Intelligence Artiﬁcielle Introduction to Description Logics · Database theory: one can translate...

Intelligence ArtificielleIntroduction to Description Logics

Master Recherche, Marseille

Nicola OlivettiINCA-LSIS

Universite Paul Cezanne

2009-10

Intelligence ArtificielleIntroduction to Description Logics – p. 1/90

Plan of The Course

History and Motivation of Description Logics

The language ALCN

Relations with first-order logic

Structure and properties of TBox

Structure and properties of ABox

Reasoning problems and Mechanisms

Extensions of ALCN

Relation with OWL and semantic web languages


Main References:

F. Baader, D. Calvanese, D. L. McGuinness, D. Nardiand P. F. Patel-Schneider (eds.)The Description Logic Handbook: Theory,Implementation, and Applications.Cambridge University Press, 2003.

DL, web resources: http://dl.kr.org/

OWL, web resources:http://www.w3.org/TR/owl-features/

(Various authors) An informal description of OIL-Coreand Standard OIL: a layered proposal for DAML-O.Posted athttp://www.ontoknowledge.org/oil/downl/dialects.pdf.


Some history: past and present

Description logics: a logical formalism fordefining Concepts and their Relations (Terminologies )specifying properties of individuals (Assertions )

Main advantages:well-defined semantic: subset of first-order logicfeasibility: decidability and efficient reasoningalgorithmsflexibility: DL-systems form a hierarchy of languagesof increasing expressivity (but also of increasingcomplexity)



The ancestors of DL are the languages of AI for knowledgerepresentation:

Semantic networks

Frame based systems (i.e. KEE)

The system KL-ONE (Brachman and Schmolze 1985)distinction between concept and rolesclear semantic based on first-order logica reacher set of primitives than first order logic (thenotion of value and number restrictions on roles)



Today DL represent one the main language ofKnowledge representation, with applications in anumber of areas of computer science

Semantic web: DL are a foundation of ontologylanguages (OWL) which have been recentlystandarized by WC3 group

Software engineering: one can translate UML diagramsinto DL

Database theory: one can translate E/R-models into DL


Principles of DL in a nutshell

A Knowledge base KB is made of two parts:

TBox : collection of definitions of concepts and theirrelations, it may contain inclusion relations betweenconcepts

ABox : specification of properties of individuals, wherethe properties are those defined in the TBox


Principles of DL in a nutshell

There are primitive concepts (eg. "Female, Place,Product, Vehicle, Publication..." which are not defined,all other concept are defined ultimately in terms of theprimitive concepts

Further Information is extracted from a KB by inferencemechanisms:

subsumptionclassificationconsistencynon-emptiness of conceptsinference of individual properties


Basic elements of a TBox

Atomic concepts : denote subsets of the domain,

Roles : denotes binary relations on the domain (relationsbetween individuals), examples "HasChild, HasCost,HasPart, IsAuthorOf, Eats, Owns..."

Concepts constructors : operation for combining andbuilding complex concepts, example: boolean operators

Role constructors : operation for building complex roles,example: transitive closure

The description logics are a family of systems forming ahierarchy determined by the set of concept constructorsand role constructors admitted in the language


The basic System ALC

Notation:A,Ai . . .: atomic conceptsC,D,Ci . . . : arbitrary conceptsR,S,Ri . . .: primitive rolesno complex roles

Defined concepts:

C := ⊤ | ⊥ | A | ¬C | C ⊓D | C ⊔D | ∀R.C | ∃R.C


Examples

Atomic concepts: Person, Female,Rich, Roles:HasChild

Person ⊓ Female

Person ⊓ ¬Female

Person ⊓ ∃HasChild.⊤

Person ⊓ ∀HasChild.⊥

Person ⊓ ∃HasChild.⊤ ⊓ ∀HasChild.Female

Person ⊓ (Rich ⊔ ∃HasChild.Rich)

Person ⊓ ∃HasChild.∃HasChild.⊤


Semantic

An interpretation I is a pair I = 〈∆, I〉, where∆ is a non-empty setI : AtomicConcepts → Pow(∆)

I : Roles → Pow(∆2)

⊤I = ∆

⊥I = ∅

(¬C)I = ∆− CI

(C ⊓D)I = CI ∩DI

(C ⊔D)I = CI ∪DI


Semantic (continued)

(∃R.C)I = {a ∈ ∆ | ∃b ∈ ∆(a, b) ∈ RI ∧ b ∈ CI}

(∀R.C)I = {a ∈ ∆ | ∀b ∈ ∆(a, b) ∈ RI → b ∈ CI}

Semantic Equivalences:¬(C ⊓D) = ¬C ⊔ ¬D¬(C ⊔D) = ¬C ⊓ ¬D¬(∃R.C) = ∀R.¬C¬(∀R.C) = ∃R.¬C¬¬C = C

Negation Normal Form : by the equivalences every conceptdefinition can be transformed into a concept definitionwhere negation only occurs in front of atomic concepts.


