A Tableau-based Federated Reasoning Algorithm for Modular Ontologies

This research was supported by grants from the US NSF (0219699, 0639230) 1

Iowa State University Department of Computer Science

Artificial Intelligence Research Laboratory

A Tableau-based Federated Reasoning Algorithm for

Modular Ontologies

Jie Bao and Vasant Honavar

Artificial Intelligence Research Laboratory,

Department of Computer Science,

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

{baojie, honavar}@cs.iastate.edu

International Conference on Web Intelligence (WI 2006),

Hong Kong, China, Dec 21st, 2006




Outline

• Ontology and Description Logics (DL)

• Modular Ontology and Package-based DL

• Distributed Reasoning with P-DL




Description Logics (DL)

• A family of Knowledge Representation (KR) formalisms – About Concepts (Classes), Properties (Roles,

Relationships) and Individuals (Instances)

– With formal semantics and well-understood computational behavior (decidability and complexity)

• Example

Students are People Student ⊑ PeopleStudent ⊑ People

some Students attend Classes Student ⊑ ∃attends.ClassesStudent ⊑ ∃attends.Classes

Bob is a Student Student(Bob)Student(Bob)

Concept

Property

Individual




DL as Ontology Language

• ALC: the basic DL

• Many extensions– Number restrictions: a core family has at least 1 child

– Role hierarchy: hasBrother is less general than hasSibling

– …

Conjunction Male ⊓⊓⊓⊓HumanMan :=

Disjunction Child := Boy ⊔⊔⊔⊔Girl

Negation Woman := Human ⊓ ¬¬¬¬Man

Exists Restrictions Human := ∃hasParent.Human

Universal Restrictions Human := ∀∀∀∀hasBrother.Man




DL Semantics

• An interpretation I=<∆I,(.)I >

– Concept à subset of ∆I

– Role à binary relations over ∆I ∆I×

– Individual à elements of ∆I

• Interpretation function is extended to concept expressions

(.)I




DL Model

• An interpretation I satisfies an subsumption iff

• A model of an ontology O is an interpretation that satisfies every axiom in O

CI ⊆DIC⊑D

Student ⊑ PeopleStudent ⊑ People

Student ⊑ ∃attends.ClassesStudent ⊑ ∃attends.Classes

Student(Bob)Student(Bob)

∃attends.Class

Class

Student, People,

Bob

x

attends




Outline







One or Many Web Ontologies?

• One single, universal ontology ?

A formal “encyclopedia” of all

knowledge on the web

• Or multiple, inter-connected ontologies ?




Call or Modularity

• Decentralization

– Web is decentralized, so will be for ontologies

– No ontology can capture the “full” knowledge for Web

• Context

– Ontologies represent local points of view– E.g. People ontology: ¬Male⊑ Female (an individual who is not a

Male is a Female) – implicit context “people”

– If a University ontology reuses the People ontology, will a “University” be a Male or Female?

• Scalability (for reasoning)

– Naive approach: download and integrate all ontologies

– Problem 1: There may be millions of axioms involved

– Problem 2: Global knowledge may not be available, e.g. in P2P




Package-based DL (P-DL)

• P-DL: Package-based Description Logics– A formal modular ontology language

– Extend DL with organizational modules called “package”

• Basic Intuitions – Syntax: a module may reuse knowledge from other

modules by importing foreign terms

– Semantics: localized (each module has local interpretation) and contextualized (axioms has scoped meaning)

– Reasoning: allow a federation of local reasoners collaborate with each other based on their local knowledge.




P-DL SyntaxPeople Package (P1)

University Package (P2)

¬Male ⊑ FemaleMan ⊑ People ⊓Male

Woman ⊑ People ⊓ Female

People, Man, Woman

ALCPC: ALC extended with concept importing

Student ⊑ PeopleFaculty ⊑ People

Class ⊑ ∃taughtBy.People ⊓ ∀taughtBy.FacultyCoEd ⊑ University ⊓ ∃hasStudent.Man ⊓ ∃hasStudent.Woman




P-DL Semantics

• Each package has a local interpretation

• Individuals in different domains can be associated

by domain relations

Man, People, Male

Woman, People, Female

People, Male

People, Female

Class

People, Faculty

CoEd, University

Man, People

Woman, People

hasStudent

hasStudent

taughtBy

r12∆I1 ∆I2




P-DL Semantics

• Domain relations are

– one-to-one and

– compositional consistent

University ⊑Male ⊔ Female

• An axiom is always kept in its context:

M

FU

CIj = rij(CIi)CIj = rij(CIi)

• For any concept i:C :

CIi CIj




Outline







Tableau

• A tableau represents a model of a DL ontology

• We can use “ABox” (assertion set) to represent tableau

Class

People, Faculty

CoEd, University

Man, People

Woman, People

hasStudent

hasStudent

taughtBy

x1

x2

x3

x4

x5

Concept AssertionsMan(x1), People(x1)Woman(x2), People(x2)Class(x3)Faculty(x4),People(x4)CoEd(x5), University(x5)

