Basics of Reasoning in Description Logics

Jie Bao

Iowa State University

Feb 7, 2006

An ontology of this talk

People

Student

Jie Bao

present

Description Logic

DL reasoning

Knowledge Representation

Topic

Roadmap

What is Description Logics (DL)

Semantics of DL

Basic Tableau Algorithm

Advanced Tableau Algorithm

Description Logics

A formal logic-based knowledge representation language

“Description" about the world in terms of concepts (classes), roles (properties, relationships) and individuals (instances)

Decidable fragments of FOL

Widely used in database (e.g., DL CLASSIC) and semantic web (e.g., OWL language)

A “Family” Knowledge Base

Person include Man(Male) and Woman(Female),

A Man is not a Woman

A Father is a Man who has Child

A Mother is a Woman who has Child

Both Father and Mother are Parent

Grandmother is a Mother of a Parent

A Wife is a Woman and has a Husband( which as Man)

A Mother Without Daughter is a Mother whose all Child(ren) are not Women

DL for Family KB

DL Basics

Concepts (unary predicates/formulae with one free variable)E.g., Person, Father, Mother

Roles (binary predicates/formulae with two free variables)E.g., hasChild, hasHudband

Individual names (constants)E.g., Alice, Bob, Cindy

Subsumption (relations between concepts)E.g. Female Person

Operators (for forming concepts and roles) And(Π) , Or(U), Not (¬)Universal qualifier ( Existent qualifier()Number restiction : Inverse role (-), transitive role (+), Role hierarchy

More for “Family” Ontology

(Inverse Role) hasParent = hasChild-

hasParent(Bob,Alice) -> hasChild(Alice, Bob)

(Transitive Role)hasBrother hasBrother(Bob,David), hasBrother(David, Mack) -> hasBrother(Bob,Mack)

(Role Hierarchy) hasMother hasParenthasMother(Bob,Alice) -> hasParent(Bob, Alice)

HappyFather Father Π hasChild.Woman Π hasChild.Man

DL Architecture

Knowledge Base

Tbox (schema)

Abox (data)

HappyFather Person Π hasChild.Woman Π hasChild.Man

Happy-Father(Bob)

Infe

ren

ce S

yste

m

Inte

rface

(Example from Ian Horrocks, U Manchester, UK)

DL Representives

ALC: the smallest DL that is propositionally closed

Constructors include booleans (and, or, not),

Restrictions on role successors

SHOIQ = OWL DLS=ALCR+: ALC with transitive role

H = role hierarchy

O = nomial .e.g WeekEnd = {Saturday, Sunday}

I = Inverse role

Q = qulified number restriction e.g. >=1 hasChild.Man N = number restriction e.g. >=1 hasChild

Roadmap

What is Description Logic (DL)

Semantics of DL

Basic Tableau Algorithm

Advanced Tableau Algorithm

Interpretations

DL Ontology: is a set of terms and their relations

Interpretation of a DL Ontology: A possible world ("model") that materalizes the ontology

People

Student

Jie Bao

present

Description Logic

DL reasoning

Knowledge Representation

Topic

Ontology:

Student PeopleStudent Present.TopicKR TopicDL KR

Interpretation

DL Semantics

DL semantics defined by interpretations: I = (I, .I), where

I is the domain (a non-empty set) .I is an interpretation function that maps:

Concept (class) name A -> subset AI of I

Role (property) name R -> binary relation RI over I

Individual name i -> iI element of I

Interpretation function .I tells us how to interpret atomic concepts, properties and individuals.

The semantics of concept forming operators is given by extending the interpretation function in an obvious way.

DL Semantics: example

I = (I, .I)I = {Jie_Bao, DL_Reasoning}PeopleI=StudentI={Jie_Bao}TopicI=KRI=DLI={DL_Reasoning}PresentI={(Jie_Bao, DL_Reasoning)}

An interpretation that satisifies all axioms in an DL ontology is also called a model of the ontology.

CSE-291: Ontologies in Data & Process Integration

Source: Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002,

CSE-291: Ontologies in Data & Process Integration

Source: Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002,

Roadmap

What is Description Logic (DL)

Semantics of DL

Basic Tableau Algorithm

Advanced Tableau Algorithm

What is Reasoning?

"Machine Understanding"

Find facts that are implicit in the ontology given explicitly stated facts

Find what you know, but you don't know you know it - yet.

ExampleA is father of B, B is father of C, then A is ancestor of C.

