Introduction Conservative Extensions Locality-based Module Summary

Conservative Extensions and Modularity inOntologies

Jie Bao1

1Iowa State University, Ames, IAmailto:baojie@cs.iastate.edu

based on work by Bernardo Cuenca Grau, Ian Horrocks, Yevgeny Kazakov,Ulrike Sattler Carsten Lutz, Dirk Walther, Frank Wolter and Silvio Ghilardi

Semantic Web Seminar, Spring 2008

Introduction Conservative Extensions Locality-based Module Summary

Outline

1 IntroductionModule and OntologyBasic Approaches

2 Conservative ExtensionsBasic NotionsComplexity Result

3 Locality-based ModuleLocality: Basic NotionsSafety, Modularity and MCELocality

Introduction Conservative Extensions Locality-based Module Summary

Module and Ontology

What is an ontology module and why it is important?

Scalability ChallengeMyth: OWL is decidable thus it is guaranteed to answer aquery, e.g., a web search queryReality: a typical web user will close a page if it does notload in 10 seconds.

Partial Reuse ChallengeMyth: Ontologies can be reused as we referring web pagesusing hyperlinksReality: With an OWL ontology, reuse all of it, or nothing ofit.

Introduction Conservative Extensions Locality-based Module Summary

Module and Ontology

What is an ontology module and why it is important?

A Module of An Ontologyis in manageable size for parse, storage and queryeasy to understand, easy to maintainhas black-box behaviorhas controlled interaction with other modulesthus, supports faster query and partial resuse· · ·

Introduction Conservative Extensions Locality-based Module Summary

Basic Approaches

Approaches to Support Ontology Modules

1 Modular Ontology Language: use specially designed logiclanguage with modular (and contextual) semantics

Distributed Description Logics (DDL)[2]E-Connections[8]Package-based Description Logics (P-DL)[1]

2 Design Pattern: still use the standard DL with the (global)first order semantics, but restrict its usage to obtainmodularity

Conservative Extension (CE)[3, 10]Locality (as an approximation to CE)[9, 7, 5, 6, 4]

Introduction Conservative Extensions Locality-based Module Summary

Basic Notions

Conservative Extension

Deductive Conservative Extension (DCE)Let O and O1 ⊆ O be two L-ontologies, and S a signature overL. We say that O is a deductive S-conservative extension of O1w.r.t. L, if for every axiom α over L with Sig(α) ⊆ S, we haveO |= α iff O1 |= α. We say that O is a deductive conservativeextension of O1 w.r.t. L if O is a deductive S-conservativeextension of O1 w.r.t. L for S = Sig(O1).

ExampleO1 := {C v D}O2 := {C v ∃R.D, C v ∀R.¬C}S := {C, D}:

Introduction Conservative Extensions Locality-based Module Summary

Basic Notions

Conservative Extension

Model Conservative Extension (MCE)Let O and O1 ⊆ O be two L-ontologies, and S a signature overL. We say that O is a model S-conservative extension of O1, iffor every model I of O1, there exists a model J of O that isobtained from I by modifying the interpretation of thepredicates in Sig(O)\S while leaving the predicates in S fixed,denoted as J |S = I|S. We say that O is a model conservativeextension of O1 if O is a model S-conservative extension of O1for S = Sig(O1).

ExampleO1 := {C v D}O2 := {C v ∃R.D}S := {C, D}:

Introduction Conservative Extensions Locality-based Module Summary

Basic Notions

Relation between DCE and MCE.

Theorem 1 [10]If O is a model S-conservative extension of O1, then O is adeductive S-conservative extension of O1, but not the converse.

Proof sketch.1 If S-MCE(O, O1),then ∀ I |= O1, ∃J |= O such that

∆I ⊆ ∆J and XI = XJ for every X ∈ S. Using inductionon the structure of concepts, for every concept C,Sig(C) ∈ S, we have that either CI = CJ orCJ = CI ∪ (∆J \∆I). Thus, if CI 6= ∅, then CJ 6= ∅;therefore, ∀J |= O s.t. CJ = ∅ ⇒ ∀I |= O1 s.t. CI = ∅,which implies S-DCE(O, O1).

2 S-DCE(O, O1) 6⇒ S-MCE(O, O1) by example.

Introduction Conservative Extensions Locality-based Module Summary

Complexity Result

Deciding DCE(O1 ∪O2, O1) in ALC.

Recall that concepts in ALC are constructed using thegrammar C|¬C|C t C|∃R.CProof strategy: try to construct a witness concept C in thesignature Sig(O1) that is satisfiable w.r.t. O1 but isunsatisfiable w.r.t. O1 ∪O2. If such a C is found, then notDCE(O1 ∪O2, O1).

Theorem 2 [3]Given two ALC TBoxes O1 and O2, it is2EXPTIME-complete to decide whether O1 ∪O2 is a DCEof O1

There are algorithms whose runtime is exponential in |O1|,but double exponential in |O2|, by constructing a tripleexponential witness concepts (w.r.t. |O1 ∪O2|).

