A Distributed Tableau Algorithm for Package-based Description Logics

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  • A Distributed Tableau Algorithm for Package-based Description Logics

    1. 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. 2. Dr. D. Caragea Dr. V. Honavar Jie Bao
    3. 3. Outline <ul><li>Requirements for reasoning with modular ontologies </li></ul><ul><li>Package-based Description Logics (P-DL): features and semantics </li></ul><ul><li>A tableau algorithm for (P-DL) ALCP C </li></ul><ul><li>Discussions </li></ul>
    4. 4. Modularity
    5. 5. The Need for Modular Ontologies(MO) <ul><li>Collaborative Ontology Building </li></ul><ul><li>Distributed Data Management </li></ul><ul><li>Large Ontology Management </li></ul><ul><li>Partial Ontology Reuse </li></ul>
    6. 6. Reasoning with MO <ul><li>If GraduateOK(Jie) is consistent with the ontology? </li></ul><ul><li>(If Jie can graduate?) </li></ul>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. 7. Reasoning with MO (2) <ul><li>Major Consideration: should not require the integration of ontology modules. </li></ul><ul><ul><li>High communication cost </li></ul></ul><ul><ul><li>High local memory cost </li></ul></ul><ul><ul><li>May violate module autonomy, e.g., privacy </li></ul></ul><ul><li>Question: can we do reasoning for modular ontologies without </li></ul><ul><ul><li>(syntactic level) an integrated ontology ? </li></ul></ul><ul><ul><li>(semantic level) a (materialized) global tableau ? </li></ul></ul>
    8. 8. Outline <ul><li>Requirements for reasoning with modular ontologies </li></ul><ul><li>Package-based Description Logics (P-DL): features and semantics </li></ul><ul><li>A tableau algorithm for (P-DL) ALCP C </li></ul><ul><li>Discussions </li></ul>
    9. 9. Package <ul><li>A package is an ontology module that captures a sub-domain; </li></ul><ul><li>Each term has a home package </li></ul><ul><li>A package can import terms from other packages </li></ul><ul><li>Package extension is denoted as P </li></ul><ul><ul><li>P C :Package extension with only concept name importing </li></ul></ul><ul><ul><li>E.g., ALCP C = ALC + P C </li></ul></ul>General Pet Wild Livestock Animal ontology PetDog Pet Dog General
    10. 10. Package: Example O 1 (General Animal) O 2 (Pet) It uses ALCP, but not ALCP C
    11. 11. Semantics of Importing <ul><li>Domain relations are compositionally consistent : r 13 =r 12 O r 23 </li></ul><ul><ul><li>Therefore domain relations are transitively reusable. </li></ul></ul><ul><li>Domain relation : individual correspondence between local domains </li></ul><ul><li>Importing establishes one-to-one domain relations </li></ul><ul><ul><li>“ Copies” of individuals are shared </li></ul></ul>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. 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. 13. Model Projection x C I x C I 1 x’ C I 2 x’’ C I 3 Global model local models
    14. 14. Outline <ul><li>Requirements for reasoning with modular ontologies </li></ul><ul><li>Package-based Description Logics (P-DL): features and semantics </li></ul><ul><li>A tableau algorithm for (P-DL) ALCP C </li></ul><ul><li>Discussions </li></ul>
    15. 15. Tableau Algorithm <ul><li>A tableau is a representation of a model </li></ul><ul><li>Basic idea: </li></ul><ul><ul><li>start with some initial facts for an ontology </li></ul></ul><ul><ul><li>use tableau expansion rules to infer new facts, </li></ul></ul><ul><ul><ul><li>until no rule can be applied, or inconsistencies are found among those facts. </li></ul></ul></ul><ul><ul><li>If a clash-free fact set is found, a model of the ontology is constructed </li></ul></ul>
    16. 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. 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. 18. Federated Reasoning for P-DL <ul><li>Basic strategy </li></ul><ul><li>Use multiple local reasoners, each for a single package </li></ul><ul><li>Each local reasoner creates and maintains a local tableau based on local knowledge </li></ul><ul><li>A local reasoner may query other reasoners if its local knowledge is incomplete </li></ul><ul><li>Global relation among tableaux is created by messages </li></ul>(1) (2) (3) (4)