D is mother of B, then D is female

Reasoning Tasks

Knowledge is correct (captures intuitions)C subsumes D w.r.t. K iff for every model I of K, CI µ DI

Knowledge is minimally redundant (no unintended synonyms)C is equivallent to D w.r.t. K iff for every model I of K, CI = DI

Knowledge is meaningful (classes can have instances)C is satisfiable w.r.t. K iff there exists some model I of K s.t. CI ;

Querying knowledgex is an instance of C w.r.t. K iff for every model I of K, xI CI

hx,yi is an instance of R w.r.t. K iff for, every model I of K, (xI,yI) RI

Knowledge base consistencyA KB K is consistent iff there exists some model I of K

Reasoning Tasks(2)

Many inference tasks can be reduced to subsumption reasoning

Subsumption can be reduced to satisfiability

Tableau Algorithm

Tableau Algorithm is the de facto standard reasoning algorithm used in DLBasic intuitions

Reduces a reasoning problem to concept satisfiability problemFinds an interpretation that satisfies concepts

in question.The interpretation is incrementally constructed

as a "Tableau"

Short Example

given: WifeWoman, WomanPerson question: if WifePerson

Reasoning processTest if there is a individual that is a Woman but not a Person, i.e. test the satisfiability of concept C0=(WifeЬPerson)

C0(x) -> Wife(x), (¬Person)(x)

Wife(x)->Woman(x)

Woman(x) ->Person(x)

Conflict!

C0 is unsatisfiable, therefore WifePerson is true with the given ontology.

General Process

Transform C into negation normal form(NNF), i.e. negation occurs only in front of concept names.

Denote the transformed expression as C0, the algorithm starts with an ABox A0 = {C0(x0)}, and apply consistency-preserving transformation rules (tableaux expansion) to the ABox as far as possible.

If one possible ABox is found, C0 is satisfiable.

If not ABox is found under all search pathes, C0 is unsatisfiable.

NNF

Tableaux Expansion(Selected)

Clash

Termination Rules

An ABox is called complete if none of the expansion rules applies to it.

An ABox is called consistent if no logic clash is found.

If any complete and consistent ABox is found, the initial ABox A0 is satisfiable

The expansion terminates, either when finds a complete and consistent ABox, or try all search pathes ending with complete but inconsistent ABoxes.

Internalisation

Embed the TBox in the initial ABox concept

CD is equivalent T¬C U D (T is the "top" concept. It imeans ¬C U D is the super concept for ANY concepts)

E.g. Given ontology: Mother Woman Π Parent, Woman Person

Query: Mother Person

The intitial ABox is : ¬Mother U(Woman Π Parent) Π (¬Woman U Person) Π (Mother Π¬Person)

A Expansion Example

Search

Tree Model

Another explanation of tableaux algorithm is that it works on a finite completion tree whose

individuals in the tableau correspond to nodes

and whose interpretation of roles is taken from the edge labels.

Requirments for Tab. Alg.

Similar tableaux expansions can be designed for more expressive DL languages.

A tableau algorithm has to meet three requirements

Soundness: if a complete and clash-free ABox is found by the algorithm, the ABox must satisfies the initial concept C0.

Completeness: if the initial concept C0 is satisfiable, the algorithm can always find an complete and clash-free ABox

Termination: the algorithm can terminate in finite steps with specific result.

Roadmap

What is Description Logic (DL)

Semantics of DL

Basic Tableau Algorithm

Advanced Tableau Algorithm

Advanced Tableau Alg.

Rich literatures in the past decade.

Advanced techniquesBlocking (Subset Blocking,Pair Locking, Dynamic Blocking)

For more expressive languages: number restriction, transitive role, inverse role, nomial, data type

Detailed analysis of complexities.

Refer to references at the end of this presentation for details

SHIQ Expansion Rules

References

F. Baader, W. Nutt. Basic Description Logics. In the Description Logic Handbook, edited by F. Baader, D. Calvanese, D.L. McGuinness, D. Nardi, P.F. Patel-Schneider, Cambridge University Press, 2002, pages 47-100.

Ian Horrocks and Ulrike Sattler. Description Logics Tutorial, ECAI-2002, Lyon, France, July 23rd, 2002.

Ian Horrocks and Ulrike Sattler. A tableaux decision procedure for SHOIQ. In Proc. of the 19th Int. Joint Conf. on Artificial Intelligence (IJCAI 2005), 2005.

I. Horrocks and U. Sattler. A description logic with transitive and inverse roles and role hierarchies. Journal of Logic and Computation, 9(3):385-410, 1999.