Number Restrictions: ALCN

ALCN is the extension of ALC obtained by adding thefollowing concept constructors called number restrictions :

≤ n R, where R is a role and n ≥ 0.Reading "Entity that has at most n entities related via R"

≥ nR where R, where is a role and n ≥ 0.Reading "Entity that has at least n entities related via R"

Interpretation:(≤ n R)I = {a ∈ ∆s.t. | {b ∈ ∆(a, b) ∈ RI} |≤ n}

(≥ n R)I = {a ∈ ∆s.t. | {b ∈ ∆(a, b) ∈ RI} |≥ n}

one can also introduce = nR ≡ ≥ nR ⊓ ≤ nR


Examples and Exercises

Person ⊓ ≥ 3HasChild

Person ⊓ ≥ 1HasChild ⊓ ≤ 1HasChild

Exercise1:write "Course with at most 20 partecipants, all of whichare master or Ph.D students", (primitive concepts:Course, MasterStudent, PhDStudent, roles:HasPartecipant)

Exercise2:write "Grandmother with a daughter who has exactlytwo sons (males)"

Exercise3: Extend the Negation Normal Form toNumber Restrictions


Qualified Number Restrictions

Qualified Number Restrictions ≤ nR.C and ≥ nR.C

(≤ n R.C)I = {a ∈ ∆s.t. | {b ∈ ∆(a, b) ∈ RI ∧ b ∈ CI} |≤ n}

(≥ n R.C)I = {a ∈ ∆s.t. | {b ∈ ∆(a, b) ∈ RI ∧ b ∈ CI} |≥ n}

Qualified number restrictions with arbitrary cardinalitycannot be expressed by using simple number restrictions


First-Order Translation

To each concept C we can associate a first-order formulaφC(x), with one free-variable x:

for an atomic concept A, φA(x) = A(x)

for roles R, we have binary relation symbols R(x, y)

φ⊤(x) = P (x) ∨ ¬P (x) and φ⊥(x) = P (x) ∧ ¬P (x) whereP is a fixed predicate

φC⊓D(x) = φC(x) ∧ φD(x)

φC⊔D(x) = φC(x) ∨ φD(x)

φ¬C(x) = ¬φC(x)

φ∃R.C(x) = ∃y(R(x, y) ∧ φC(y))

φ∀R.C(x) = ∀y(R(x, y) → φC(y))


First-Order Translation (continued)

φ≥n R(x) = ∃y1 . . . yn(R(x, y1)∧. . .∧R(x, yn)∧∧

i<j yi 6= yj)

φ≤n R(x) = ∀y1...yn+1(R(x, y1) ∧ ... ∧ R(x, yn+1) →∨i<j yi = yj)

Exercise4: translate the following in first order logic:Person ⊓ ≤ 3HasChildPerson ⊓ ∃HasChild.∃HasChild.Female

Propositionfor all interpretations I = 〈∆, I〉:

CI = {a ∈ ∆ | I |= φC(a)}

Exercise5: prove the proposition aboveIntelligence ArtificielleIntroduction to Description Logics – p. 18/90

Modal Interpretation of ALC

ALC is nothing but a notational variant of multi-modallogic Km (K with m modalities)

we have modal operators �R,♦R for each role R

we define the modal translation Cm of a concept C asfollows:

Am = A (a propositional variable)

(C ⊓D)m = Cm ∧Dm

(C ⊔D)m = Cm ∨Dm

(¬C)m = ¬Cm

(∃R.C)m = ♦RCm

(∀R.C)m = �RCm



Let the language contain A1, . . . , An atomic conceptsand R1, . . . , Rm roles.

Given an Interpretation I = 〈∆, I〉, let MI be the Kripkemodel:

MI = (∆, RI1, . . . , R

Im, V )

where the evaluation functionV : ∆ → Pow({A1, . . . , An}) is defined by

V (a) = {Aj | a ∈ Aj}



Propositionfor all interpretations I = 〈∆, I〉 we have:

a ∈ CI iff MI , a |= Cm

We can also establish the opposite mapping, given aKripke model M = (W,R1, . . . , Rm, V ) we can define aninterpretation IM = (W, IM ) whereAIM = {w ∈ W | A ∈ V (w)}

Propositionfor all Kripke models M we have:

w ∈ CIM iff M,w |= Cm


Terminological axioms-TBox

C ⊑ D concept inclusionan interpretation I satisfies C ⊑ D (I |= C ⊑ D) ifCI ⊆ DI

C ≡ D concept equivalence(stands for C ⊑ D and D ⊑ C),an interpretation I satisfies C ≡ D (I |= C ≡ D) ifCI = DI

DefinitionsA definition is an equivalence A ≡ C where A is anatomic concept, we say that the name A is defined by C


