Horrocks - Reasoning With Expressive Description Logics

8/8/2019 Horrocks - Reasoning With Expressive Description Logics

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Reasoning with Expressive Description Logics

Logical Foundations for the Semantic Web

Ian Horrocks

[email protected]

University of Manchester

Manchester, UK

Reasoning with Expressive Description Logics p. 1/


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Talk Outline



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Talk Outline

Introduction to Description Logics



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Talk Outline


The Semantic Web: Killer App for (DL) Reasoning?Web Ontology Languages

DAML+OIL Language



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Talk Outline



DAML+OIL Language

Reasoning with DAML+OIL

OilEd Demo



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Talk Outline



DAML+OIL Language


OilEd Demo

Description Logic Reasoning



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Talk Outline



DAML+OIL Language


OilEd Demo

Description Logic Reasoning

Research Challenges



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What are Description Logics?



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A family of logic based Knowledge Representation formalisms

Descendants of semantic networks and KL-ONE

Describe domain in terms of concepts (classes), roles(relationships) and individuals



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A family of logic based Knowledge Representation formalisms

Descendants of semantic networks and KL-ONE

Describe domain in terms of concepts (classes), roles(relationships) and individuals

Distinguished by:

Formal semantics (model theoretic) Decidable fragments of FOL Closely related to Propositional Modal & Dynamic Logics

Provision of inference services Sound and complete decision procedures for key problems Implemented systems (highly optimised)



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DL Architecture

Tbox (schema)

Abox (data)

Knowledge Base

InferenceSy

stem

Interface

Man.

= Human Male

Happy-Father.

= Man has-child.Female . . ..

.

.

.

.

.

John : Happy-Father

John, Mary : has-child



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Short History of Description Logics



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Phase 1:

Incomplete systems (Back, Classic, Loom, . . . )

Based on structural algorithms



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Phase 1:



Phase 2:

Development of tableau algorithms and complexity results

Tableau-based systems for Pspace logics (e.g., Kris, Crack)

Investigation of optimisation techniques



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Phase 1:



Phase 2:

Development of tableau algorithms and complexity results

Tableau-based systems for Pspace logics (e.g., Kris, Crack)

Investigation of optimisation techniquesPhase 3:

Tableau algorithms for very expressive DLs

Highly optimised tableau systems for ExpTime logics (e.g.,

FaCT, DLP, Racer)

Relationship to modal logic and decidable fragments of FOL



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Latest Developments

Phase 4:



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Latest Developments

Phase 4:

Mature implementations



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Latest Developments

Phase 4:


Mainstream applications and Tools

Databases Consistency of conceptual schemata (EER, UML etc.) Schema integration Query subsumption (w.r.t. a conceptual schema)

Ontologies and Semantic Web (and Grid) Ontology engineering (design, maintenance, integration) Reasoning with ontology-based markup (meta-data) Service description and discovery



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Latest Developments

Phase 4:


Mainstream applications and Tools

Databases Consistency of conceptual schemata (EER, UML etc.) Schema integration Query subsumption (w.r.t. a conceptual schema)

Ontologies and Semantic Web (and Grid) Ontology engineering (design, maintenance, integration) Reasoning with ontology-based markup (meta-data) Service description and discovery

Commercial implementations Cerebra system from Network Inference Ltd



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The Semantic Web



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The Semantic Web Vision



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Web made possible through established standards

TCP/IP for transporting bits down a wire

HTTP & HTML for transporting and rendering hyperlinked text



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Applications able to exploit this common infrastructure

Result is the WWW as we know it



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1st generation web mostly handwritten HTML pages



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2nd generation (current) web often machine generated/active



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Both intended for direct human processing/interaction



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In next generation web, resources should be more accessible to

automated processes



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In next generation web, resources should be more accessible toautomated processes

To be achieved via semantic markup

Metadata annotations that describe content/function



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2nd generation (current) web often machine generated/active Both intended for direct human processing/interaction




