Tableau-based Federated Reasoning Algorithm for Modular Ontologies

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


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 ⊑ P eople
Property
some Students attend Classes Student ⊑ ∃attends.Classes
Bob is a Student Student(Bob) Individual
Concept


DL as Ontology Language

• ALC: the basic DL
Conjunction M an := M ale ⊓ Human
Disjunction Child := Boy ⊔ Girl

Negation W oman := Human ⊓ ¬M an
Exists Restrictions Human := ∃hasP arent.Human

Universal Restrictions Human := ∀hasBrother.M an

• Many extensions
– Number restrictions: a core family has at least 1 child
– Role hierarchy: hasBrother is less general than hasSibling
– …



DL Semantics

• An interpretation I =<∆I,(.)I >
– Concept subset of ∆I
– Role binary relations over ∆I × ∆I
– Individual elements of ∆I
• Interpretation function (.)I is extended to concept
expressions



DL Model
• An interpretation I satisfies an subsumption C ⊑ D iff CI ⊆ DI
• A model of an ontology O is an interpretation that satisfies
every axiom in O

Bob
Student ⊑ P eople Student, People,
∃attends.Class
Student ⊑ ∃attends.Classes
Student(Bob) attends

x Class



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 Syntax
People Package (P1)

¬M ale ⊑ F emale
M an ⊑ P eople ⊓ M ale
W oman ⊑ P eople ⊓ F emale

University Package (P2) People, Man, Woman

Student ⊑ P eople
F aculty ⊑ P eople
Class ⊑ ∃taughtBy.P eople ⊓ ∀taughtBy.F aculty
CoEd ⊑ U niversity ⊓ ∃hasStudent.M an ⊓ ∃hasStudent.W oman

ALCPC: ALC extended with concept importing


P-DL Semantics

• Each package has a local interpretation
• Individuals in different domains can be associated
by domain relations
Man, People
Man, People, Male hasStudent
CoEd, University
hasStudent
Woman, People, Female
Woman, People

People, Male Class
taughtBy

People, Female People, Faculty

∆I1 r12 ∆I2


P-DL Semantics
• Domain relations are
– one-to-one and
– compositional consistent

• For any concept i:C :
CIi CIj
CIj = rij (C Ii )

• An axiom is always kept
in its context:
M
University ⊑ M ale ⊔ F emale U
F



Outline




Tableau

• A tableau represents a model of a DL ontology
• We can use “ABox” (assertion set) to represent tableau

Concept Assertions
Man, People Man(x1), People(x1)
x1 hasStudent
Woman(x2), People(x2)
CoEd, University
x5 Class(x3)
hasStudent
x2 Faculty(x4),People(x4)
Woman, People
CoEd(x5), University(x5)
x3 Class
taughtBy Role Assertions
hasStudent(x5,x1)
x4 People, Faculty
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)
Student ⊑ ¬F aculty (∃taughtBy.Student)(x0)
EasyClass ⊑ ∃taughtBy.Student taughtBy(x0,x1)
Class ⊑ ∀taughtBy.F aculty Student(x1)
EasyClass ⊑ Class ¬Faculty(x1)
Class(x0)
(∀taughtBy.Faculty)(x0)
Check: Satisfability of EasyClass Faculty(x1)

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

Incremental Storage

((C⊔D)⊓∃R.D)(x),¬C(x),
(∀R.¬D)(x)

⊓ (C⊔D)(x),∃R.D(x)

∃ R(x,y),D(y)

Choice!
⊔ C(x) D(x)

Inconsistent

∀ ¬D(y)
Inconsistent



Distributed Tableaux

– 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
(Virtually)
Package A Package B
Integrated Ontology

(Virtually)
Global Tableau

Local ABox A Local ABox B


Example
Package A Package B

A1 ⊑ A2
A2 ⊑ ∃RA .B1 B1 ⊑ ¬B2
A2 ⊑ ∀RA .B2

A1(x0)

A2(x0),(∃RA.B1)(x0)

⊥
RA(x0,x1), B1(x1) B1(x1) , B2(x1)

(∀RA.B2)(x0) ¬ B2(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
Package A Package B

• 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



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

nA1
nA2 nB0 lst= nA1

nB1 lst= nA1

nB2
lst= 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

nA1
nA2 nB0 lst= nA1

nB1
nA3
nB2 nB3
lst= nA1


Example 2
Time 1 TA
(A1⊓¬A3 ⊓(¬A1⊔B1)⊓(¬A2⊔B2))(x)
¬ ⊓¬

Package A Package B



Example 2
Time 2 TA

(A1⊓¬A3 ⊓(¬A1⊔B1)⊓(¬A2⊔B2))(x)

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



Example 2
Time 3 TA

(A1⊓¬A3 ⊓(¬A1⊔B1)⊓(¬A2⊔B2))(x)

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

¬A1(x) B1(x)



Example 2
Time 4 TA TB

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

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

¬A1(x) B1(x)



Example 2
Time 5 TA TB

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

¬

¬A1(x) B1(x)
A2(x)

A2(x)

¬A2(x) B2(x)



Example 2
Time 6 TB
TA
(A1⊓¬A3 ⊓(¬A1⊔B1)⊓(¬A2⊔B2))(x) B1(x),(¬B1⊔A2⊔A3)(x),(¬B2⊔A3)(x)


¬A1(x) B1(x) B2(x)

A2(x) A3(x) ¬B2(x)

¬A2(x) B2(x)



Example 2
Time 7 TA TB
(A1⊓¬A3 ⊓(¬A1⊔B1)⊓(¬A2⊔B2))(x) B1(x),(¬B1⊔A2⊔A3)(x),(¬B2⊔A3)(x)


¬A1(x) B1(x) B2(x)

A3(x)
A2(x) A3(x) ¬B2(x)

clash
¬A2(x) B2(x)

A3(x)



Example 2
Time 8 (Hide some unsuccessful branches)

TA TB
(A1⊓¬A3 ⊓(¬A1⊔B1)⊓(¬A2⊔B2))(x) ¬
B1(x),(¬B1⊔A2⊔A3)(x),(¬B2⊔A3)(x)


¬A1(x) B1(x)

A3(x)
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 SHOIQ
– More expressive semantic connections
• Concept importing 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 Class
less general than attend

Student is a
Bob

• Ontology Languages: Frame Logics, Description
Logics,…


Web Ontology Language

• OWL: a syntactical variation of the DL SHOIQ(D)
• Used to represent knowledge on the Semantic
Web

Web Data

P hD S tudent(J ieB ao)
P hD S tudent ⊑ Graduate
Meta Data
Graduate ⊑ S tudent
S tudent ⊑ P eople (Ontology)



Contextualized Negation

(¬C)Ij = rij (∆Ii )rij (C Ii )

Not

(¬C)Ij = ∆Ij rij (C Ii )


Tableau-based Federated Reasoning Algorithm for Modular Ontologies

Recomendados

Recomendados

Más contenido relacionado

La actualidad más candente

La actualidad más candente (9)

Similar a Tableau-based Federated Reasoning Algorithm for Modular Ontologies

Similar a Tableau-based Federated Reasoning Algorithm for Modular Ontologies (20)

Más de Jie Bao

Más de Jie Bao (20)

Último

Último (20)

Tableau-based Federated Reasoning Algorithm for Modular Ontologies