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Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
Conservative Extensions and Modularity in Ontologies
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Conservative Extensions and Modularity in Ontologies

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  • 1. Introduction Conservative Extensions Locality-based Module Summary Conservative Extensions and Modularity in Ontologies Jie Bao1 1 Iowa State University, Ames, IA mailto: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
  • 2. Introduction Conservative Extensions Locality-based Module Summary Outline 1 Introduction Module and Ontology Basic Approaches 2 Conservative Extensions Basic Notions Complexity Result 3 Locality-based Module Locality: Basic Notions Safety, Modularity and MCE Locality
  • 3. Introduction Conservative Extensions Locality-based Module Summary Module and Ontology What is an ontology module and why it is important? Scalability Challenge Myth: OWL is decidable thus it is guaranteed to answer a query, e.g., a web search query Reality: a typical web user will close a page if it does not load in 10 seconds. Partial Reuse Challenge Myth: Ontologies can be reused as we referring web pages using hyperlinks Reality: With an OWL ontology, reuse all of it, or nothing of it.
  • 4. Introduction Conservative Extensions Locality-based Module Summary Module and Ontology What is an ontology module and why it is important? A Module of An Ontology is in manageable size for parse, storage and query easy to understand, easy to maintain has black-box behavior has controlled interaction with other modules thus, supports faster query and partial resuse ···
  • 5. Introduction Conservative Extensions Locality-based Module Summary Basic Approaches Approaches to Support Ontology Modules 1 Modular Ontology Language: use specially designed logic language 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 obtain modularity Conservative Extension (CE)[3, 10] Locality (as an approximation to CE)[9, 7, 5, 6, 4]
  • 6. 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 over L. We say that O is a deductive S-conservative extension of O1 w.r.t. L, if for every axiom α over L with Sig(α) ⊆ S, we have O |= α iff O1 |= α. We say that O is a deductive conservative extension of O1 w.r.t. L if O is a deductive S-conservative extension of O1 w.r.t. L for S = Sig(O1 ). Example O1 := {C D} O2 := {C ∃R.D, C ∀R.¬C} S := {C, D}:
  • 7. 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 over L. We say that O is a model S-conservative extension of O1 , if for every model I of O1 , there exists a model J of O that is obtained from I by modifying the interpretation of the predicates in Sig(O)S while leaving the predicates in S fixed, denoted as J |S = I|S . We say that O is a model conservative extension of O1 if O is a model S-conservative extension of O1 for S = Sig(O1 ). Example O1 := {C D} O2 := {C ∃R.D} S := {C, D}:
  • 8. 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 a deductive 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 X I = X J for every X ∈ S. Using induction on the structure of concepts, for every concept C, Sig(C) ∈ S, we have that either C I = C J or C J = C I ∪ (∆J ∆I ). Thus, if C I = ∅, then C J = ∅; therefore, ∀J |= O s.t. C J = ∅ ⇒ ∀I |= O1 s.t. C I = ∅, which implies S-DCE(O, O1 ). 2 S-DCE(O, O1 ) ⇒ S-MCE(O, O1 ) by example.
  • 9. Introduction Conservative Extensions Locality-based Module Summary Complexity Result Deciding DCE(O1 ∪ O2 , O1 ) in ALC. Recall that concepts in ALC are constructed using the grammar C|¬C|C C|∃R.C Proof strategy: try to construct a witness concept C in the signature Sig(O1 ) that is satisfiable w.r.t. O1 but is unsatisfiable w.r.t. O1 ∪ O2 . If such a C is found, then not DCE(O1 ∪ O2 , O1 ). Theorem 2 [3] Given two ALC TBoxes O1 and O2 , it is 2EXPTIME-complete to decide whether O1 ∪ O2 is a DCE of O1 There are algorithms whose runtime is exponential in |O1 |, but double exponential in |O2 |, by constructing a triple exponential witness concepts (w.r.t. |O1 ∪ O2 |).
  • 10. 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 C|∃R.C| ∃R − .C| ≤ nR.C| ≤ nR − .C Theorem 3 [10] It is 2-EXPTIME-complete to decide DCE in ALCQI. In the case that O1 ∪ O2 is not a DCE of O1 , there exists a witness concept C of length at most 3-exponential in |O1 ∪ O2 |. This bound is optimal. Proof sketch. Using the tree model property of ALCQI, O1 ∪ O2 is not a DCE of O1 iff there is a tree (correspondent to a witness concept) which is embeddable into a model of O1 but not into any model of O1 ∪ O2
  • 11. Introduction Conservative Extensions Locality-based Module Summary Complexity Result Deciding DCE(O1 ∪ O2 , O1 ) in ALCQIO. Recall that ALCQIO allows the grammar C|¬C|C C|∃R.C|∃R − .C| ≤ nR.C| ≤ nR − .C| o, where o stands for nominal (concept of a single instance). Also recall that a problem P is undecidable if a known undecidable problem can be reduced to it. Theorem 4 [10] DCE in ALCQIO is undecidable. Proof sketch. By reducing the undecidable domino tiling problem to a DCE problem D in ALCQIO: constructing O1 , O2 s.t. D is solvable iff O1 ∪ O2 is not a DCE of O1 . A solution to D (a grid of infinite plane) corresponds to a witness concept.