Coincides with Tim Berners-Lees vision of a Semantic Web



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2nd generation (current) web often machine generated/active Both intended for direct human processing/interaction




Coincides with Tim Berners-Lees vision of a Semantic Web


O


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Ontologies



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O t l i


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Ontologies

Semantic markup must be meaningful to automated processes

Ontologies will play a key role

Source of precisely defined terms (vocabulary)

Can be shared across applications (and humans)


O t l i


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Ontologies





Ontology typically consists of:

Hierarchical description of important concepts in domain

Descriptions of properties of instances of each concept


Ontologies


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Ontologies







Descriptions of properties of instances of each concept Degree of formality can be quite variable (NLlogic)


Ontologies


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Ontologies








Increased formality and regularity facilitates machine understanding


Ontologies


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Ontologies









Ontologies can be used, e.g.: To facilitate buyerseller communication in e-commerce

In semantic based search

To provide richer service descriptions that can be more flexibly

interpreted by intelligent agents


Ontologies


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Ontologies














Ontologies


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Ontologies














Ontologies


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Ontologies














Web Ontology Languages


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OIL and DAML-ONT web ontology languages developed inEuropean and DARPA projects




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Efforts merged to produce DAML+OIL




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Submitted to W3C as basis for standardisation WebOnt working group developing OWL language standard



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eb O to ogy a guages




DAML+OIL/OWL layered on top of RDFS

RDFS based syntax and ontological primitives (subclass etc.)

Adds much richer set of primitives (transitivity, cardinality, . . . )



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gy g g




DAML+OIL/OWL layered on top of RDFS

RDFS based syntax and ontological primitives (subclass etc.)

Adds much richer set of primitives (transitivity, cardinality, . . . )

Describes class/property structure of domain (Tbox)

E.g., Person subclass of Animal whose parents are all Persons

Uses RDF for class/property membership assertions (Abox)

E.g., john instance of Person; john,mary instance of parent


Logical Foundations of DAML+OIL


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g




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g

DAML+OIL equivalent to very expressive Description Logic



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More precisely, DAML+OIL is (extension of) SHIQ DL

DAML+OIL benefits from many years of DL research

Well defined semantics Formal properties well understood (complexity, decidability)

Known reasoning algorithms

Implemented systems (highly optimised)




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Implemented systems (highly optimised) DAML+OIL classes can be names (URIs) or expressions

Various constructors provided for building class expressions




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Implemented systems (highly optimised) DAML+OIL classes can be names (URIs) or expressions

Various constructors provided for building class expressions

Expressive power determined by

Kinds of constructor provided

Kinds of axiom allowed


DAML+OIL Class Constructors


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Constructor DL Syntax Example (Modal Syntax)

intersectionOf C1 . . . Cn Human Male C1 . . . CnunionOf C1 . . . Cn Doctor Lawyer C1 . . . Cn

complementOf C Male C

oneOf {x1 . . . xn} {john,mary} x1 . . . xntoClass P.C hasChild.Doctor [P]C

hasClass P.C hasChild.Lawyer PCmaxCardinalityQ nP.C 1hasChild.Male [P]n+1C

minCardinalityQ nP.C 2hasChild.Lawyer PnC




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XMLS datatypes as well as classes in P.C and P.C E.g., hasAge.nonNegativeInteger




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XMLS datatypes as well as classes in P.C and P.C E.g., hasAge.nonNegativeInteger

Arbitrarily complex nesting of constructors

E.g., Person hasChild.(Doctor hasChild.Doctor)


