Knowledge Representation Knowledge Representation and Reasoning using and Reasoning using

Description LogicDescription Logic

PresenterPresenterShamima MithunShamima Mithun

OverviewOverview Introduction to Description Logics (DL)Introduction to Description Logics (DL) DL Syntax and SemanticDL Syntax and Semantic DL Systems ArchitectureDL Systems Architecture Reasoning TechniquesReasoning Techniques KB Construction and Access using PowerLoomKB Construction and Access using PowerLoom

IntroductionIntroduction Description Logics are a family of logic based knowledge Description Logics are a family of logic based knowledge

representation formalisms based on representation formalisms based on conceptsconcepts and and rolesroles:: ConceptsConcepts (classes) are interpreted as sets of (classes) are interpreted as sets of

objects objects e.g. person.e.g. person. Roles Roles are interpreted as binary relations on are interpreted as binary relations on

objects objects e.g. has-child.e.g. has-child. Key featuresKey features of DLs are of DLs are

a well defined semantics a well defined semantics Inference servicesInference services

BackgroundBackgroundDescription Logics:Description Logics: Introduced to resolve the ambiguities of Introduced to resolve the ambiguities of Semantic Semantic

NetworksNetworks Descendants of Semantic Networks, Descendants of Semantic Networks, frame based frame based

systemssystems and and KL-ONEKL-ONE Decidable fragment of FOLDecidable fragment of FOL [2] [2] Useful for Semantic Web Language (DAML +OIL) [4]Useful for Semantic Web Language (DAML +OIL) [4]

DL Applications & SystemsDL Applications & SystemsApplication AreasApplication Areas Terminological KR and OntologiesTerminological KR and Ontologies Semantic WebSemantic Web Software Information SystemsSoftware Information Systems Database ApplicationsDatabase Applications

SystemsSystems LOOM/PowerLoom, RACER, CLASSIC, and othersLOOM/PowerLoom, RACER, CLASSIC, and others

DL ConstructorsDL Constructors

DLs are characterized by a DLs are characterized by a set of constructorsset of constructors. . These are used to construct complex concepts These are used to construct complex concepts and roles from simple ones.and roles from simple ones.

Common constructors:Common constructors:

ConjunctionConjunction ((ПП), ), disjunction disjunction ((),), negationnegation (())

Restricted forms of quantification (Restricted forms of quantification (, , ) ) Example in ALC: the concept Happy-FatherExample in ALC: the concept Happy-Father

Happy-FatherHappy-Father ManMan ПП has- child. Malehas- child. Male

DL Syntax and SemanticDL Syntax and Semantic

Figure is taken from Ian Horrocks [2]Ian Horrocks [2]

Other DL ConstructorsOther DL Constructors

DL and Other Logical DL and Other Logical Formalism: FOPLFormalism: FOPL

Syntactic feature of DL: Syntactic feature of DL: variable free notationvariable free notation. . Most DLs are fragments of FOL, e.g. ACL.Most DLs are fragments of FOL, e.g. ACL. ACL expressions can be translated into FOL: ACL expressions can be translated into FOL:

A unary predicate A unary predicate ΦΦA A is introducedis introduced for each concept C, for each concept C, and a binary relation and a binary relation ρρRR for each role R. for each role R.

Translation ACL →FOL:Translation ACL →FOL:

artist artist ПП (( CREATES. song ) → CREATES. song ) →

xx y: artist ( x ) y: artist ( x ) ΛΛ (CREATES ( x, y ) (CREATES ( x, y ) ΛΛ song ( y )) song ( y ))

Why not use FOL?Why not use FOL?The expressive power is too high for having good The expressive power is too high for having good computational properties and efficient inference computational properties and efficient inference procedures.procedures.