Role AssertionshasStudent(x5,x1)hasStudent(x5,x2)taughtBy(x3,x4)




Tableau Algorithm

• Satisfiability of a concept C w.r.t. a DL Ontology TBox (set of concept inclusions) O can be checked by constructing a

common model of C and O

EasyClass(x0)

(∃taughtBy.Student)(x0)taughtBy(x0,x1)Student(x1)

¬Faculty(x1)Class(x0)

(∀taughtBy.Faculty)(x0)Faculty(x1)

Student ⊑ ¬FacultyEasyClass ⊑ ∃taughtBy.StudentClass ⊑ ∀taughtBy.Faculty

EasyClass ⊑ Class

Check: Satisfability of EasyClass

Note: we simplify the presentation (and in some following slides) by omitting some

facts due to “TBox internalization”, e.g., (EasyClass ⊔ ¬Class)(x0)




ALC Expansion

((C⊔D)⊓∃R.D)(x),¬C(x), (∀R.¬D)(x)

(C⊔D)(x),∃R.D(x)⊓

∃ R(x,y),D(y)

C(x)

Inconsistent

⊔

¬D(y)

Inconsistent

∀

Incremental Storage

D(x)

Choice!




Distributed Tableaux

• Distributed Reasoning with P-DL– Syntactically: no integration of ontology modules is

needed

– Semantically: no (materialized) global tableau (or model) is needed

• How to make it possible?– Instead of using a global reasoner (with access to full

knowledge), we use a federation of local reasoners, each for a package, with only local knowledge of that package.

– Local reasoners communicate with each other to create a distributed tableau (distributed ABox)




Distributed Tableau

Package A Package B(Virtually)

Integrated Ontology

(Virtually)

Global Tableau

Local ABox A Local ABox B




Example

Package A Package B

B1 ⊑ ¬B2

A1 ⊑ A2A2 ⊑ ∃RA.B1A2 ⊑ ∀RA.B2

A1(x0)

A2(x0),(∃RA.B1)(x0)

RA(x0,x1), B1(x1)

(∀RA.B2)(x0)

B2(x1)

¬ B2(x1)

B1(x1) , B2(x1)

⊥⊥⊥⊥




Messages

• A fact of the form C(x) or ¬C(x) may be shared by two local tableaux– C is an atomic concept name– We don’t allow role name importing, hence role instances are never

shared

• Destination of facts– C(x) or ¬C(x) will always be sent to the reasoner for the home

package of C (where C is defined)

• Termination with acyclic concept importing [Bao et al. CRR 2006]

– Subset blocking can be locally applied to avoid non-termination.• E.g. {C(x),D(x),C(y)} then y is blocked by x

– Synchronous reasoning: local expansions are blocked until a remote answer (clash or consistency) is returned (i.e., only one branch of ABox tree is under expanding at any time)

– Hence there is no cyclic message between local reasoners




Handle Cyclic Importing

• Cyclic Importing

• Difficulty

– How to ensure no cyclic messages or deadlock between local reasoners

– How to maximize the usage of computational resources by parallel, asynchronous reasoning: local reasoners

may work on different (search) branches simultaneously

Package A Package B




Handle Cyclic Importing (2)

• Key: different search branches are kept globally separated • Contact List: every node has one and only one contact node from each

local ABox tree. – Can be locally inherited

– Updated after receiving messages (only most recent contacts are kept)

• If a fact in node n of Tj is sent to tableau Ti, it is added to – lsti(n), if no local branches created since last message from lsti(n)

–

nA0

nA1nA2 nB0

nB1

nB2

lst= nA1

lst= nA1

nA0

nA1nA2 nB0

nB1

nB2

lst= nA1

lst= nA1

lst= nA1lst= nA1




Handle Cyclic Importing (2)

• Key: different search branches are kept globally separated • Contact List: every node has one and only one contact node from each

local ABox tree. – Can be locally inherited

– Updated after receiving messages (only most recent contacts are kept)

• If a fact in node n of Tj is sent to tableau Ti, it is added to – lsti(n), if no local branches created since last message from lsti(n)

– A new node under lsti(n), otherwise

nA0

nA1nA2

nA3

nB0

nB1

nB3nB2

lst= nA1

lst= nA1

nA0

nA1nA2

nA3

nB0

nB1

nB3nB2

lst= nA1

lst= nA1




Example 2

(A1⊓⊓⊓⊓¬¬¬¬A3 ⊓⊓⊓⊓(¬¬¬¬A1⊔⊔⊔⊔B1)⊓⊓⊓⊓(¬¬¬¬A2⊔⊔⊔⊔B2))(x)

Time 1 TA

Package A Package B




Example 2

(A1⊓¬A3 ⊓(¬A1⊔B1)⊓(¬A2⊔B2))(x)