RDFS Syntax


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Semantics


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Semantics defined by interpretations: I = (I, I)

concepts subsets of I

roles binary relations over I (subsets of I I)

individuals elements of I


Semantics


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Semantics defined by interpretations: I = (I, I)

concepts subsets of I

roles binary relations over I (subsets of I I)

individuals elements of I

Interpretation function I extended to concept expressions

(C D)I = CI DI (C D)I = CI DI (C)I = I \ CI

{xn, . . . , xn}I = {xIn, . . . , xIn

}

(R.C)I = {x | y.(x, y) RI y CI}

(R.C)I = {x | y.x, y RI y CI}

(nR.C)I = {x | #{y | x, y RI y CI} n} (nR.C)I = {x | #{y | x, y RI y CI} n}


DAML+OIL Axioms


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DAML+OIL Axioms


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Axiom DL Syntax Example

subClassOf C1 C2 Human Animal Biped

sameClassAs C1 C2 Man Human Male

disjointWith C1 C2 Male FemalesameIndividualAs {x1} {x2} {President_Bush} {G_W_Bush

differentIndividualFrom {x1} {x2} {john} {peter}

subPropertyOf P1 P2 hasDaughter hasChild

samePropertyAs P1 P2 cost price

inverseOf P1 P

2 hasChild hasParent

transitiveProperty P+ P ancestor+ ancestor

uniqueProperty 1P 1hasMotherunambiguousProperty 1P 1hasSSN



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DAML+OIL Axioms


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Axiom DL Syntax Example

subClassOf C1 C2 Human Animal Biped

sameClassAs C1 C2 Man Human Male

disjointWith C1 C2 Male FemalesameIndividualAs {x1} {x2} {President_Bush} {G_W_Bush

differentIndividualFrom {x1} {x2} {john} {peter}

subPropertyOf P1 P2 hasDaughter hasChild

samePropertyAs P1 P2 cost price

inverseOf P1 P

2 hasChild hasParent

transitiveProperty P+ P ancestor+ ancestor

uniqueProperty 1P 1hasMotherunambiguousProperty 1P 1hasSSN

I satisfies C1 C2 iff CI1 CI2 ; satisfies P1 P2 iff P

I1 P

I2

I satisfies ontology O (is a model of O) iff satisfies every axiom in OReasoning with Expressive Description Logics p. 16/


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XML Datatypes in DAML+OIL


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DAML+OIL supports XML Schema datatypes

Primitive (e.g., decimal) and derived (e.g., integer sub-range)



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Clean separation between object classes and datatypes

Disjoint interpretation domain: dI

D, and D I

= Disjoint datatype properties: PI

D I D

Philosophical reasons:

Datatypes structured by built-in predicates Not appropriate to form new datatypes using ontology language




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Clean separation between object classes and datatypes

Disjoint interpretation domain: dI

D, and D I

= Disjoint datatype properties: PI

D I D

Philosophical reasons:

Datatypes structured by built-in predicates Not appropriate to form new datatypes using ontology language

Practical reasons:

Ontology language remains simple and compact

Semantic integrity of ontology language not compromised

Implementability not compromised can use hybrid reasoner Only need sound and complete decision procedure for

dI1 . . . dIn

, where di is a (possibly negated) datatype



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Reasoning


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Reasoning


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Why do we want it?

Semantic Web aims at machine understanding

Understanding closely related to reasoning

What can we do with it?

Design and maintenance of ontologies Check class consistency and compute class hierarchy Particularly important with large ontologies/multiple authors

Integration of ontologies Assert inter-ontology relationships Reasoner computes integrated class hierarchy/consistency

Querying class and instance data w.r.t. ontologies

Determine if set of facts are consistent w.r.t. ontologies Determine if individuals are instances of ontology classes Retrieve individuals/tuples satisfying a query expression Check if one description more general than another w.r.t.

ontology . . .


Basic Inference Problems


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Consistency check if knowledge is meaningful

Is O consistent? There exists some model I of O

Is C consistent? CI = in some model I of O




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Subsumption structure knowledge, compute taxonomy

CO D ? CI DI in all models I of O




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Equivalence check if two classes denote same set of instances

CO D ? CI

= DI

in all models I of O




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Equivalence check if two classes denote same set of instances

CO D ? CI

= DI

in all models I of O Instantiation check if individual i instance of class C

i O C? i CI in all models I of O



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Reasoning Support for Ontology Design: OilEd


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Tableaux Algorithms Basics


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T bl l ith d t t t ti fi bilit


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Tableaux algorithms used to test satisfiability

Try to build tree-like model I of input concept C

Work on concepts in negation normal form

Push in negation using de Morgans, R.C R.Cetc.



