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A Distributed Tableau Algorithm for Package-based Description Logics
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A Distributed Tableau Algorithm for Package-based Description Logics

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  • 1. A Distributed Tableau Algorithm for Package-based Description Logics Jie Bao 1 , Doina Caragea 2 and Vasant G Honavar 1 1 Artificial Intelligence Research Laboratory, Department of Computer Science, Iowa State University, Ames, IA 50011-1040, USA. {baojie, honavar}@cs.iastate.edu 2 Department of Computing and Information Sciences Kansas State University, Manhattan, KS 66506, USA dcaragea@ksu.edu 2nd International Workshop on Context Representation and Reasoning (CRR 2006) @ ECAI 2006, Aug 29, 2006, Riva del Garda, Italy
  • 2. Dr. D. Caragea Dr. V. Honavar Jie Bao
  • 3. Outline
    • Requirements for reasoning with modular ontologies
    • Package-based Description Logics (P-DL): features and semantics
    • A tableau algorithm for (P-DL) ALCP C
    • Discussions
  • 4. Modularity
  • 5. The Need for Modular Ontologies(MO)
    • Collaborative Ontology Building
    • Distributed Data Management
    • Large Ontology Management
    • Partial Ontology Reuse
  • 6. Reasoning with MO
    • If GraduateOK(Jie) is consistent with the ontology?
    • (If Jie can graduate?)
    Computer Science Dept Ontology Registration Office Ontology Semantic Relations Bob = 3304 G r a d u a t e O K v : 9 f a i l s : C o r e C o u r s e G r a d u a t e O K v P r e l i m O K P r e l i m O K ( J i e ) C s C o r e C o u r s e v C o r e C o u r s e C s C o r e C o u r s e ( c s 5 1 1 ) f a i l s ( 3 3 0 4 ; c s 5 1 1 ) S S N ( 3 3 0 4 ; 1 2 3 4 5 6 7 8 9 )
  • 7. Reasoning with MO (2)
    • Major Consideration: should not require the integration of ontology modules.
      • High communication cost
      • High local memory cost
      • May violate module autonomy, e.g., privacy
    • Question: can we do reasoning for modular ontologies without
      • (syntactic level) an integrated ontology ?
      • (semantic level) a (materialized) global tableau ?
  • 8. Outline
    • Requirements for reasoning with modular ontologies
    • Package-based Description Logics (P-DL): features and semantics
    • A tableau algorithm for (P-DL) ALCP C
    • Discussions
  • 9. Package
    • A package is an ontology module that captures a sub-domain;
    • Each term has a home package
    • A package can import terms from other packages
    • Package extension is denoted as P
      • P C :Package extension with only concept name importing
      • E.g., ALCP C = ALC + P C
    General Pet Wild Livestock Animal ontology PetDog Pet Dog General
  • 10. Package: Example O 1 (General Animal) O 2 (Pet) It uses ALCP, but not ALCP C
  • 11. Semantics of Importing
    • Domain relations are compositionally consistent : r 13 =r 12 O r 23
      • Therefore domain relations are transitively reusable.
    • Domain relation : individual correspondence between local domains
    • Importing establishes one-to-one domain relations
      • “ Copies” of individuals are shared
    x x’ Δ I 1 Δ I 2 C I 1 C I 2 r 12 Δ I 3 r 13 r 23 x’’ C I 3
  • 12. Partially Overlapping Models x x’ Δ I 1 Δ I 2 C I 1 C I 2 Δ I 3 r 13 r 23 x’’ C I 3 x C I Global interpretation obtained from local Interpretations by merging shared individuals r 12
  • 13. Model Projection x C I x C I 1 x’ C I 2 x’’ C I 3 Global model local models
  • 14. Outline
    • Requirements for reasoning with modular ontologies
    • Package-based Description Logics (P-DL): features and semantics
    • A tableau algorithm for (P-DL) ALCP C
    • Discussions
  • 15. Tableau Algorithm
    • A tableau is a representation of a model
    • Basic idea:
      • start with some initial facts for an ontology
      • use tableau expansion rules to infer new facts,
        • until no rule can be applied, or inconsistencies are found among those facts.