Terminological axioms-TBox

A terminology T (TBox) is a finite set of definitions wherefor each defined name A there is exactly one definition

an interpretation I satisfies T (or is a model of T ),denoted by

I |= T

if it satisfies every axiom in T

Two TBox T , T ′ are equivalent iff for all interpretation II |= T iff I |= T ′


Example

Woman ≡ Person ⊓ Female

Man ≡ Person ⊓ ¬Female

Mother ≡ Woman ⊓ ∃HasChild.Person

Father ≡ Man ⊓ ∃HasChild.Person

Parent ≡ Mother ⊔ Father

Grandmother ≡ Mother ⊓ ∃HasChild.Parent

MotherWithManyChildren ≡ Mother ⊓ ≥ 3HasChild

MotherWithoutDaughter ≡Mother ⊓ ∀HasChild.¬Woman

Wife ≡ Woman ⊓ ∃HasHusband.Man


Exercises

Suppose the following atomic concept are given:DegreeType, Course,Professor,Student and roles:HasPartecipant,HasDegree, IsTaughtBy

Define GraduatedStudent as Student who has exactlyone degree of type DegreeType

Define FundamentalCourse as a Course taught by aprofessor

Define AuxiliaryCourse as a Course taught by aprofessor or by a graduate student

Define AdvancedCourse as a Course whosepartecipants are all graduated student and has no morethan 20 partecipants


Definitorial TBox

Given a TBox T we can distinguish between:

Base symbols : atomic concepts which occur only in theright part of definitions of T

Base interpretation : Interpretation of base symbols of T

Definitorial TBox : A TBox is definitorial if for every baseinterpretation I there is exactly one extension J of I of Tthat is a model of T

(J is an extension of I if they have the same domainand the same interpretation of the base symbols androles)


Definitorial TBox

Not every TBox is definitorial:

Suppose we want to define "Man with only maledescendants" (MoMD): we can define it as

MoMD ≡ Man ⊓ ∀HasChild.MoMD

The TBox T containing this definition is not definitorial.Let I with:

∆ = {Joe1, Joe2, . . .} ∪ {Bob1, . . . , Bobn}

ManI = ∆

HasChildI = {(Joei, Joei+1) | i ≥ 1} ∪∪ {(Bobi, Bobi+1) | 1 ≤ i < n}


Definitorial TBox

The extension J of I with MoMDJ = {Bob1, . . . , Bobn} isa model of T

The extension J ′ of I with MoMDJ ′

= ∆ is also a modelof T .

Exercise:show that C ≡ A ⊓ ∀R.C is not definitorial, byconsidering any interpretation I with A = ∆ andRI = ∆2.

Another example of non-definitorial is:

BinTree ≡ Tree⊓ ≤ 2HasBranch⊓∀HasBranch.BinTree


Definitorial TBox

Theorem A TBox is definitorial iff it is equivalent to anacyclic TBox.(proof of the non-trivial part: by the interpolation and theconsequent Beth-definability property of modal logic)

acyclic: in the graph induced by concept occurrences indefinitions (arc (A,B) if A occurs in the right-hand of thedefinition of B).

Let T be definitorial then for each interpretation I ofbase symbols there exists an extension I ′ of I such thatI ′ |= T

Fixpoint Semantics for non-definitorial TBox

We restrict our concern to definitorial (or acyclic) TBox


Expansion of an acyclic TBox

If T is acyclic, it is possible to expand T in order toobtain a terminology where in the right part there areonly primitive concepts, example:

Mother ≡ (Person ⊓ Female) ⊓ ∃HasChild.Person

Father ≡ (Person ⊓ ¬Female) ⊓ ∃HasChild.Person

Parent ≡ ((Person ⊓ Female) ⊓ ∃HasChild.Person) ⊔⊔ ((Person ⊓ ¬Female) ⊓ ∃HasChild.Person)

The expansion T E of T is equivalent to T

The expansion T E of T can be exponentially larger thanT .


Expansion of an acyclic TBox

For a concept C = F [A1, . . . Ak] we let

CE = F [A1/DE1 , . . . , Ak/D

Ek ]

where Ai ≡ Di ∈ T

We define the expansion T E ofT = {A1 ≡ C1, . . . , An ≡ Cn} as

T E = {A1 ≡ CE1 , . . . , An ≡ CE

n }

Exercise: (i) define a measure of length of a concept;(ii) find a concept whose expansion is exponentiallylonger than the original concept.