DL and Other Logical Formalism:DL and Other Logical Formalism: Modal LogicsModal Logics

DLs are notational DLs are notational variants of Modal Logicsvariants of Modal Logics ALC ALC multi-modal K: multi-modal K:

C C ПП D D C C ΛΛ D, D, C C D D C C νν D D C C C, C, R.C R.C <R>C, <R>C, R.C R.C [R]C [R]C

transitive roles transitive roles transitive frames (e.g., K4) transitive frames (e.g., K4) inverse roles inverse roles converse programs (e.g., C-PDL) converse programs (e.g., C-PDL) number restrictions number restrictions deterministic programs (e.g., D-PDL) deterministic programs (e.g., D-PDL)

No TBoxesNo TBoxes available in modal logics: available in modal logics:internalise" axioms using a universal role u: C internalise" axioms using a universal role u: C D D [u] (C [u] (CD)D)

No ABoxesNo ABoxes available in modal logics: Use nominals available in modal logics: Use nominals

DL Systems ArchitectureDL Systems Architecture

Knowledge BaseKnowledge Base

TBox (schema)TBox (schema)

Man ≡ Human ПП Male

Happy-Father ≡ Man П П has-child. Female

ABox (data)ABox (data)

John : Happy-Father

John, Mary> : has-child

Infe

ren

ce S

yste

m

Inte

rface

Inte

rface

Figure after Horrocks [3] Figure after Horrocks [3]

DL TBoxDL TBox

DL ABoxDL ABox

Knowledge to ReasoningKnowledge to Reasoning

Knowledge BaseKnowledge Base

TBoxTBox

ABoxABox

ReasoningReasoning

Reasoning about theReasoning about the

knowledgeknowledge Add new knowledge to Add new knowledge to

the KB that follows the KB that follows logically. logically.

Ask KB if a statement is Ask KB if a statement is valid.valid.

Figure is taken from Ian Horrocks [3]Ian Horrocks [3]

Reasoning / InferenceReasoning / InferenceBasic Inference Problems, Basic Inference Problems, for TBox T:

Consistency:Consistency:

““A concept C is consistent with respect to T, if there exists A concept C is consistent with respect to T, if there exists a model a model II of T with C of T with CII . [. [II is a model of C]”. is a model of C]”.

Inconsistent:Inconsistent:

songwriter songwriter artist artist ПП ((CREATES. song )CREATES. song )

song song IS _ CREATED _ BY. songwriter IS _ CREATED _ BY. songwriter

Subsumption:Subsumption:

““A concept C is subsumed by a concept D with respect to A concept C is subsumed by a concept D with respect to T if CT if CII D DII for every model for every model II of T”.of T”.

male male personperson

Definitions taken from [6], p. 66.Definitions taken from [6], p. 66.

ClassificationClassification““ClassificationClassification is the task ofis the task ofinserting new concepts or inserting new concepts or instances in a taxonomy” [ 3].instances in a taxonomy” [ 3].

Classification of conceptsClassification of concepts allows to structure the allows to structure the

terminology in the form of a terminology in the form of a subsumption hierarchysubsumption hierarchy

Classification of instances Classification of instances determines whether an determines whether an

individual is an instance of a individual is an instance of a certain concept.certain concept.

Fixed-Furniture

Movable-Furniture

Furniture

Door Window Chair Sofa

Is-a

Reasoning AlgorithmsReasoning Algorithms Structural subsumption algorithms

Subsumption of concepts can be computed.Subsumption of concepts can be computed. They are complete for simple languages with little They are complete for simple languages with little

expressivity.expressivity. Used by KL-ONE, LOOM and other systems.Used by KL-ONE, LOOM and other systems.

Tableau–based algorithms Turned out to be very efficient reasoning algorithms.Turned out to be very efficient reasoning algorithms. Sound, complete and decidable.Sound, complete and decidable. Used e.g. in RACER.Used e.g. in RACER.

Structural Subsumption Structural Subsumption AlgorithmAlgorithm

Proceed in two phasesProceed in two phases

1)1) The descriptions to be tested for subsumption are The descriptions to be tested for subsumption are normalized.normalized.

2)2) Then the syntactic structure of the normal forms is Then the syntactic structure of the normal forms is compared with each other.compared with each other.