      • If a clash-free fact set is found, a model of the ontology is constructed
  • 16. Tableau Algorithm: Example Dog(goofy) Animal(goofy) ( eats.DogFood)(goofy) eats(goofy,foo) DogFood(foo) goofy L(goofy)={Dog, Animal, eats.DogFood } foo L(foo)={DogFood } eats ABox Representation Completion Tree Representation Note: both representations are simplified for demostration purpose
  • 17. Federated Reasoning Chef: Hello there, children! Where does Kyle move to? Chef: We are in South Park, Colorado; San Francisco is in California; Colorado is far from California. Stan: So they are far from us. Too Bad. Stan: Hey, Chef . Is Kyle’s new home far from us? Cartman: San Francisco, I guess.
  • 18. Federated Reasoning for P-DL
    • Basic strategy
    • Use multiple local reasoners, each for a single package
    • Each local reasoner creates and maintains a local tableau based on local knowledge
    • A local reasoner may query other reasoners if its local knowledge is incomplete
    • Global relation among tableaux is created by messages
    (1) (2) (3) (4)
  • 19. Tableau Projection x 1 {A 1 } {A 2 } {A 3 } x 2 x 4 x 1 {B 1 } {B 3 } {B 2 } x 3 x 4 The (conceptual) global tableau Local Reasoner for package A Local Reasoner for package B Shared individuals mean partially overlapped local models x 1 {A 1 ,B 1 } {A 2 } {A 3 ,B 3 } {B 2 } x 2 x 3 x 4
  • 20. Model Projection x C I x C I 1 x’ C I 2 x’’ C I 3 Global model local models
  • 21. Tableau Expansion Tableau Expansion for ALCP C with acyclic importing
  • 22. Communication among Local Tableaux
    • Membership m ( y,C ):
    • Reporting r ( y,C ):
    • Clash bottom ( y ):
    • Model top ( y ):
    y y {C?} y y {C} C(y) y y {…} y y {…} X Query if y is an instance of C Notify that y is an instance of C Notify that y has local inconsistency Notify that no more rule can be applied locally on y T 1 T 2
  • 23. ALCP C Expansion Example
    • Consistency of the ontology is witnessed by P 1
    • y is the shared individual
    • Subset blocking is still applicable
      • E.g. L 1 (y)  L 1 (x)
    x L 1 (x)={A,  R.B} y y z L 2 (y)={B,  P.C} L 2 (z)={C,  P.C} R P T 1 T 2 L 1 (y)={A,  R.B} w L 2 (w)={C,  P.C} P P 1 P 2 > v 1 : A ; > v 9 ( 1 : R ) : ( 2 : B ) > v ( 2 : P ) : ( 2 : C )
  • 24. ALCP C Expansion Example (2)
    • P 1 : 1:A 1:B
    • P 2 : 1:B 2:C
    • P 3 : 2:C 3:D
    • Query: if A D (from the point of view of P 3 )
    • (it is not answerable by either DDL nor E-Connection in their current forms)
    • Reasoning: if A D is not true, then there will be clash. Hence, it must be true
    L 3 (x)={ A⊓  D ,  C⊔D A,  C,  D} Transitive Subsumption Propagation T 3 x r(x,  C ) x x r(x,A) T 2 T 1 L 2 (x)={  B⊔C  C ,  B} L 1 (x)={  A⊔B A ,  B , B } r(x,  B )  (x)  (x)  (x)
  • 25. ALCP C Expansion Example (3) L 2 (x)={ P,  P⊔B,  P⊔  F,B,  F} x x L 1 (x)={ B,  F ,  B⊔F, F } T 2 T 1 r(x,B) r(x,  F)  (x) L 1 (x)={A,  A⊔C,C} y z L 2 (y)={A,  A⊔  R.B,  B⊔(A⊓  C),  R.B,  B} P T 1 T 2 L 2 (z)={B,  A⊔  R.B,  B⊔(A⊓  C),  R.B, A⊓  C, A,  C} y L 1 (z)={A,  C ,  A⊔C, C } z r(z,A) r(z,  C)  (x) r(z,A) (x)  Detect Inter-module Unsatisfiability 2:P is unsatisfiable Reasoning from Local Point of View 1:A is unsatisfiable witnessed by P 2 is satisfiable witnessed by P 1 P 1 : f 1 : B v 1 : F g , P 2 : f 1 : P v 1 : B ; 2 : P v : 1 : F g P 1 : f 1 : A v 1 : C g P 2 : f 1 : A v 9 2 : R : ( 2 : B ) ; 2 : B v 1 : A u ( : 1 : C ) g