Elimination of Inclusions

Generalized TBox can contain also inclusions of theform

A ⊑ C

where A is an atomic concept, there is at most oneinclusion axioms for each concept A.

example:

Shark ⊑ Fish ⊓ CanBite ⊓ IsDangerous

for efficient reasoning, it is necessary to eliminateinclusion axioms


Elimination of Inclusions

Each inclusion A ⊑ C is replaced by an equality:

A ≡ A ⊓ C

where A is a new atomic concept. (it represents theadditional unknown features of A)

Let us denote by T the terminology obtained from T byreplacing in the described way all inclusion axioms

Theorem For each interpretation I

If I |= T then I |= T ;If I |= T then there exists an interpretation I ′ whichcoincides with I on base symbols and roles suchthat I ′ |= T


General Terminology

A General terminology may contain not only definitions, but also

arbitrary inclusion axioms like the following:

Elephant ⊑ Mammal

Student ⊓ ∃IsEnrolled.MIAGE ⊑ ∃MakesStage.Company

RealityShowTV Program ⊑ ¬SportTV Program

Worker ⊑ TaxPayer

Worker ⊓ ≥ 3HasChild ⊑ ¬TaxPayer

Student ⊑ ∀HasFriend.Student


ABox

One ABox A contains one finite set of instances ofconcept and roles defined in a TBox thus has the form

A = {Ci(aj), Rs(ak, bh)}

An interpretation I is a model of an ABox A ifI satisfies the Unique Name assumption: if a 6= b

(two different constants) then aI 6= bI .

I |= C(a) if aI ∈ CI

I |= R(a, b) if (aI , bI) ∈ RI


Example of ABox

Mother(mary)

Father(john)

HasChild(mary, charles)

HasChild(mary, daniel)

MotherWithoutDaughter(ann)

Mother(jane)

HasChild(ann, jane)


Reasoning Tasks

(satisfiability) A concept C is satisfiable if there exists aninterpretation I such that CI 6= ∅

(subsumption) A concept C is subsumed by D (C ⊑ D) iffor all interpretation I, CI ⊆ DI

(equivalence) Two concept C and D are equivalent if forall interpretations I, CI = DI

(disjointness) Two concept C and D are disjoint if for allinterpretations I, CI ∩DI = ∅

Proposition(subsumption), (equivalence), and (disjointness) arereducible to testing (un)satisfiability


Reasoning Tasks wrt. the TBOX

(satisfiability) A concept C is satisfiable wrt. a TBox T ifthere exists an interpretation I |= T such that CI 6= ∅

(subsumption) A concept C is subsumed by D (C ⊑T D)if for all interpretations I, if I |= T then CI ⊆ DI

(equivalence) Two concept C and D are equivalent wrt. Tif for all interpretations I, if I |= T then CI = DI

(disjointness) Two concept C and D are disjoint wrt. T iffor all interpretation I, if I |= T then CI ∩DI = ∅

Proposition(subsumption), (equivalence), and (disjointness) wrt. Tare reducible to testing (un)satisfiability wrt. T .


Eliminating the TBox

If the TBox is large, it is not feasible to performdeduction with respect to it

One can eliminate the TBox from the reasoning tasksprovided it is definitorial

This is achieved by considering in place of a concept Cits expansion CE with respect to T


Eliminating the TBox

PropositionC is satisfiable wrt. T iff CE is satisfiable

ExampleParent ⊓ ∀HasChild.⊥is satisfiable wrt. the Family TBox if and only if(((Person ⊓ Female) ⊓ ∃HasChild.Person) ⊔ ((Person ⊓¬Female) ⊓ ∃HasChild.Person)) ⊓ ∀HasChild.⊥ issatisfiable (which is not)


Reasoning wrt. the ABox

(satisfiablility) A is satisfiable wrt. T if there exists aninterpretation I |= T ∪ A

if T is definitorial it can be eliminated and we obtain thatA is consistent wrt. T iff the expansion AE is satisfiable.

(property inference) A |= C(a) wrt. T iff A ∪ {¬C(a)} is notsatisfiable wrt T , where

A |= C(a) wrt T means: for all interpretation I such thatI |= T ∪ A we have I |= C(a)Example: {Grandmother(ann)} |= Parent(ann)

(query) find all x such that A |= C(x)


ABox vs Database

Analogy with Database querythe TBox corresponds to the database schema

the ABox corresponds to the database

However, there is a basic difference with databasesemantic:

A DB is interpreted as complete information thus oneadopt the Closed World Assumption : e.g. if the onlyavailable information is Doctor(ann), F rench(marc),by CWA it follows: ¬Doctor(marc),¬French(ann)

Description Logics, in contrast adopt, an Open WorldAssumption , thus from Doctor(ann), F rench(marc)nothing can be concluded about the profession ofmarc and the nationality of ann


ABox vs Database

Moreover in DB the existing individuals are restricted tothe ones denoted by a constant, that is the DomainClosure Assumption (DCA) is assumed: e.g. if the onlyavailable information is

A = {Doctor(joe), Doctor(marc),

F riend(ann, joe), F riend(ann,marc)}

by CWA and DCA one can infer:

(∀Friend.Doctor)(ann)

In contrast this conclusion is not valid in descriptionlogics, thus

A 6|= (∀Friend.Doctor)(ann)


ABox vs Database

Let KB′ = KB + CWA+DCA+ UNA we have:If KB′ ⊢ A ∨ B then KB′ ⊢ A or KB′ ⊢ B

If KB′ ⊢ ∃xA[x] then for some term t KB′ ⊢ A[t]