Time 2

A1(x),¬A3(x),(¬¬¬¬A1⊔⊔⊔⊔B1)(x),(¬¬¬¬A2⊔⊔⊔⊔B2)(x)

TA




Example 2

(A1⊓¬A3 ⊓(¬A1⊔B1)⊓(¬A2⊔B2))(x)

Time 3

A1(x),¬A3(x),(¬A1⊔B1)(x),(¬¬¬¬A2⊔⊔⊔⊔B2)(x)

¬A1(x) B1(x)

TA




Example 2

(A1⊓¬A3 ⊓(¬A1⊔B1)⊓(¬A2⊔B2))(x)

Time 4

A1(x),¬A3(x),(¬A1⊔B1)(x),(¬¬¬¬A2⊔⊔⊔⊔B2)(x)

¬A1(x)

B1(x),(¬B1⊔A2⊔A3)(x),(¬¬¬¬B2⊔⊔⊔⊔A3)(x)

¬B1(x) A2(x) A3(x)

B1(x)

TA TB

B1(x)




Example 2

(A1⊓¬A3 ⊓(¬A1⊔B1)⊓(¬A2⊔B2))(x)

Time 5

A1(x),¬A3(x),(¬A1⊔B1)(x),(¬¬¬¬A2⊔⊔⊔⊔B2)(x)

¬A1(x)

B1(x),(¬B1⊔A2⊔A3)(x),(¬¬¬¬B2⊔⊔⊔⊔A3)(x)

¬B1(x) A3(x)

¬A2(x) B2(x)

A2(x)

A2(x)

B1(x)

TA TB

A2(x)




Example 2

(A1⊓¬A3 ⊓(¬A1⊔B1)⊓(¬A2⊔B2))(x)

Time 6

A1(x),¬A3(x),(¬A1⊔B1)(x),(¬A2⊔B2)(x)

¬A1(x)

B1(x),(¬B1⊔A2⊔A3)(x),(¬B2⊔A3)(x)

¬B1(x) A3(x)

A3(x) ¬B2(x)

¬A2(x) B2(x)

A2(x)

B1(x)

A2(x)

B2(x)

TATB




Example 2

(A1⊓¬A3 ⊓(¬A1⊔B1)⊓(¬A2⊔B2))(x)

Time 7

A1(x),¬A3(x),(¬A1⊔B1)(x),(¬A2⊔B2)(x)

¬A1(x)

B1(x),(¬B1⊔A2⊔A3)(x),(¬B2⊔A3)(x)

¬B1(x) A3(x)

A3(x) ¬B2(x)

¬A2(x) B2(x)

A2(x)

B1(x)

A2(x)

B2(x)

A3(x)

clash

A3(x)

TA TB




Example 2

(A1⊓¬A3 ⊓(¬A1⊔B1)⊓(¬A2⊔B2))(x)

Time 8 (Hide some unsuccessful branches)

A1(x),¬A3(x),(¬A1⊔B1)(x),(¬A2⊔B2)(x)

¬A1(x)

B1(x),(¬B1⊔A2⊔A3)(x),(¬¬¬¬B2⊔⊔⊔⊔A3)(x)

¬B1(x) A3(x)

B1(x)

A2(x)

A3(x)

TATB

A3(x)

clash




Summary

We presented a federated, asynchronous reasoning

algorithm for modular ontologies such that

• No global knowledge is required

• Cyclic concept name importing is allowed

• Reasoning can be performed in asynchronous,

peer-to-peer fashion

• Can handle both inter-module subsumption (like

DDL[Borgida and Serafini, 2002]) and roles with foreign

range (like E-Connections [Grau et al. 2004])




Ongoing Work

• Reasoning with expressive modular ontologies

– More expressive component languages

• ALC à SHOIQ

– More expressive semantic connections

• Concept importing à Concept + Role + Nominal importing

• Theoretical investigation

– Contextualized negation

– Locally closed world semantics

– Controlled axiom propagation (partial ontology reuse)




Thanks




Ontology

• Science of Being (Aristotle, Metaphysics, IV, 1)

• Some formal descriptions about

– A vocabulary

– Relations between terms in the vocabulary

People

Student

Class

Bob

• Ontology Languages: Frame Logics, Description

Logics,…

is a

attendless general than




Web Ontology Language

• OWL: a syntactical variation of the DL SHOIQ(D)

• Used to represent knowledge on the Semantic Web

Web Data

Meta Data

(Ontology)

P hDStudent(J ieBao)

P hDStudent ⊑ Graduate

Graduate ⊑ Student

Student ⊑ P eople




Contextualized Negation

(¬C)Ij = rij(∆Ii)\rij(C

Ii)(¬C)Ij = rij(∆Ii)\rij(C

Ii)

(¬C)Ij = ∆Ij\rij(CIi)

Not

Technology

A Tableau-based Federated Reasoning Algorithm for Modular Ontologies