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