An FLAn FLoo- concept description is in - concept description is in normal formnormal form iff iff it is of the form it is of the form

AA11 ПП… … ППAAmm ПП RR11.C.C11 ПП… … П П RRnn.C.Cnn

Where Where AA11,.., ,.., AAmm are distinct concept names, are distinct concept names, RR11,..., R,..., Rn n are are

distinct roles names, and Cdistinct roles names, and C11,…,…, , CCn n are concept are concept

descriptions in normal from.descriptions in normal from.

Structural Subsumption Structural Subsumption

AlgorithmAlgorithm (contd.)(contd.)

Given is the normal form of the concept description CGiven is the normal form of the concept description C

AA11 ПП… … ППAAmm ПП RR11.C.C11 ПП… … П П RRnn. C. Cnn

and the normal form of the concept description Dand the normal form of the concept description D

BB11 ПП… … ППBBkk ПП SS11.D.D11 ПП… … П П SSll. D. Dll

D subsumes C: C D iff

i, 1≤ i ≤k, j, 1 ≤j ≤m: Bi = Aj

i, 1≤ i ≤l, j, 1 ≤j ≤n: Si = Rj and Cj Di

Tableau-based AlgorithmsTableau-based Algorithms Construct a Construct a modelmodel for the input concept description for the input concept description

CC0.0.

Model is represented by Model is represented by tree formtree form.. The formula has been translated into The formula has been translated into Negation Negation

Normal FormNormal Form (NNM). (NNM). To decide satisfiability of the concept CTo decide satisfiability of the concept C0 0 , start with , start with

the initial tree (root node).the initial tree (root node). Repeatedly apply expansion rules until find Repeatedly apply expansion rules until find

contradiction or expansion completed.contradiction or expansion completed. Satisfiable Satisfiable iff iff a complete and contradiction-free tree a complete and contradiction-free tree

is found.is found.

Tableau-based Algorithms - Tableau-based Algorithms - ExampleExample

Determine the satisfiability of the concept-definition:Determine the satisfiability of the concept-definition:

( (( ( CHILD. Male ) CHILD. Male ) ПП ( ( CHILD. CHILD. Male ) ) Male ) )

( (( ( CHILD. Male ) CHILD. Male ) ПП ( ( CHILD. CHILD. Male ) ) <x> Male ) ) <x>(( CHILD. Male ) <x> CHILD. Male ) <x> ПП -rule -rule(( CHILD. CHILD. Male ) <x> Male ) <x> ПП –rule –ruleCHILD <x, y > CHILD <x, y > -rule -rule Male < y> Male < y> -rule -ruleMale <y > Male <y > -rule -rule<CLASH ><CLASH >

ReasoningReasoning (contd.)(contd.)

Reasoning services like subsumption and consistencyReasoning services like subsumption and consistency Speed-upSpeed-up the inference procedures for query. the inference procedures for query. Help to infer implicitly represented knowledge from the Help to infer implicitly represented knowledge from the

explicitly contained knowledge of KB.explicitly contained knowledge of KB.

T-BoxT-Box A-BoxA-Box

Female Female Male Male Human Human Mary: MotherMary: MotherMother Mother Female Female John: FatherJohn: FatherFather Father Male Male Mary: Mary: parent.Childparent.ChildChild Child has.Mother has.Mother ПП has.Father has.Father John: John: parent.Childparent.Child

Able to deduce implicit knowledge, like Mary is a Human.Able to deduce implicit knowledge, like Mary is a Human.

Reasoning: Decidability vs. Reasoning: Decidability vs. ExpressivityExpressivity

KR system should KR system should answer queries in a reasonable time.answer queries in a reasonable time. The reasoning procedures should terminate.The reasoning procedures should terminate.

Trade-off between the Trade-off between the expressivity of DLsexpressivity of DLs and the and the complexity of their reasoningcomplexity of their reasoning.. Very expressive DLs can have inference problems of Very expressive DLs can have inference problems of

high complexity, they may even be undecidable.high complexity, they may even be undecidable. Very Weak DLs my not be sufficiently expressive to Very Weak DLs my not be sufficiently expressive to

represent the important concepts of an application. represent the important concepts of an application. Quest for a highly expressive DL with decidable/ Quest for a highly expressive DL with decidable/

practical inference problems.practical inference problems.