T bl l ith d t t t satisfiabilit


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Break down C syntactically, inferring constraints on elements of I





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Decomposition uses tableau rules corresponding to constructors in

logic (e.g., , ) Some rules are nondeterministic (e.g., , )

In practice, this means search



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Decomposition uses tableau rules corresponding to constructors in

logic (e.g., , ) Some rules are nondeterministic (e.g., , )

In practice, this means search

Stop when clash occurs or when no rules are applicable

Blocking (cycle check) used to guarantee termination



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Tableaux Algorithms Details


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Work on tree T representing model I of concept C


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p g p Nodes represent elements of I; labeled with subconcepts of C

Edges represent role-successorships between elements of I





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T initialised with single root node labeled {C}





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T initialised with single root node labeled {C}

Tableau rules repeatedly applied to node labels

Extend labels or extend/modify T structure

Rules can be blocked, e.g, if predecessor has superset label

Nondeterministic rules search possible extensions



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Tableaux Rules for ALC

! !

'

'

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x {C1 C2, C1, C2, . . .}x {C1 C2, . . .}


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for C {C1, C2}

x {C1 C2, C, . . .}

{ }

x {C1 C2, . . .}

{ }



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Tableaux Rule for Transitive Roles

e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e

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p

p

p

p

px

R

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R

x {R.C,.. .}

{. . .}

{R.C,.. .}

{R.C, . . .}

+

Where R is a transitive role (i.e., (RI)+ = RI)

No longer naturally terminating (e.g., if C= R.)

Need blocking

Simple blocking suffices for ALC plus transitive roles

I.e., do not expand node label if ancestor has superset label More expressive logics (e.g., with inverse roles) need more

sophisticated blocking strategies



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Tableaux Algorithm Example

Test satisfiability of S.C S.(C D) R.C R.(R.C)} where R isa transitive role


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w

L(w) = {S.C, S.(C D), R.C, R.(R.C)}



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w

L(w) = {S.C, S.(C D), R.C, R.(R.C)}

L(x) = {C} x

S





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w

L(w) = {S.C, S.(C D), R.C, R.(R.C)}

L(x) = {C} x

S



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w

L(x) = {C, C D} x

S

L

(w) = {S.C, S.(C D), R.C, R.(R.C)}



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w

L(x) = {C, C D} x

S

L

(w) = {S.C, S.(C D), R.C, R.(R.C)}



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w

x y L(y) = {C}L(x) = {C, (C D), D}

RS

L

(w

) = {S.C,

S.

(C

D

),

R.C,

R.

(R.C

)}



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w

x y L(y) = {C, R.C, R.(R.CL(x) = {C, (C D), D}

RS

L(w) = {S.C, S.(C D), R.C, R.(R.C)}





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w


zL(z) = {C}

RS

R

L(w) = {S.C, S.(C D), R.C, R.(R.C)}





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w


zL(z) = {C}

RS

R

L(w) = {S.C, S.(C D), R.C, R.(R.C)}





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w


zL(z) = {C, R.C, R.(R.C

RS

R

L(w) = {S.C, S.(C D), R.C, R.(R.C)}





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w


zL(z) = {C, R.C, R.(R.C

RS

R

L(w) = {S.C, S.(C D), R.C, R.(R.C)}

blocked





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w


zL(z) = {C, R.C, R.(R.C

RS

R

L(w) = {S.C, S.(C D), R.C, R.(R.C)}

blocked

Concept is satisfiable: T corresponds to model





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w


RS

L(w) = {S.C, S.(C D), R.C, R.(R.C)}

R

Concept is satisfiable: T corresponds to model


More Advanced Techniques


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Satisfiability w.r.t. a Terminology For each axiom C D T, add C D to every node label


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More expressive DLs


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p




More expressive DLs


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Basic technique can be extended to deal with

Role inclusion axioms (role hierarchy) Number restrictions Inverse roles Concrete domains and datatypes

Aboxes etc.