This means that the under CWA+DCA+UNA (thusunder the logic underlying databases) one cannotrepresent information that is genuinely disjunctive, norexistential


Decidability and Tableaux Procedures

In contrast with full first-order logic, ALCN is decidable

Its complexity is:PSPACE-complete for concept satisfiability withrespect to acyclic terminologiesEXPTIME-complete for concept satisfiability withrespect to arbitray terminologies

We present a tableau decision procedure for conceptsatisfiability, as we have seen all reasoning tasks arereducible to satisfiability)


Decidability and Tableaux Procedures

To test if C is satisfiable the tableau is initialized withA = {C(x)} where x is an individual constant

A tableau configuration S is a finite set of ABox’esA1, . . . ,Ak

A Tableau T for A = {C(x)} is a sequence ofconfigurations

We use the notation S,A for S ∪ {A}


Tableaux Rules

We assume that the concept C tested for satisfiability:

is in expanded form CE

is in Negation Normal Form

We write A[X] to denote that the formulas X occur in A

The rules have the following format:

S,A[PREM ]

S,A[CONS1] . . . A[CONSn]

the rule prescribe to add CONSi to A if PREM ⊆ Aand it is applied provided

CONS1 6⊆ A . . . CONSn 6⊆ A


Tableaux Rules

Rule for ⊓S,A[(C ⊓D)(x)]

S,A[C(x), D(x)]

Rule for ⊔S,A[(C ⊔D)(x)]

S,A[C(x)],A[D(x)]


Tableaux Rules

Rule for ∃S,A[(∃R.C)(x)]

S,A[R(x, y), C(y)](y 6∈ A)

The rule is applied if there is not y such that R(x, y) ∈ Aand C(y) ∈ A

Rule for ∀S,A[(∀R.C)(x), R(x, y)]

S,A[C(y)]


Tableaux Rules (follows)

Rule for ≥ n

S,A[(≥ nR)(x)]

S,A[R(x, y1), . . . , R(x, yn), yi 6= yj | 1 ≤ i < j ≤ n]

where y1, . . . , yn are new constants

Rule for ≤ n

S,A[(≤ nR)(x), R(x, y1), . . . R(x, yn+1)]

S, {Ayi/yj | yi 6= yj 6∈ A ∧ i < j}}

if for some yi, yj , we have yi 6= yj 6∈ A


Clashes

A contains a clash if one of the following conditionsapplies:

⊥(x) ∈ A

C(x),¬C(x) ∈ A

{(≤ nR)(x)} ∪ {R(x, yi) | 1 ≤ i ≤ n+ 1}∪∪{yi 6= yj | 1 ≤ i < j ≤ n+ 1} ⊆ A

Example of the last condition:(≤ 2R)(x), R(x, y1), R(x, y2), R(x, y3),y1 6= y2, y1 6= y3, y2 6= y3

is a clash


Example 1

∃R.A ⊓ ∃R.B ⊓ ≤ 1R ⊑ ∃R.(A ⊓ B)

show that the following is unsatisfiable (NNFtransformation)

∃R.A ⊓ ∃R.B ⊓ ≤ 1R ⊓ ∀R.(¬A ⊔ ¬B)

(∃R.A ⊓ ∃R.B⊓ ≤ 1R ⊓ ∀R.(¬A ⊔ ¬B))(x)

(∃R.A)(x), (∃R.B)(x), (≤ 1R)(x), (∀R.(¬A ⊔ ¬B))(x)

R(x, y), A(y), (∃R.B)(x), (≤ 1R)(x), (∀R.(¬A ⊔ ¬B))(x)

R(x, y), A(y), R(x, z), B(z), (≤ 1R)(x), (∀R.(¬A ⊔ ¬B))(x)

R(x, y), A(y), R(x, z), B(z), (≤ 1R)(x),(¬A ⊔ ¬B)(y), (¬A ⊔ ¬B)(z), (∀R.(¬A ⊔ ¬B))(x)


Example 1(continued)

Let

A = {R(x, y), A(y), R(x, z), B(z), (≤ 1R)(x), (¬A ⊔ ¬B)(z)},

(∀R.(¬A ⊔ ¬B))(x)}

A ∪ {(¬A)(y)},A ∪ {(¬B)(y)}

The first one contains a clash, thus it is eliminated

Expanding (¬A ⊔ ¬B)(z) we get:

A ∪ {(¬B)(y), (¬A)(z)},A∪ {(¬B)(y), (¬B)(z)}

The second one contains a clash, thus it is eliminated

R(x, y), A(y), R(x, z), B(z), (¬B)(y), (¬A)(z),(≤ 1R)(x), (∀R.(¬A ⊔ ¬B))(x)



let

A′ = {R(x, y), A(y), R(x, z), B(z), (¬B)(y), (¬A)(z),

(≤ 1R)(x), (∀R.(¬A ⊔ ¬B))(x)}

we apply the rule for ≤ 1R and we compute Ay/z

R(x, z), A(z), R(x, z), B(z), (¬B)(z), (¬A)(z),≤ 1R,∀R.(¬A ⊔ ¬B))(x)

Clash!