ConclusionConclusion DL are logic based knowledge representation formalisms.DL are logic based knowledge representation formalisms. DL systems provide efficient inference services like DL systems provide efficient inference services like

consistency, subsumption.consistency, subsumption. DLs are effective in a range of applications.DLs are effective in a range of applications.

PowerLoomPowerLoom

PowerLoom CommandsPowerLoom Commands Defines concept, relation, function and rules using Defines concept, relation, function and rules using

defconcept, defrelation, deffunction, defrule.defconcept, defrelation, deffunction, defrule. Add/Remove facts from KB withAdd/Remove facts from KB with

assert and retractassert and retract Query KBQuery KB

ask, retrieveask, retrieve

NoteNote: Relations have to defined, before they are used in : Relations have to defined, before they are used in assertions or queries.assertions or queries.

Model DomainModel Domain

Color

Fixed-Furniture

Movable-Furniture

Furniture

Age

Size

BigSmall

GreenRed NewOld

IS-A

Door Window Chair Sofa

Define ConceptDefine Concept|=(defconcept furniture) |=(defconcept furniture) |c| FURNITURE|c| FURNITURE

|= (defconcept movable-furniture (?f furniture))|= (defconcept movable-furniture (?f furniture))|c| MOVABLE-FURNITURE|c| MOVABLE-FURNITURE

|= (defconcept fixed-furniture (?f furniture))|= (defconcept fixed-furniture (?f furniture))|c| FIXED-FURNITURE|c| FIXED-FURNITURE

|=(defconcept chair (?f movable-furniture)) |=(defconcept chair (?f movable-furniture)) |c| CHAIR|c| CHAIR

|= (defconcept sofa (?f movable-furniture)) |= (defconcept sofa (?f movable-furniture)) |c| SOFA|c| SOFA

Define ConceptDefine Concept (contd.)(contd.)

|= (defconcept window (?ff fixed-furniture))|= (defconcept window (?ff fixed-furniture))|c| WINDOW|c| WINDOW

|= (defconcept door (?ff fixed-furniture))|= (defconcept door (?ff fixed-furniture))|c| DOOR|c| DOOR

|= (defconcept age (?a) :<=> (member-of ?a (setof new old))) |= (defconcept age (?a) :<=> (member-of ?a (setof new old))) |c| AGE|c| AGE

|= (defconcept color (?c) :<=> (member-of ?c (setof green red |= (defconcept color (?c) :<=> (member-of ?c (setof green red blue)))blue)))

|c| COLOR|c| COLOR

|= (defconcept size (?s) :<=> (member-of ?s (setof small big)))|= (defconcept size (?s) :<=> (member-of ?s (setof small big)))|c| SIZE|c| SIZE

Model DomainModel Domain

Color

Fixed-Furniture

Movable-Furniture

Furniture

Age

Size

BigSmall

GreenRed NewOld

IS-A

Door Window Chair Sofa

has-colorhas-age

has-size

Define Relation and Define Relation and FunctionFunction

|= (defrelation has-age ((?f furniture) (?a age))) |= (defrelation has-age ((?f furniture) (?a age))) |r| HAS-AGE|r| HAS-AGE

|= (defrelation has-color ((?f furniture) (?c color))) )|= (defrelation has-color ((?f furniture) (?c color))) )|r| HAS-COLOR|r| HAS-COLOR

|= (defrelation has-size ((?f furniture) (?s size)))|= (defrelation has-size ((?f furniture) (?s size)))|r| HAS-SIZE|r| HAS-SIZE

|= (deffunction has-price ((?f furniture)):-> (?n INTEGER))|= (deffunction has-price ((?f furniture)):-> (?n INTEGER))|r| HAS-PRICE|r| HAS-PRICE

