Merge soundness and completeness, termination slides
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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
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 )
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
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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
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Model Projection x C I x C I 1 x’ C I 2 x’’ C I 3 Global model local models
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
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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
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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.
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)
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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
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Model Projection x C I x C I 1 x’ C I 2 x’’ C I 3 Global model local models
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Tableau Expansion Tableau Expansion for ALCP C with acyclic importing
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
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 )
(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)
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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
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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
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Completeness P-DL model can be constructed from a distributed Tableau
Requirements for reasoning with modular ontologies
Package-based Description Logics (P-DL): features and semantics
A tableau algorithm for (P-DL) ALCP C
Discussions
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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
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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 }
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.
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