    19. 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. 20. Model Projection x C I x C I 1 x’ C I 2 x’’ C I 3 Global model local models
    21. 21. Tableau Expansion Tableau Expansion for ALCP C with acyclic importing
    22. 22. Communication among Local Tableaux <ul><li>Membership m ( y,C ): </li></ul><ul><li>Reporting r ( y,C ): </li></ul><ul><li>Clash bottom ( y ): </li></ul><ul><li>Model top ( y ): </li></ul>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. 23. ALCP C Expansion Example <ul><li>Consistency of the ontology is witnessed by P 1 </li></ul><ul><li>y is the shared individual </li></ul><ul><li>Subset blocking is still applicable </li></ul><ul><ul><li>E.g. L 1 (y)  L 1 (x) </li></ul></ul>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. 24. ALCP C Expansion Example (2) <ul><li>P 1 : 1:A 1:B </li></ul><ul><li>P 2 : 1:B 2:C </li></ul><ul><li>P 3 : 2:C 3:D </li></ul><ul><li>Query: if A D (from the point of view of P 3 ) </li></ul><ul><li>(it is not answerable by either DDL nor E-Connection in their current forms) </li></ul><ul><li>Reasoning: if A D is not true, then there will be clash. Hence, it must be true </li></ul>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. 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. 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. 27. Completeness P-DL model can be constructed from a distributed Tableau
    28. 28. Termination <ul><li>Acyclic importing ensures no message loop </li></ul><ul><li>Blocking </li></ul><ul><ul><li>Subset blocking </li></ul></ul><ul><ul><li>Reporting blocking: A node is temporarily blocked after sending a reporting message </li></ul></ul>x y y z T 1 T 2 w T 3 z v P 1 P 3 P 2 import import Tableaux Ontology
    29. 29. Outline <ul><li>Requirements for reasoning with modular ontologies </li></ul><ul><li>Package-based Description Logics (P-DL): features and semantics </li></ul><ul><li>A tableau algorithm for (P-DL) ALCP C </li></ul><ul><li>Discussions </li></ul>
    30. 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. 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. 32. Ongoing Work <ul><li>Working with cyclic importing </li></ul>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. 33. Ongoing Work (2) <ul><li>Asynchronous reasoning: </li></ul><ul><ul><li>local reasoners don’t need to wait after a reporting message </li></ul></ul><ul><ul><li>Thus they can concurrently search on different branches for a possible global tableau. </li></ul></ul><ul><li>Working with OWL </li></ul><ul><ul><li>Support SHOIQ(D) </li></ul></ul><ul><li>Implementation based on Pellet </li></ul>
    34. 34. References <ul><li>P-DL: </li></ul><ul><li>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. </li></ul><ul><li>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. </li></ul><ul><li>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) </li></ul><ul><li>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) </li></ul><ul><li>Related work: </li></ul><ul><li>L. Serafini and A. Tamilin. Local tableaux for reasoning in distributed description logics. In Description Logics Workshop 2004, CEUR-WS Vol 104 , 2004. </li></ul><ul><li>L. Serafini and A. Tamilin. Drago: Distributed reasoning architecture for the semantic web. In ESWC , pages 361-376, 2005. </li></ul><ul><li>B. C. Grau. Combination and Integration of Ontologies on the Semantic Web . PhD thesis, Dpto. de Informatica, Universitat de Valencia, Spain, 2005. </li></ul>
    35. 35. <ul><li>Thanks ! </li></ul>
    36. 36. Reasoning by Model Construction Model x Man I Human I <ul><li>If such a model is not possible in any situation, Man <= Human is true </li></ul>Reasoning <ul><li>Suppose it is not true, then at least one individual x in a world (model) is Man but not Human </li></ul>To query Man Human <ul><li>If such a model can be constructed, then Man <= Human is not true </li></ul>

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