Example 2

∃R.A ⊓ ∃R.B ⊑ ∃R.(A ⊓ B)

show that the following is unsatisfiable (NNFtransformation)

∃R.A ⊓ ∃R.B ⊓ ∀R.(¬A ⊔ ¬B)

(∃R.A ⊓ ∃R.B ⊓ ∀R.(¬A ⊔ ¬B))(x)

(∃R.A)(x), (∃R.B)(x), (∀R.(¬A ⊔ ¬B))(x)

R(x, y), A(y), (∃R.B)(x), (∀R.(¬A ⊔ ¬B))(x)

R(x, y), A(y), R(x, z), B(z), (∀R.(¬A ⊔ ¬B))(x)

R(x, y), A(y), R(x, z), B(z),(¬A ⊔ ¬B)(y), (¬A ⊔ ¬B)(z), (∀R.(¬A ⊔ ¬B))(x)



Let

A = {R(x, y), A(y), R(x, z), B(z), (¬A ⊔ ¬B)(z)},

(∀R.(¬A ⊔ ¬B))(x)}

A ∪ {(¬A)(y)},A ∪ {(¬B)(y)}

The first one contains a clash, thus it is eliminated

Expanding (¬A ⊔ ¬B)(z) we get:

A ∪ {(¬B)(y), (¬A)(z)},A∪ {(¬B)(y), (¬B)(z)}

The second one contains a clash, thus it is eliminated

R(x, y), A(y), R(x, z), B(z), (¬B)(y), (¬A)(z),(∀R.(¬A ⊔ ¬B))(x)


Example 2(end)

let I = (∆, I) be an interpretation, with

∆ = {x, y, z},

RI = {(x, y), (x, z)},

AI = {y},

BI = {z}

We can easily check that

I |= ∃R.A ⊓ ∃R.B ⊓ ∀R.(¬A ⊔ ¬B)

Thus∃R.A ⊓ ∃R.B 6⊑ ∃R.(A ⊓ B)


Soundness, Completeness, Termination

Satisfiable ConfigurationA tableau configuration S = {A1, . . . ,Ak} it is satisfiableif exists Ai ∈ S and exist an intepretation I |= Ai

Open configurationLet S = {A1, . . . ,Ak} be a tableau configuration. S isopen if some Ai ∈ S is clash-free.

Complete configurationLet S = {A1, . . . ,Ak} be a tableau configuration. Ai iscomplete if no rule can be applied to it. S is complete ifevery Ai ∈ S is complete.


Termination

Theorem1 (Termination)Any tableau T for {C(x0)} terminates with a finite andcomplete configuration S.Proof (hint)

(finite length ) There are only finite role-chainsR(x, y1), R(y1, y2), . . . , R(yn, yn+1) ∈ A, where thelength n is bounded by the level of nesting of roleconstructs(finite width ) There are only finitely many successorsof each element, i.e. the number of yi such thatR(x, yi) ∈ A is finite and bounded by the cardinalityof ≥ nR and the number of ∃R


Soundness

Lemma1Let S be a satisfiable configuration and let S′ beobtained by applying any rule to S, then S′ is alsosatisfiable

Theorem2 (Soundness)Let C be satisfiable concept and let T be any tableaufor {C(x0)}. All configurations of T are open.

CorollaryLet C be satisfiable concept and let T be any tableaufor {C(x0)}, T terminates with an open and completeconfiguration.


Completeness

Lemma2Let S be any complete and open configuration, then S issatisfiable

Theorem3 (Completeness)Let T be a tableau for {C(x0)}, if T terminates with anopen and complete configuration S, then C issatisfiable.


Proof of Lemma 2

Proof of Lemma2Let A ∈ S be complete and open, define aninterpretation as follows: I = (∆, I) where

∆ = {z | z occurs in A},

RI = {(x, y) | R(x, y) ∈ A},

AI = {y | A(y) ∈ A}

Show that I |= C(u) for every C(u) ∈ A

Extension of Tableaux for ABoxes: add the UniqueName Assumption


Tableaux for general terminologies

given a terminology T , add the rule

S,A

S,A[¬C(x)],A[D(x)](x ∈ A)

if C ⊑ D ∈ T

to get termination we need a suitable loop-checkingmechanism

consider for instanceT = {Student ⊑ ∃HasFriend.Student}, a tableaustarting with Student(joe) can be expanded infinitely

the complexity becomes exponential (EXPTIMEcomplete)


Tableaux as Trees

The rules have the following format:

A[PREM ]

A[CONS1] | . . . | A[CONSn]

the rule prescribe to replace PREM by CONSi ifPREM ⊆ A and it is applied provided

CONS1 6⊆ A . . . CONSn 6⊆ A

each A[CONS1] represent a branching in the tableau


Tableaux Rules

Rule for ⊓A[(C ⊓D)(x)]

