A Distributed Tableau Algorithm for Package-based Description Logics

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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

[email protected]

2nd International Workshop on Context Representation and Reasoning (CRR 2006) @ ECAI 2006, Aug 29, 2006, Riva del Garda, Italy

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Dr. D. Caragea Dr. V. HonavarJie Bao

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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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Modularity

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The Need for Modular Ontologies(MO)

• Collaborative Ontology Building

• Distributed Data Management

• Large Ontology Management

• Partial Ontology Reuse

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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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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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Outline




• Discussions

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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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Package: Example

O1 (General Animal) O2 (Pet)

It uses ALCP, but not ALCPC

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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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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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Model Projection

x

CI

x

CI1

x’

CI2

x’’CI3

Global model

local models

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Outline




• Discussions

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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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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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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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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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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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Model Projection

x

CI

x

CI1

x’

CI2

x’’CI3

Global model

local models

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Tableau Expansion

Tableau Expansion for ALCPC with acyclic importing

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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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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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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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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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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

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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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Outline




• Discussions

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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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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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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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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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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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Thanks !

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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

A Distributed Tableau Algorithm for Package-based Description Logics

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