Modular Ontologies - A Formal Investigation of Semantics and Expressivity

ASWC, Sept 7, 2006, Beijing, China 1

Iowa State University Department of Computer ScienceArtificial Intelligence Research Laboratory

Modular Ontologies - A Formal Investigation of Semantics and

ExpressivityJie Bao, Doina Caragea and Vasant G Honavar

Artificial Intelligence Research Laboratory, Department of Computer Science,

Iowa State University, Ames, IA 50011-1040, USA.

{baojie,dcaragea, honavar}@cs.iastate.edu



Outline

• Desiderata of Modular Ontologies

• Abstract Modular Ontology (AMO)

• Semantics & Expressivity Comparison

• Summary & Conclusion



Modularity



A Modular Semantic Web

Visualising the Semantic Web by Juan C. Dürsteler



Modular Ontologies

• What is modular ontology?– An ontology that is composed by a set of smaller

(semantically) connected component ontologies

• Why modular ontology ?– Collaborative Ontology Building– Selective Ontology Reuse– Selective Knowledge Hiding– Distributed Data Management– Large Ontology Storage and Reasoning



OWL Limitations• owl:imports: syntactic

modularization• No localized semantics

– Reasoning is possible only with the integrated ontology

• No partial reuse– Reuse all or nothing– E.g. OpenCyc OWL file

needs 9 hours to load into Protege

owl:imports

Syntactic import:“copy and paste”



Modular Ontology Approaches

CЄ (SHOIN(D))

OWL

1998 2002 2003 2004 2005 2006

C-OWLC-OWLCTXML

E-Connections

P-OWL

(Planning)

P-DL

DDLDFOLRole<->Concept

Mapping

CЄ(SHIF(D)) IHN+s



Requirements

• Semantic soundness– Reasoning correctness

– Module autonomy

• Needed language features– Concept relations

– Role relations

– …



Localized Semantics

Integrated ontology

Materialized Global Model

Modular ontology

Local Models



Exact Reasoning

vv

Integrated ontology Modular ontology

C D vvC D



Directional Semantic Relations

vvD E

vvD E



Transitive Reusability

vvC D vvD E vvE F

vvC F



Decidability

is answerable in finite steps

vvC D



Language Features• Concept Subsumption• Concept Construction with

Foreign Concepts• Concept Construction with Role

Restrictions.• Role Inclusion• Role Inversion• Role Construction• Transitive Role• Nominal Correspondence

1:C v 2: D1:C v 2: D1:C u2: D1:C u2: D

9(1: R):(2 : D)9(1 : R):(2 : D)

1:P v 2: R1:P v 2: R1:P = 2:R¡1:P = 2:R¡

1:P u2: R1:P u2: R

1:x=2:y1:x=2:yTrans(1:P), 1:P used in 2



Outline




• Conclusion



Local Points of View

The domain

agents agents

Multiple observers of a domain



Abstract Modular Ontology (AMO)

DL2DL1

DL3

r13

r231

r132

1

Semantics

Δ1 Δ2

Δ3



(General) Domain Relations

r13neighbourOf

r13friendOf

Δ1 Δ3



Image Domain Relation

r13

Δ1

Δ3



Concept Image

r13

Agent3: "these objects in my mind state correspond to the conceptLeg from agent 1’s mind state"

Legm1Legm1

Leg1! 3 : r!13(Legm1 )Leg1! 3 : r!13(Legm1 )



Role Image

P1! 3P1! 3

r13

P

Δ1 Δ3

Agent3: "these object pairs in my mind state correspond to object pairs

P from agent 1’s mind state"



Possible AMO Expressivity Features



Semantic Soundness Definitions

• Localized Semantics: local domains {Δi} are not necessarily identical

• Decidability (of concept C w.r.t AMO O): there is an algorithm to decide in finite steps whether there is a common model <{mi}, {rij}> of C and O.

