoning for description logics. Journal of AI Research 36: o [KPHS’07] Kalyanpur A., Parsia B., Horridge M., Sirin E Finding all justifications of OWL DL entailments. In Proc. of ISWC, References