  • 12. 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 in modal logic. Full proof is in the TR http: //www.csc.liv.ac.uk/~frank/publ/ijcai02.ps
  • 13. Introduction Conservative Extensions Locality-based Module Summary Complexity Result Deciding DCE and MCE in EL. Recell that EL allows the grammar |C|C 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 D and C ⇒1 D (We write C ⇒ 1D if, for all sig(O1 )-concepts E, O1 ∪ O2 |= D E implies O1 |= C 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.
  • 14. 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 are decidable. Locality Syntactical Locality (SynL) ⇒ Semantic Locality (SemL) ⇒ MCE ⇒ DCE SynL is decidable in polynominal time SemL is decidable in the same complexity of the logic for concept satisfiability (NExpTime for OWL).
  • 15. 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 the interpretation of names in S. Example O1 := {∃R.C 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 imported ontology will not be changed by O.
  • 16. Introduction Conservative Extensions Locality-based Module Summary Safety, Modularity and MCE Safety and MCE. Safety Given L-ontologies O1 and O2 , we say that O2 is safe for O1 w.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 is a model S-conservative extension of the empty ontology O1 = ∅; that is, for every interpretation I there exists a model J of O such that J |S = I|S . Then O is safe for S w.r.t. L. Proof sketch by showing that for any O s.t. Sig(O) ∩ Sig(O ) ⊆ S, O ∪ O is a DCE of O w.r.t. L
  • 17. Introduction Conservative Extensions Locality-based Module Summary Safety, Modularity and MCE Module Module Let O, O andO1 ⊆ O be L-ontologies. We say that O1 is a module for O in O w.r.t. L, if O ∪ O is a deductive S-conservative extension of O ∪ O1 for S = Sig(O) w.r.t. L. S-Module Let O and O1 ⊆ O be L-ontologies and S a signature over L. We say that O1 is a S-module in O w.r.t. L, if for every L-ontology O with Sig(O) ∩ Sig(O ) S, we have that O1 is a module for O in O w.r.t. L.
  • 18. 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 O1 ⊆ O be ontologies over L. Then: 1 O is safe for O w.r.t. L iff the empty ontology ∅ is a module for O in O w.r.t. L. 2 If O O1 is safe for O ∪ O1 then O1 is a module for O in O w.r.t. L. We also has a similar theorem for S-module.
  • 19. Introduction Conservative Extensions Locality-based Module Summary Locality Complexity Recall that MCE ⇒ Safety ⇒ Modularity ⇒ DCE Theorem 1 Given ontologies O and O over L, the problem of determining whether O is safe for O w.r.t. L is EXPTIME-complete for L = EL, 2-EXPTIME-complete for L = ALC and L = ALCIQ, and undecidable for L = ALCIQO. 2 Given ontologies O, O , andO1 ⊆ O over L, the problem of determining whether O1 is a module for O in O is EXPTIME-complete for L = EL, 2-EXPTIME-complete for L = ALC and L = ALCIQ, and undecidable for L = ALCIQO
  • 20. Introduction Conservative Extensions Locality-based Module Summary Locality Semantic Locality Semantic Locality Let E ⊆ S. A SHIQ axiom α with Sig(α) ⊆ S is semantically local w.r.t. E if the trivial expansion I of every E-interpretation I to S is a model of α. A SHIQ-TBox T is semantically local w.r.t. S if every axiom in T is semantically local w.r.t. S. T is semantically local if it is local w.r.t. an empty S. Example O1 := {∃R.C 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 .
  • 21. 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 any O ∈ 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 in NExpTime. There is a syntactical testing algorithm for semantic locality, which can be done in polynomial time.
  • 22. Introduction Conservative Extensions Locality-based Module Summary Summary DCE is undecidable for ALCQIO, MCE is decidable for EL Decide modularity of an ontology can be reduced to MCE Semantical Locality is an approximation of modularity, which is decidable in NExpTime for SHOIQ
  • 23. References J. Bao, G. Slutzki, and V. Honavar. A semantic importing approach to knowledge reuse from multiple ontologies. In AAAI, pages 1304–1309, 2007. A. Borgida and L. Serafini. Distributed description logics: Assimilating information from peer sources. Journal of Data Semantics, 1:153–184, 2003. S. Ghilardi, C. Lutz, and F. Wolter. Did i damage my ontology? a case for conservative extensions in description logics. In KR, pages 187–197, 2006. B. C. Grau, C. Halaschek-Wiener, and Y. Kazakov. History matters: Incremental ontology reasoning using modules. In ISWC/ASWC, pages 183–196, 2007.
  • 24. 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 Web Conference (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
  • 25. References 3298 of Lecture Notes in Computer Science, pages 620–634. 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. C. Lutz and F. Wolter. Conservative extensions in the lightweight description logic el. In CADE, pages 84–99, 2007.

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