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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
A Distributed Tableau Algorithm for Package-based Description Logics
Jie Bao1, Doina Caragea2 and Vasant G Honavar 1
1Artificial Intelligence Research Laboratory, Department of Computer Science,
Iowa State University, Ames, IA 50011-1040, USA. {baojie, honavar}@cs.iastate.edu
2Department of Computing and Information SciencesKansas State University, Manhattan, KS 66506, USA
2nd International Workshop on Context Representation and Reasoning (CRR 2006) @ ECAI 2006, Aug 29, 2006, Riva del Garda, Italy
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Dr. D. Caragea Dr. V. HonavarJie Bao
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Outline
• Requirements for reasoning with modular ontologies
• Package-based Description Logics (P-DL): features and semantics
• A tableau algorithm for (P-DL) ALCPC
• Discussions
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Modularity
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
The Need for Modular Ontologies(MO)
• Collaborative Ontology Building
• Distributed Data Management
• Large Ontology Management
• Partial Ontology Reuse
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
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
GraduateOK v : 9f ails:CoreCourseGraduateOK v PrelimOKPrelimOK(J ie)
CsCoreCourse v CoreCourseCsCoreCourse(cs511)f ails(3304;cs511)SSN (3304;123456789)
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
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 ?
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Outline
• Requirements for reasoning with modular ontologies
• Package-based Description Logics (P-DL): features and semantics
• A tableau algorithm for (P-DL) ALCPC
• Discussions
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
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– PC :Package extension with only
concept name importing
– E.g., ALCPC = ALC + PC
General Pet
Wild Livestock
Animal ontology
PetDogPet
DogGeneral
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Package: Example
O1 (General Animal) O2 (Pet)
It uses ALCP, but not ALCPC
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Semantics of Importing
• Domain relations are compositionally consistent: r13=r12
O
r23– Therefore domain relations are
transitively reusable.
x x’
ΔI1 ΔI2
CI1 CI2
r12
ΔI3
r13 r23
x’’CI3
• Domain relation: individual correspondence between local domains
• Importing establishes one-to-one domain relations – “Copies” of individuals are
shared
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Partially Overlapping Models
x x’
ΔI1 ΔI2
CI1 CI2
r12
ΔI3
r13 r23
x’’CI3
x
CI
Global interpretation obtained from localInterpretations by merging shared individuals
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Model Projection
x
CI
x
CI1
x’
CI2
x’’CI3
Global model
local models
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Outline
• Requirements for reasoning with modular ontologies
• Package-based Description Logics (P-DL): features and semantics
• A tableau algorithm for (P-DL) ALCPC
• Discussions
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
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
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Tableau Algorithm: Example
Dog(goofy)
Animal(goofy)( eats.DogFood)(goofy)
eats(goofy,foo)DogFood(foo)
goofyL(goofy)={Dog, Animal, eats.DogFood }
fooL(foo)={DogFood }
eats
ABox Representation Completion Tree Representation
Note: both representations are simplified for demostration purpose
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Federated Reasoning
Stan: Hey, Chef. Is Kyle’s new home far from us?
Chef: Hello there, children! Where does Kyle move to?
Cartman: San Francisco, I guess.
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.