Introduction Conservative Extensions Locality-based Module Summary

Complexity Result

Deciding DCE(O1 ∪O2, O1) in ALCQI.

Recall that ALCQI allows the grammar C|¬C|C t C|∃R.C|∃R−.C| ≤ nR.C| ≤ nR−.C

Theorem 3 [10]It is 2-EXPTIME-complete to decide DCE in ALCQI. In thecase that O1 ∪O2 is not a DCE of O1, there exists a witnessconcept C of length at most 3-exponential in |O1 ∪O2|. Thisbound is optimal.

Proof sketch.Using the tree model property of ALCQI, O1 ∪O2 is not a DCEof O1 iff there is a tree (correspondent to a witness concept)which is embeddable into a model of O1 but not into any modelof O1 ∪O2

Introduction Conservative Extensions Locality-based Module Summary

Complexity Result

Deciding DCE(O1 ∪O2, O1) in ALCQIO.

Recall that ALCQIO allows the grammarC|¬C|C t C|∃R.C|∃R−.C| ≤ nR.C| ≤ nR−.C| o, where ostands for nominal (concept of a single instance).Also recall that a problem P is undecidable if a knownundecidable problem can be reduced to it.

Theorem 4 [10]DCE in ALCQIO is undecidable.

Proof sketch.By reducing the undecidable domino tiling problem to aDCE problem D in ALCQIO: constructing O1, O2 s.t. D issolvable iff O1 ∪O2 is not a DCE of O1.A solution to D (a grid of infinite plane) corresponds to awitness concept.

Introduction Conservative Extensions Locality-based Module Summary

Complexity Result

Deciding MCE(O1 ∪O2, O1) in ALC.

Theorem 5 [10]MCE in ALC is undecidable.

Proof sketch.By a reduction from the semantic consequence problem inmodal logic. Full proof is in the TR http://www.csc.liv.ac.uk/~frank/publ/ijcai02.ps

Introduction Conservative Extensions Locality-based Module Summary

Complexity Result

Deciding DCE and MCE in EL.

Recell that EL allows the grammar >|C|C u C|∃R.C

Theorem 6 [11]1 DCE in EL is decidable (ExpTime-complete).2 MCE in EL is undecidable.

Proof sketch.1 DCE decidability: construct C in Sig(O1) and D in

Sig(O1 ∪O2), such that O1 ∪O2 |= C v D and C 6⇒1 D(We write C ⇒ 1D if, for all sig(O1)-concepts E ,O1 ∪O2 |= D v E implies O1 |= C v E .)

2 DCE hardness: by reduction of the two-player game Peek.3 MCE undecidability: by reduction of halting problem for

deterministic Turing machines on the empty tape.

Introduction Conservative Extensions Locality-based Module Summary

Locality: Basic Notions

Both DCE and MCE are undecidable for OWL (SHOIN (D)), but. . .

There exist approximations of DCE and MCE that aredecidable.

LocalitySyntactical Locality (SynL) ⇒ Semantic Locality (SemL) ⇒MCE ⇒ DCESynL is decidable in polynominal timeSemL is decidable in the same complexity of the logic forconcept satisfiability (NExpTime for OWL).

Introduction Conservative Extensions Locality-based Module Summary

Locality: Basic Notions

Locality

Informally, an axiom (or an ontology) is semantically local w.r.t.a signature S if it imposes no restrictions between theinterpretation of names in S.

ExampleO1 := {∃R.C v D}S1 := {C, D}, S2 := {C, D, R}O1 is local w.r.t. S1, is not local w.r.t. S2.

if O is local w.r.t. S, then S is an importing “interface" of O,such that the “original meaning” of S from any importedontology will not be changed by O.

Introduction Conservative Extensions Locality-based Module Summary

Safety, Modularity and MCE

Safety and MCE.

SafetyGiven L-ontologies O1 and O2, we say that O2 is safe for O1w.r.t. L if O2 ∪O1 is a DCE of O1 w.r.t. L.

Theorem 7: MCE means Safety [7]Let O be an L-ontology and S a signature over L such that O isa model S-conservative extension of the empty ontologyO1 = ∅; that is, for every interpretation I there exists a model Jof O such that J |S = I|S. Then O is safe for S w.r.t. L.

Proof sketchby showing that for any O′ s.t. Sig(O) ∩ Sig(O′) ⊆ S, O ∪O′ isa DCE of O′ w.r.t. L

Introduction Conservative Extensions Locality-based Module Summary

Safety, Modularity and MCE

Module

ModuleLet O, O′andO′

1 ⊆ O′ be L-ontologies. We say that O′1 is a

module for O in O′ w.r.t. L, if O ∪O′ is a deductiveS-conservative extension of O ∪O′

1 for S = Sig(O) w.r.t. L.