• Directional Semantic Relations: mj ² Cmj µ Dmjmj ² Cmj µ Dmj mi ² C iÃ j µ D iÃ jmi ² C iÃ j µ D iÃ j



Semantic Soundness Definitions (2)

• Reusability

Ci ! j µ D i ! jCi ! j µ D i ! jCmi µ DmiCmi µ Dmi

C D

• Transitive ReusabilityΔ1 Δ2 Δ3

(an agent can infer local constraints based on observing constraints in other agents’ points of view)



Semantic Soundness Definitions (3)

• Exact Reasoning– Compatible

beliefs of agents may be combined.

– Local models M can be merged into an integrated model M' s.t.

Physical World

LocalModels

Integrated Model(consensus)



Outline




• Conclusion



DDL Semantics

DDL : C into D

DI

CI

(R.D)I

r(CI) DI

r(CI)= (R-.C)I

C onto D

DI

CI

r(CI) DI

r(CI)= (R-.C)I

1:Dog into 2:Animal

1:Dog onto 2:Hound

implicit domain disjointness

[Borgida and L. Serafini, 2002]



Subsumption Propagation Problem

DDL domain relations are not transitively reusable

Cm1 C into D Dm2

Em3

D into EC into E ?



Inter-module Unsatisfiability Problem

DDL allows arbitrary domain relations: loss of reasoning exactness

Flym1

~Fly onto Penguin

Penguinm2

Birdm1

Bird onto Penguin



DDL: Pros & Cons

• Pros– Localized Semantics– Directional Relation– Decidability transfer

• Cons– No support for role relations– No general module transitive reusability– No general reasoning exactness



E-Connections

• Local domains are disjoint

• It allows multiple “link” relations between two local domains

• Links can be used to construct local concepts

PetOwner

Petowns

PetOwner

Petowns

[Grau, 2005]



E-Connections Semantics

DI

(R.D)I = r-(DI)

R

DI(R.D)I=Δ 1\r-(Δ 2\DI)

R



E-Connections: Pros & Cons

• Pros– Localized Semantics– Decidability Transfer– Exact Reasoning (without generalized link)

• Cons– Very limited transitive reusability– No support for inter-module concept

subsumption



P-DL • Semantic Importing

O1 (General Animal) O2 (Pet)O1 (General Animal) O2 (Pet)

[Bao et al., 2006]

35


P-DL Semantics

• 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 – “Copied” individuals are

shared

36


P-DL Semantics (2)

x x’

ΔI1 ΔI2

CI1 CI2

r12

ΔI3

r13 r23

x’’CI3

x

CI

Global model obtained from localmodels by merging shared individuals

37


Partially Overlapping Domains• Ensure unambiguous

communication between local models– satisfiability transfer– transitive reusability

• Overlapped domains represent the consensus of agents

• Non-sharing domains are still kept local.

x

CI



P-DL: Pros & Cons

• Pros– Localized Semantics– Exact Reasoning – Stronger Expressivity– Transitive Reusability

• Cons– Directional Semantic Relation does not always

hold– Decidable only if all modules use the same

decidable DL (e.g. OWL).



Summary: Semantic Soundness



Summary: Expressivity



Outline



• Semantics & Expressvity Comparison

• Conclusion



Summary

• We discussed– Semantic soundness and expressvity

requirements for modular ontologies– Comparsion of DDL, E-Connections and P-DL

under the AMO framework– Analysis of several semantic difficulties and

expressivity limitations of existing approaches



Conclusions• There is still no language or reasoner support for

both general inter-module concept and inter-module role correspondence

• Local domain disjointedness assumption of DDL and E-Connections may be partially relaxed.– to improve expressivity– to ensure reasoning exactness and transitivity

reusability.



Open Problems• A consensus on expressive modular

ontology language• A OWL-compatible syntax for modular

ontologies• A reasoner that supports the expressive

modular ontology language– Pellet and DRAGO are complementary to each

other

To be discussed at the Modular Ontology Workshop (WoMo) at ISWC 2006 , Athens, Georgia, USA, Nov 2006.



Thanks!



Semantic SoundnessWhat are the logical consequences in an AMO?

What are the possible cause of semantic inconsistencies between two agents?

What is the "objective" way to integrate knowledge of agents?

a b

friendOf

x y

friendOf ?

r13

a b

friendOf

x y

enemyOf

r13

a/x b/y

friendOf

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