More expressive DLs


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Basic technique can be extended to deal with

Role inclusion axioms (role hierarchy) Number restrictions Inverse roles Concrete domains and datatypes

Aboxes etc.

Extend expansion rules and use more sophisticated blockingstrategy



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Highly Optimised Implementation


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Naive implementation effective non-termination


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Naive implementation

effective non-termination

Modern systems include MANY optimisations

Optimised classification (compute partial ordering)


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Use enhanced traversal (exploit information from previous tests)

Use structural information to select classification order



Naive implementation

effective non-termination

Modern systems include MANY optimisations

Optimised classification (compute partial ordering)


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Use enhanced traversal (exploit information from previous tests)

Use structural information to select classification order

Optimised subsumption testing (search for models)

Normalisation and simplification of concepts

Absorption (simplification) of general axioms Davis-Putnam style semantic branching search

Dependency directed backtracking

Caching of satisfiability results and (partial) models

Heuristic ordering of propositional and modal expansion

. . .


Research Challenges


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Research Challenges



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Challenges

Increased expressive power

Existing DL systems implement (at most) SHIQ

DAML+OIL extends SHIQ with datatypes and nominals


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Challenges




Scalability


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Scalability

Very large KBs Reasoning with (very large numbers of) individuals


Challenges




Scalability


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Scalability


Other reasoning tasks

Querying Matching

Least common subsumer

. . .


Challenges




Scalability


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Scalability


Other reasoning tasks

Querying Matching

Least common subsumer

. . .

Tools and Infrastructure

Support for large scale ontological engineering and deployment



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Increased Expressive Power: Datatypes

DAML+OIL has simple form of datatypes

Unary predicates plus disjoint object-class/datatype domains


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Well understood theoretically


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y

Existing work on concrete domains [Baader & Hanschke, Lutz] Algorithm already known for SHOQ(D) [Horrocks & Sattler]

Can use hybrid reasoning (DL reasoner + datatype oracle)





Well understood theoretically


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Existing work on concrete domains [Baader & Hanschke, Lutz] Algorithm already known for SHOQ(D) [Horrocks & Sattler]

Can use hybrid reasoning (DL reasoner + datatype oracle)

May be practically challenging

All XMLS datatypes supported (?)



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Increased Expressive Power: Nominals


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DAML+OIL oneOf constructor equivalent to hybrid logic nominals

Extensionally defined concepts, e.g., EU {France, Italy, . . .}


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DAML+OIL oneOf constructor equivalent to hybrid logic nominals

Extensionally defined concepts, e.g., EU {France, Italy, . . .}

Theoretically very challenging

Resulting logic has known high complexity (NExpTime)


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No known practical algorithm Not obvious how to extend tableaux techniques in this direction

Loss of tree model property Spy-points: R.{Spy}

Finite domains: {Spy} nR

?? automata based algorithms ??



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Scalability


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Scalability

Reasoning hard (ExpTime) even without nominals (i.e., SHIQ)

Web ontologies may grow very large


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Scalability



Good empirical evidence of scalability/tractability for DL systems

E.g., 5,000 (complex) classes; 100,000+ (simple) classes


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But evidence mostly w.r.t. SHF (no inverse)


Scalability



Good empirical evidence of scalability/tractability for DL systems

E.g., 5,000 (complex) classes; 100,000+ (simple) classes


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But evidence mostly w.r.t. SHF (no inverse)

