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Ontology Contraction:beyond Propositional Paradise
Bernardo Cuenca Grau,Computer Science Department, University of Oxford
Evgeny Kharlamov, Dmitriy ZheleznyakovKRDB research centre, Free University of Bozen-Bolzano
AMW 2012, Ouro Preto, Brazil
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o Schema provideo standard vocabularies for datao classes (concepts)o properties (roles)
o a way to structure datao means for machines
to be able to understand data
o Data is a collections of factso Instantiations of classes o Instantiations of properties
Ontologies: schema + data
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o Ontology Based Data Accesso provide unified query interface to heterogeneous data sourceso e.g., Quest, OWLIM
o Web Knowledge Baseso Wiki based Knowledge baseso e.g., Jago, DBpedia
o Clinical sciences ontologieso provide standard vocabularies to communitieso e.g., SNOMED CT, NCIt
Usage of Ontologies
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o Ontology Based Data Accesso the schema may change
o Web Knowledge Baseso Wiki changes all the time,
and so does Wiki-based knowledge bases
o Clinical sciences ontologieso from 2002 to 2008 SNOMED went from 278k to 311k concepts
Evolution of Ontologies (1)
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o At the high level ontologies are changed by o addition of informationo usually referred as revision or update
o deletion of informationo usually referred as contraction
o Evolution may affect botho schema levelo data level
Evolution of Ontologies
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o Evolution of knowledge is a classical problem in KRo intensively studied for propositional logico there are different semantics for evolutiono many complexity resultso very few results beyond propositional case
o Two main types of approaches to evolutiono Model-Based Approach (MBA)o Formula-Based Approach (FBA)
o Principal of minimal changeo a knowledge base should change as little as possible
Can Previous Works Help?
Adam in the Garden of Eden
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MBA: Contraction Process
newdata
operatorontologyin L
processing
evolvedontology
in L
o transform modelso minimal change
Contraction operator: takes models of the original ontology, transform them so they do not entail axioms to be deleted
evolvedmodelsmodels
represents
info to delete
?
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MBA: Propositional Case (1)
original models
models of new info
dist
o choose models of M2 less distanced from M1
o distance is based on symmetric difference between modelso 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 operatoro Satoh’s operatoro …
M1 M2 evolvedmodels
M3
[EG’92]
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MBA: Propositional Case (2)
original models
dist
o Is M3 axiomatizable in the propositional logic?o Yes!o The number of models
is just exponential in the size of the original ontology
M1
EXPnumber
EXPnumber
Adam in the Garden of Eden
evolvedmodels
M3
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FBA: Contraction Process
newdata
operatorontologyin L
processing
evolvedontology
in L
o add/delete axiomso minimal change
Contraction operator: takes a subset of the ontology deductive closure which does not entail axioms to be deleted
represents
info to delete
evolvedclosure
in L
closurein L
expand
?
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FBA: Propositional Case
ontology
the closure
evolved closure:a subset not
entailing new info
o What subset to choose? o WIDTIO operatoro Cross-product operatoro …
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]
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1. Languages for ontologies
2. Ontology evolution under MBA
3. Ontology evolution under FBA
4. Evolution under semantic constraints
5. Conclusion & directions
Outline
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o Languages that are natural for real-life ontologieso flexible to capture complex interactiono logic-based o propositional logic is not enougho fragments of FOL are neededo the situation becomes much more difficult
Languages for Ontologies
The Fall of Adam and Eve The Expulsion from Paradise
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o Languages that are natural for real-life ontologieso flexible to capture complex interactiono logic-based o propositional logic is not enougho fragments of FOL are neededo the situation becomes much more difficult
o Ontology Web Language: OWL 2 – W3C standardo 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
Languages for Ontologies
these are not propositional
tractable reasoning
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Description Logics DL-Lite & EL
Concepts
DL-Lite EL
Syntax Example
, The Universe, The Nothing
A, R Koala, hasFather
R– hasFather – = isFather
C1 ⊓ C2 Koala ⊓ Gourmet
∃R. ∃R.C ∃likes.FrenchFood
Axioms
DL-Lite EL
Syntax Example
C1 ⊑ C2 Koala ⊑ Mammal
C1 ⊑ ¬C2 – Koala ⊑ ¬Human
(funct R) – (funct hasFather)
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1. Languages for ontologies
2. Ontology evolution under MBA
3. Ontology evolution under FBA
4. Evolution under semantic constraints
5. Conclusion & directions
Outline
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MBA: Contraction Process
newdata
operatorontologyin L
processing
evolvedontology
in L
o transform modelso minimal change
evolvedmodelsmodels
represents
axioms to delete
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MBA: FOL Case
original models
models of new info
dist
o How to measure distance between models of a FOL theory?