A[C(x), D(x)]

Rule for ⊔A[(C ⊔D)(x)]

A[C(x)] | A[D(x)]


Tableaux Rules

Rule for ∃A[(∃R.C)(x)]

A[R(x, y), C(y)](y 6∈ A)

The rule is applied if there is not y such that R(x, y) ∈ Aand C(y) ∈ A

Rule for ∀A[(∀R.C)(x), R(x, y)]

A[(∀R.C)(x), R(x, y)], C(y)]


Tableaux Rules (follows)

Rule for ≥ n

A[(≥ nR)(x)]

A[R(x, y1), . . . , R(x, yn), yi 6= yj | 1 ≤ i < j ≤ n]

where y1, . . . , yn are new constants

Rule for ≤ n

A[(≤ nR)(x), R(x, y1), . . . R(x, yn+1)]

Ayi/yj | . . . | Ayi/yj

for all yi, yj such that yi 6= yj 6∈ A and i < j


Extensions of ALCN

Role Constructors

R,S := R ⊓ S | R ⊔ S | ¬R | R ◦ S | R+ | R−

where R ◦ S denotes the composition of R and S, R+

denotes the transitive closure of R, and R− denotes theinverse role of R

Semantics (we omit the boolean cases)

(R ◦ S)I = {(a, c) ∈ ∆2 | ∃b(a, b) ∈ RI ∧ (b, c) ∈ SI}

(R+)I = RI ∪ (R ◦R)I ∪ (R ◦R ◦ R)I ∪ . . .

(R−)I = {(b, a) | (a, b) ∈ R}


Extensions

Role InclusionsR ⊑ S

Definition of Transitive Roles

R ≡ R+

Definition of Symmetric Roles

R ≡ R−


Qualified Number Restrictions

Qualified Number Restrictions ≤ nR.C and ≥ nR.C

(≤ n R.C)I = {a ∈ ∆s.t. | {b ∈ ∆(a, b) ∈ RI ∧ b ∈ CI} |≤ n}

(≥ n R.C)I = {a ∈ ∆s.t. | {b ∈ ∆(a, b) ∈ RI ∧ b ∈ CI} |≥ n}

Number restrictions can be combined with roleconstructors:

≥ 2(HasChild ◦HasChild)

but with care! e.g. if combined with role intersection andcomposition give an undecidable language (as well as ifcombined with transitive roles).


One-of-constructor

One-of constructor: {a1, . . . , an}, where

({a1, . . . , an})I = {aI1, . . . , a

In}

Using the ∃R construct and one-of when n = 1, we candefine exact values of roles as

WhiteWine = Wine ⊓ ∃HasColor.{White}

we can rewrite it as

WhiteWine = Wine ⊓HasColor = White


Concrete Domains

Concrete Domains:Natural Numbers, Strings

Adult ≡ Person ⊓ ∃hasAge ≥ 18

the predicate ≥ 18 is interpreted on the natural numbers


Exercise

Give tableau rules for the followign extensions of ALCN

transitive roles

symmetric roles

functional roles

one-of construct


Applications to semantic web

Purpose of the semantic web project : web informationshould be described and organized according to itsmeaning independently from its presentation

The meaning is defined by an ontology which fixes theprimitive concepts and defines all the others and theirrelations

An ontology should be specified in a formalism which isprocessable by a machine: a formal language adequatefor for web exchange

The system should provide reasoning services forontologies, such as subsumption, satisfiablility,classification.


Steps to semantic web

OIL: (Ontology Interchange Layer) abstract languagewith a well-defined semantics defined by DescriptionLogics (2001-2002)

DAML+OIL: Language for Ontologies with a concretesyntax based on RDF schemas and XML

OWL: current evolution of DAML+OIL, it is a standard(2005 Recommendation by WC3 group on semanticweb)

We will see OWL DL: there is also a small and versionof OWL, called OWL Lite, and a very strong version,called OWL Fullwhich is undecidable


OWL DL

It has an RDF/XML based syntax<owl:Class rdf:ID="Student">

<owl:intersectionOf rdf:parsetype="Collection"><owl:Class rdfs:about="Person" /><owl:Restriction>

<owl:onProperty rdf:resource="enrolledIn" /><owl:minCardinality

rdfs:datatype="&xsd;Integer">1

</owl:minCardinality></owl:Restriction>

<owl:intersectionOf></owl:Class>


OWL (DL): Abstract syntax

A (URI reference) A

owl: Thing ⊤

owl: Nothing ⊥

intersectionOf(C1 C2 ...) C1 ⊓ C2 . . .

unionOf(C1 C2 ...) C1 ⊔ C2 . . .