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
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)
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Tableau Projection
x1
{A1,B1}
{A2}
{A3,B3}
{B2}x2 x3
x4
x1
{A1}
{A2}
{A3}
x2
x4
x1
{B1}
{B3}
{B2}x3
x4
The (conceptual) global tableau Local Reasoner
for package ALocal Reasonerfor package B
Shared individuals mean partially overlapped local models
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Model Projection
x
CI
x
CI1
x’
CI2
x’’CI3
Global model
local models
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Tableau Expansion
Tableau Expansion for ALCPC with acyclic importing
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
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
T1 T2
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
ALCPC Expansion Example
• Consistency of the ontology is witnessed by P1
• y is the shared individual
• Subset blocking is still applicable– E.g. L1(y)L1(x)
x L1(x)={A,R.B}
y y
z
L2(y)={B,P.C}
L2(z)={C,P.C}
R
P
T1
T2
L1(y)={A,R.B}
w L2(w)={C,P.C}P
> v 1: A;> v 9(1 : R):(2 : B)
P1
> v (2 : P ):(2 : C)
P2
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
T3
x
ALCPC Expansion Example (2)
• P1: 1:A 1:B• P2: 1:B 2:C• P3: 2:C 3:D• Query: if A D (from
the point of view of P3) (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
L3(x)={A⊓
D, C D⊔A,C, D}
r(x,C)
x x
r(x,A)
T2 T1
L2(x)={B C⊔C, B}
L1(x)={A B⊔A, B, B}
r(x,B)
(x)
(x) (x)
Transitive Subsumption Propagation
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
ALCPC Expansion Example (3)
L2(x)={P,P B, ⊔P⊔F,B,F}
x xL1(x)={B,
F,B F,⊔ F}
T2 T1r(x,B)r(x,F)
(x)
L1(x)={A, A C,C}⊔
y
z
L2(y)={A,A⊔R.B, B (A⊔ ⊓C),
R.B, B} P
T1T2
L2(z)={B,A⊔R.B,
B (A⊔ ⊓C), R.B, A⊓C, A,
C}
y
L1(z)={A, C, A C, ⊔C}
z
r(z,A)r(z,C)
(x)
r(z,A)
(x)
Detect Inter-module Unsatisfiability
P1 : f 1 : B v 1: F g,P2 : f 1 : P v 1 : B;2 : P v : 1 : F g
2:P is unsatisfiable
Reasoning from Local Point of View
P1 : f 1 : A v 1: CgP2 : f 1 : A v 92: R:(2 : B);2 : B v 1: A u (: 1 : C)g
1:A is unsatisfiable witnessed by P2
is satisfiable witnessed by P1
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
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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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Completeness
P-DL model can be constructed from a distributed Tableau
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
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
T1 T2
w
T3
z
v
P1 P3P2
import import
Tableaux
Ontology
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Outline
• Requirements for reasoning with modular ontologies
• Package-based Description Logics (P-DL): features and semantics
• A tableau algorithm for (P-DL) ALCPC
• Discussions
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Other Tableau Projections
Distributed Description Logics (DDL) [Serafini and Tamilin 2004, 2005]
x1
x2 x3
x4
x1
x2
x3
x4
x3
x5
x5
fB1 u : B2;¢¢¢g
fB1 u : B2;¢¢¢g
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Other Tableau Projections (2)x1
x2 x3
x4
x1
x2
x4
x5
x3
x6
E-Connections [Grau 2005]
x5 x6
E
E
{A1}
{A1}{A2} {A3}
{B1}
{B2} {B3}
{A2} {A3}{B1}
{B2} {B3}
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Ongoing Work
• Working with cyclic importing
x1
{A1,B1}
{A2}
{A3,B3}
{B2}x2 x3
x4
x1
{A1}
{A2}
{A3}
x2
x4
x1
{B1}
{B3}
{B2}x3
x4
{B4}{B4}
B1
A3
PA PB
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
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
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
ReferencesP-DL:1. 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.
2. 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.
3. 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)
4. 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:1. L. Serafini and A. Tamilin. Local tableaux for reasoning in distributed description
logics. In Description Logics Workshop 2004, CEUR-WS Vol 104, 2004.2. L. Serafini and A. Tamilin. Drago: Distributed reasoning architecture for the semantic
web. In ESWC, pages 361-376, 2005.3. 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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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Thanks !
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Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory
Model
x
ManI
HumanI
2. If such a model is not possible in any situation, Man <= Human is true
Reasoning by Model ConstructionReasoning
1. Suppose it is not true, then at least one individual x in a world (model) is Man but not Human
To query
Man Human
3. If such a model can be constructed, then Man <= Human is not true