Model DomainModel Domain

Color

Fixed-Furniture

Movable-Furniture

Furniture

Age

Size

has-size

has-age

has-color

BigSmall

GreenRed NewOld

IS-A

Door Window Chair Sofa

Value AssertionValue Assertion|= (assert (has-size chair big))|= (assert (has-size chair big))|P|(HAS_SIZE CHAIR BIG) |P|(HAS_SIZE CHAIR BIG)

|= (assert (has-color sofa red))|= (assert (has-color sofa red))|P|(HAS_COLOR SOFA RED) |P|(HAS_COLOR SOFA RED)

|= (assert (has-age chair new))|= (assert (has-age chair new))|P|(HAS_AGE CHAIR NEW) |P|(HAS_AGE CHAIR NEW)

|= (assert (has-age door old))|= (assert (has-age door old))|P|(HAS_AGE DOOR OLD) |P|(HAS_AGE DOOR OLD)

|= (assert (forall (?x ?y) (=> (has-color ?x ?y) (and (furniture ?x) |= (assert (forall (?x ?y) (=> (has-color ?x ?y) (and (furniture ?x) (color ?y)))))(color ?y)))))

Query the KB: Retrieve, Query the KB: Retrieve, Retract, and AskRetract, and Ask

|=(retrieve (has-color sofa ?x))|=(retrieve (has-color sofa ?x))There is 1 solution so far:There is 1 solution so far: #1: ?X=RED#1: ?X=RED

|= (retract (has-color sofa red)) |= (retract (has-color sofa red)) |P?|(HAS-COLOR SOFA RED) |P?|(HAS-COLOR SOFA RED)

|= (retrieve (has-color sofa ?x))|= (retrieve (has-color sofa ?x)) No solutions. No solutions.

|= (ask (has-size chair big))|= (ask (has-size chair big))TRUETRUE

|= (ask (has-color door red))|= (ask (has-color door red))UNKNOWNUNKNOWN

ReferencesReferences1.1. PowerLoom HomepagePowerLoom Homepage

http://www.isi.edu/isd/LOOM/PowerLoom/http://www.isi.edu/isd/LOOM/PowerLoom/2. 2. Ian Horrocks and Ulrike Sattler: Ian Horrocks and Ulrike Sattler: Description Logics - Basics, Description Logics - Basics,

Applications, and More. Applications, and More. Aavilable atAavilable at http://www.cs.man.ac.ukhttp://www.cs.man.ac.uk3.3. Ian Horrocks: Ian Horrocks: Reasoning with Expressive Description Logics: Reasoning with Expressive Description Logics:

Theory and Practice. Available at: Theory and Practice. Available at: http://www.cs.man.ac.uk/~horrocks/Slideshttp://www.cs.man.ac.uk/~horrocks/Slides

4.4. Christel Kemke. Lecture Notes on Artificial Intelligence. Christel Kemke. Lecture Notes on Artificial Intelligence. Available at http://www.cs.umanitoba.ca/~ckemke/74.419-AI/Available at http://www.cs.umanitoba.ca/~ckemke/74.419-AI/

5.5. Daniele Nardi and Ronald J. Brachman. An introduction to Daniele Nardi and Ronald J. Brachman. An introduction to Description Logics. In Baader, Calvanese, McGuinnes Nardi Description Logics. In Baader, Calvanese, McGuinnes Nardi and Patel-Schneider, (eds). The Description Logics Handbook, and Patel-Schneider, (eds). The Description Logics Handbook, chapter 1. Cambridge University Press, 2003.chapter 1. Cambridge University Press, 2003.

6.6. Franz Baader and Werner Nutt. Franz Baader and Werner Nutt. Basic Description Logics. In Basic Description Logics. In Baader, Calvanese, McGuinnes Nardi and Patel-Schneider, Baader, Calvanese, McGuinnes Nardi and Patel-Schneider, (eds). The Description Logics Handbook, chapter 2. (eds). The Description Logics Handbook, chapter 2. Cambridge University Press, 2003.Cambridge University Press, 2003.

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