complementOf(C) ¬C

oneOf(o1 ...on) {o1, . . . , on}


OWL (DL): : Abstract syntax

restriction(R someValuesFrom(C)) ∃R.C

restriction(R allValuesFrom(C)) ∀R.C

restriction(R hasValue(o)) ∃R.{o}

restriction(R minCardinality(n)) ≥ nR

restriction(R maxCardinality(n)) ≤ nR



restriction(U someValuesFrom(D)) ∃U.D

restriction(U allValuesFrom(D)) ∀U.D

restriction(U hasValue(v)) ∃U.{v}

restriction(U minCardinality(n)) ≥ nU

restriction(U maxCardinality(n)) ≤ nU



Class(A partial C1 ... Cn) A ⊑ C1 ⊓ . . . ⊓ Cn

Class(A complete C1 ... Cn) A ≡ C1 ⊓ . . . ⊓ Cn

EnumeratedClass(A o1.... on) A ≡ {o1, . . . , on}

SubClassOf(C1 C2) C1 ⊑ C2

EquivalentClasses(C1 ...Cn) C1 ≡ . . . ≡ Cn

DisjointClasses(C1 ...Cn) Ci ⊑ ¬Cj


OWL(DL) : Abstract syntax

ObjectProperty(R super(Ri) R ⊑ Ri

domain(C1) ≥ 1R ⊑ C1

range(C2) ⊤ ⊑ ∀R.C2

[inverseOf(R0)] R ≡ R−0

[Symmetric] R ≡ R−

[Functional] ⊤ ⊑ ≤ 1R[InverseFunctional] ⊤ ⊑ ≤ 1R−

[Transitive]) R ≡ R+

SubPropertyOf(R1 R2) R1 ⊑ R2

EquivalentProperties(R1...Rn) R1 ≡ . . . ≡ Rn



ObjectProperty(R super(Ri) R ⊑ Ri

domain(C1) ≥ 1R ⊑ C1

range(C2) ⊤ ⊑ ∀R.C2

[inverseOf(R0)] R ≡ R−0

[Symmetric] R ≡ R−

[Functional] ⊤ ⊑ ≤ 1R[InverseFunctional] ⊤ ⊑ ≤ 1R−

[Transitive]) R ≡ R+

SubPropertyOf(R1 R2) R1 ⊑ R2

EquivalentProperties(R1...Rn) R1 ≡ . . . ≡ Rn



DataProperty(U super(Ui) U ⊑ Ui

domain(C1) ≥ 1U ⊑ C1

range(D) ⊤ ⊑ ∀U.D[Functional] ⊤ ⊑ ≤ 1U

SubPropertyOf(U1 U2) U1 ⊑ U2

EquivalentProperties(U1...Un) U1 ≡ . . . ≡ Un



Individual(o type(C1) ...type(Cn) Ci(o)value(R1 o1)...value(Rn on) Ri(o, oi)value(U1 v1)...value(Un vn)) Ui(o, vi)

SameIndividual(o1...on) o1 = . . . = on

DifferentIndividuals(o1...on) oi 6= oj


Example: African Animals Ontology

ObjectProperty (eatsInverseOf (is-eaten-by))

ObjectProperty (has-partInverseOf(is-part-of)transitive)

Class (Animal)

DisjointClasses (Animal,UnionOf(Plant,

(restriction (is-part-of SomeValuesFrom Plant))))

Class (Tree partial Plant)

Class (Branch partialrestriction (is-part-of SomeValuesFrom Tree))



Class (Leaf partialrestriction (is-part-of SomeValuesFrom Branch))

Class (Carnivore completeAnimalrestriction (eats AllValuesFrom Animal))

Class (Herbivore completeAnimalrestriction (eats

AllValuesFrom unionOf(Plantrestriction (is-part-ofSomeValuesFrom Plant)))))



Class (Giraffe partialAnimalrestriction(eats SomeValuesFrom Leaf))

Class (Lion partialAnimalrestriction(eats AllValuesFrom Herbivore))

Class (Tasty Plant completePlantrestriction(eaten-by SomeValuesFrom Herbivore)restriction(eaten-by SomeValuesFrom Carnivore))


Some Inferences

Lion subclass of Carnivore

Giraffe subclass of Herbivore

Leaf is-part-of Tree

Lion subclass of Carnivore

Herbivore and Carnivore are disjoint

Tasty-Plant is inconsistent


DL Translation of African Animals

eats ≡ eaten-by−

is-part-of ≡ has-part−

is-part-of ≡ is-part-of+

Plant ⊑ ¬(Animal ⊔ ∃is-part-of.P lant)

Tree ⊑ Plant

Branch ⊑ ∃is-part-of.T ree

Leaf ⊑ ∃is-part-of.Branch


DL Translation of African Animals

Carnivore ≡ Animal ⊓ ∀eats.Animal

Herbivore ≡ Animal ⊓ ∀eats.(Plant ⊔ ∃is-part-of.P lant)

Giraffe ⊑ Animal ⊓ ∀eats.Leaf

Lion ⊑ Animal ⊓ ∀eats.Herbivore

TastyP lant ≡Plant ⊓ ∃eaten-by.Herbivore ⊓ ∃eaten-by.Carnivore


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