Problems can arise when SHF extended to SHIQ

Important optimisations no longer (fully) work



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Other Reasoning Tasks

Querying

Retrieval and instantiation wont be sufficient

Minimum requirement will be DB style query language

M l d h t I b t ? t l f


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May also need what can I say about x? style of query Explanation

To support ontology design

Justifications and proofs (e.g., of query results)



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Summary


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Summary

Description Logics are family of logical KR formalisms

Applications of DLs include DataBases and Semantic Web

Ontologies will provide vocabulary for semantic markup

DAML+OIL web ontology language based on SHIQ DL

Set to become W3C standard (OWL) & already widely adopted


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Set to become W3C standard (OWL) & already widely adopted Use of DL provides formal foundations and reasoning support

DL Reasoning based on tableau algorithms

Highly Optimised implementations used in DL systems


Summary

Description Logics are family of logical KR formalisms

Applications of DLs include DataBases and Semantic Web

Ontologies will provide vocabulary for semantic markup

DAML+OIL web ontology language based on SHIQ DL

Set to become W3C standard (OWL) & already widely adopted


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Set to become W3C standard (OWL) & already widely adopted Use of DL provides formal foundations and reasoning support

DL Reasoning based on tableau algorithms

Highly Optimised implementations used in DL systems Challenges remain

Reasoning with full DAML+OIL/OWL language

(Convincing) demonstration(s) of scalability

New reasoning tasks

Development of (high quality) tools and infrastructure


Acknowledgements


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Acknowledgements

Members of the OIL and DAML+OIL development teams, in

particular Dieter Fensel and Frank van Harmelen (Amsterdam) andPeter Patel-Schneider (Bell Labs)


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Acknowledgements



Franz Baader, Uli Sattler and Stefan Tobies (Dresden)


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Acknowledgements



Franz Baader, Uli Sattler and Stefan Tobies (Dresden)

Members of the Information Management, Medical Informatics andFormal Methods Groups at the University of Manchester


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e be s o t e o at o a age e t, ed ca o at cs a dFormal Methods Groups at the University of Manchester



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Select Bibliography

I. Horrocks. DAML+OIL: a reason-able web ontology language. In Proc. of

EDBT 2002, number 2287 in Lecture Notes in Computer Science, pages213. Springer-Verlag, Mar. 2002.

I. Horrocks, P. F. Patel-Schneider, and F. van Harmelen. Reviewing thedesign of DAML+OIL: An ontology language for the semantic web. In Proc.of AAAI 2002, 2002. To appear.


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of AAAI 2002, 2002. To appear.

I. Horrocks and S. Tessaris. Querying the semantic web: a formalapproach. In I. Horrocks and J. Hendler, editors, Proc. of the 2002

International Semantic Web Conference (ISWC 2002), number 2342 inLecture Notes in Computer Science. Springer-Verlag, 2002.

C. Lutz. The Complexity of Reasoning with Concrete Domains. PhDthesis, Teaching and Research Area for Theoretical Computer Science,

RWTH Aachen, 2001.


Select Bibliography

I. Horrocks and U. Sattler. Ontology reasoning in the SHOQ(D)

description logic. In B. Nebel, editor, Proc. of IJCAI-01, pages 199204.Morgan Kaufmann, 2001.

F. Baader, S. Brandt, and R. Ksters. Matching under side conditions indescription logics. In B. Nebel, editor, Proc. of IJCAI-01, pages 213218,Seattle, Washington, 2001. Morgan Kaufmann.


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Seattle, Washington, 2001. Morgan Kaufmann.

A. Borgida, E. Franconi, and I. Horrocks. Explaining ALC subsumption. InProc. of ECAI 2000, pages 209213. IOS Press, 2000.

D. Calvanese, G. De Giacomo, M. Lenzerini, D. Nardi, and R. Rosati. Aprincipled approach to data integration and reconciliation in datawarehousing. In Proceedings of the International Workshop on Designand Management of Data Warehouses (DWDM99), 1999.


Documents

Horrocks - Reasoning With Expressive Description Logics