S-ModuleLet O′ and O′

1 ⊆ O′ be L-ontologies and S a signature over L.We say that O′

1 is a S-module in O′ w.r.t. L, if for everyL-ontology O with Sig(O) ∩ Sig(O′) v S, we have that O′

1 is amodule for O in O′ w.r.t. L.

Introduction Conservative Extensions Locality-based Module Summary

Safety, Modularity and MCE

Safety ⇒ Modularity

Theorem 8: Safety vs. Modules [7]Let L be an ontology language, and let O, O′, and O′

1 ⊆ O′ beontologies over L. Then:

1 O′ is safe for O w.r.t. L iff the empty ontology ∅ is a modulefor O in O′ w.r.t. L.

2 If O′\O′1 is safe for O ∪O′

1 then O′1 is a module for O in O′

w.r.t. L.

We also has a similar theorem for S-module.

Introduction Conservative Extensions Locality-based Module Summary

Locality

Complexity

Recall that MCE ⇒ Safety ⇒ Modularity ⇒ DCE

Theorem1 Given ontologies O and O′ over L, the problem of

determining whether O is safe for O′ w.r.t. L isEXPTIME-complete for L = EL, 2-EXPTIME-complete forL = ALC and L = ALCIQ, and undecidable forL = ALCIQO.

2 Given ontologies O, O′, andO′1 ⊆ O′ over L, the problem of

determining whether O′1 is a module for O in O′ is

EXPTIME-complete for L = EL, 2-EXPTIME-complete forL = ALC and L = ALCIQ, and undecidable forL = ALCIQO

Introduction Conservative Extensions Locality-based Module Summary

Locality

Semantic Locality

Semantic LocalityLet E ⊆ S. A SHIQ axiom α with Sig(α) ⊆ S is semanticallylocal w.r.t. E if the trivial expansion I of every E-interpretationI′ to S is a model of α. A SHIQ-TBox T is semantically localw.r.t. S if every axiom in T is semantically local w.r.t. S. T issemantically local if it is local w.r.t. an empty S.

ExampleO1 := {∃R.C v D}S1 := {C, D}, S2 := {C, D, R}O1 is local w.r.t. S1 by setting ∃R.C = ⊥,O1 is not local w.r.t. S2.

Introduction Conservative Extensions Locality-based Module Summary

Locality

Semantic Locality

Theorem 9 [6]Let O be a of set of semantically local ontologies, then for anyO ∈ O, O is a module of the union of any set of O.

Theorem 10 [6]Deciding semantical locality of an SHOIQ TBox is decidable inNExpTime.

There is a syntactical testing algorithm for semantic locality,which can be done in polynomial time.

Introduction Conservative Extensions Locality-based Module Summary

Summary

DCE is undecidable for ALCQIO, MCE is decidable for ELDecide modularity of an ontology can be reduced to MCESemantical Locality is an approximation of modularity,which is decidable in NExpTime for SHOIQ

References

J. Bao, G. Slutzki, and V. Honavar.A semantic importing approach to knowledge reuse frommultiple ontologies.In AAAI, pages 1304–1309, 2007.

A. Borgida and L. Serafini.Distributed description logics: Assimilating information frompeer sources.Journal of Data Semantics, 1:153–184, 2003.

S. Ghilardi, C. Lutz, and F. Wolter.Did i damage my ontology? a case for conservativeextensions in description logics.In KR, pages 187–197, 2006.

B. C. Grau, C. Halaschek-Wiener, and Y. Kazakov.History matters: Incremental ontology reasoning usingmodules.In ISWC/ASWC, pages 183–196, 2007.

References

B. C. Grau, I. Horrocks, Y. Kazakov, and U. Sattler.Just the right amount: Extracting modules from ontologies.In Proc. of the Sixteenth International World Wide WebConference (WWW 2007), 2007.

B. C. Grau, I. Horrocks, Y. Kazakov, and U. Sattler.A logical framework for modularity of ontologies.In IJCAI, pages 298–303, 2007.

B. C. Grau, I. Horrocks, Y. Kazakov, and U. Sattler.Modular reuse of ontologies: Theory and practice.JConservativeournal of Artificial Intelligence Research(JAIR), 31:to appear, 2008.

B. C. Grau, B. Parsia, and E. Sirin.Working with multiple ontologies on the semantic web.In S. A. McIlraith, D. Plexousakis, and F. van Harmelen,editors, International Semantic Web Conference, volume

References

3298 of Lecture Notes in Computer Science, pages620–634. Springer, 2004.

B. C. Grau, B. Parsia, E. Sirin, and A. Kalyanpur.Modularity and web ontologies.In KR, pages 198–209, 2006.

C. Lutz, D. Walther, and F. Wolter.Conservative extensions in expressive description logics.In IJCAI, pages 453–458, 2007.

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