  • 26. Soundness β α α α α β α or or α A A A B A’ A’’ A’ A B’ infer (a) Augmenting (c) Reporting (b) Searching A is consistent iff A’ is consistent A is consistent iff A’ is consistent or A’’ is consistent (A,B) is consistent iff (A,B’) is consistent send
  • 27. Completeness P-DL model can be constructed from a distributed Tableau
  • 28. Termination
    • Acyclic importing ensures no message loop
    • Blocking
      • Subset blocking
      • Reporting blocking: A node is temporarily blocked after sending a reporting message
    x y y z T 1 T 2 w T 3 z v P 1 P 3 P 2 import import Tableaux Ontology
  • 29. Outline
    • Requirements for reasoning with modular ontologies
    • Package-based Description Logics (P-DL): features and semantics
    • A tableau algorithm for (P-DL) ALCP C
    • Discussions
  • 30. Other Tableau Projections Distributed Description Logics (DDL) [ Serafini and Tamilin 2004, 2005] x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 3 x 5 x 5 f B 1 u : B 2 ; ¢ ¢ ¢ g f B 1 u : B 2 ; ¢ ¢ ¢ g
  • 31. Other Tableau Projections (2) x 1 x 2 x 3 x 4 x 1 x 2 x 4 x 5 x 3 x 6 E-Connections [ Grau 2005] x 5 x 6 E E {A 1 } {A 1 } {A 2 } {A 3 } {B 1 } {B 2 } {B 3 } {A 2 } {A 3 } {B 1 } {B 2 } {B 3 }
  • 32. Ongoing Work
    • Working with cyclic importing
    x 1 {A 1 ,B 1 } {A 2 } {A 3 ,B 3 } {B 2 } x 2 x 3 x 4 x 1 {A 1 } {A 2 } {A 3 } x 2 x 4 x 1 {B 1 } {B 3 } {B 2 } x 3 x 4 {B 4 } {B 4 } B 1 A 3 P A P B
  • 33. Ongoing Work (2)
    • Asynchronous reasoning:
      • local reasoners don’t need to wait after a reporting message
      • Thus they can concurrently search on different branches for a possible global tableau.
    • Working with OWL
      • Support SHOIQ(D)
    • Implementation based on Pellet
  • 34. References
    • P-DL:
    • J. Bao, D. Caragea, and V. Honavar. Towards collaborative environments for ontology construction and sharing. In International Symposium on Collaborative Technologies and Systems (CTS 2006) . 2006.
    • J. Bao, D. Caragea, and V. Honavar. Modular ontologies - a formal investigation of semantics and expressivity. 2006. In the Asian Semantic Web Conference (ASWC), LNCS 4185, pp. 616–631, 2006.
    • J. Bao, D. Caragea, and V. Honavar. On the Semantics of Linking and Importing in Modular Ontologies. accepted by the International Semantic Web Conference (ISWC) 2006. (In Press)
    • J. Bao, D. Caragea, and V. Honavar. A tableau-based federated reasoning algorithm for modular ontologies. Submitted to 2006 IEEE/WIC/ACM International Conference on Web Intelligence, 2006 (under reviewing)
    • Related work:
    • L. Serafini and A. Tamilin. Local tableaux for reasoning in distributed description logics. In Description Logics Workshop 2004, CEUR-WS Vol 104 , 2004.
    • L. Serafini and A. Tamilin. Drago: Distributed reasoning architecture for the semantic web. In ESWC , pages 361-376, 2005.
    • B. C. Grau. Combination and Integration of Ontologies on the Semantic Web . PhD thesis, Dpto. de Informatica, Universitat de Valencia, Spain, 2005.
  • 35.
    • Thanks !
  • 36. Reasoning by Model Construction Model x Man I Human I
    • If such a model is not possible in any situation, Man <= Human is true
    Reasoning
    • Suppose it is not true, then at least one individual x in a world (model) is Man but not Human
    To query Man Human
    • If such a model can be constructed, then Man <= Human is not true