o There are two ways to generalize the propositional approacho propositional case: I = {a, b} J = {b, c} diffp(I,J) = {a, c}
o FOL case 1: I = {A(a), B(b)} J = {B(b), A(c)} diff1(I,J) = {A(a), A(c)}
o FOL case 2: I = {A(a), B(b)} J = {B(b), A(c)} diff2(I,J) = {A}
o Each of the propositional operators can be generalized in two ways
M1 M2 evolvedmodels
M3
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MBA: DL-Lite & EL Cases
original models
dist
o Theorem: In general, M3 is not axiomatizable in DL-Lite, nor in ELo the number of models is
continuumo evolved models are “too
many” & “too irregular”to capture them
M1
infinitenumber
infinitenumber
evolvedmodels
M3
The Flood
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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]
a bRA B
a b
A B
a bR
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
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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]
a bRA B
a b
A B
a bR
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
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1. Languages for ontologies
2. Ontology evolution under MBA
3. Ontology evolution under FBA
4. Evolution under semantic constraints
5. Conclusion & directions
Outline
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evolvedclosure
in L
closurein L
FBA: Evolution Process
newinfo
operatorontologyin L
processing
evolvedontology
in L
expandrepresent
axioms to delete
o add/deleteo minimal change
Contraction operator: takes a maximal subset (w.r.t. set inclusion)of the ontology deductive closure which does not entail axioms to be deleted
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FBA: DL-Lite Case
ontology
DL-Lite closure
evolved closure:a subset not
entailing new info
o What subset to choose? o WIDTIO operatoro Cross-product operatoro …
o Theorem: DL-Lite is closed under FBAo closure is finite
[EG’92]
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FBA: EL Case
ontology
EL closure
evolved closure:a subset not
entailing new info
o What subset to choose? o WIDTIO operatoro Cross-product operatoro …
Theorem: : In general, EL is not closed under FBAo too many (infinite number of) formulas to preserveo not always possible
[EG’92]
Tower of Bable
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1. Languages for ontologies
2. Ontology evolution under MBA
3. Ontology evolution under FBA
4. Evolution under semantic constraints
5. Conclusion & directions
Outline
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o Our view of principle of minimal changeo maximize preservation of ontology structureo maximize preservation of ontology entailments
o Preservation language (LP) tells us which class of entailments should be maximized
Our Proposal in a Nutshell [GJRKZ’12]
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Evolution under Semantic Constraints
SA
FBA
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evolvedclosure
in L
closurein L
Contraction Process [GJRKZ’12]
operatorontologyin L
processing
evolvedontology
in L
expandrepresent
evolvedclosurein LP
closurein LP
sub-ontology
in L
o add/deleteo minimal change
newinfo
axioms to delete
Choosing relevant LP allows too achieve expressibility (for any language) o reduce computational hardness
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1. Languages for ontologies
2. Ontology evolution under MBA
3. Ontology evolution under FBA
4. Evolution under semantic constraints
5. Conclusion & directions
Outline
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Expressibility & exponentiality
Inexpressibility even in simple settingsFOL case: classical
wayFOL case: evolution under SC
Way to go!
Conclusion & DirectionsPropositional case:
FOL case: classical way
… sometimes FOL inexpressibility
Handling inexpressibility by tuning LP.Practical and logically sound.
Evolution under SC:
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o [HKR’08] Hartung, M.; Kirsten, T.; and Rahm, E. 2008. 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. 2000 Nov;17(6):29.
o [SM-1] http://www.ihtsdo.org/snomed-ct/snomed-ct0/adoption-of-snomed-ct/
o [SM-2] http://www.ihtsdo.org/fileadmin/user_upload/doc/download/doc_UserGuide_Current-en-US_INT_20120131.pdf
o [FMA] http://sig.biostr.washington.edu/projects/fm/AboutFM.html
o [NCI] https://wiki.nci.nih.gov/display/EVS/NCI+Thesaurus+versus+NCI+Metathesaurus
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) 170–184
References
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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. 2010. Evolution of DL-Lite Knowledge Bases. In Proc. of ISWC, 112-128.
o [CJKZ’12] Cuenca Grau B., Jiménez-Ruiz E., Kharlamov E., Zheleznyakov D. 2012. Ontology evolution under semantic constraints. In Proc. of KR.
o [MSH’09] Motik B., Shearer R., Horrocks I. 2009. Hyper-tableau reasoning for description logics. Journal of AI Research 36: 165-228.
o [KPHS’07] Kalyanpur A., Parsia B., Horridge M., Sirin E. 2007. Finding all justifications of OWL DL entailments. In Proc. of ISWC, 267-280.
References