phase 1 - lehrstuhl für automatentheories for role names dresden, ws 2013-2014 description logics 9...

History

Phase 1implementation of systems based on incomplete structural subsumption algorithms

Phase 2• tableau-based algorithms and complexity results

• first tableau-based systems (Kris, Crack)

• first formal study of optimization methods

Phase 3• tableau-based algorithms for very expressive DLs

• highly-optimized tableau-based systems (FaCT, Racer)

• relationship to modal logic and FOL

Dresden, WS 2013-2014 Description Logics 2

History

Phase 4• Web Ontology Language (OWL) based on DL

• industrial-strength reasoners and ontology editors

• light-weight (tractable) DLs

Phase 5• non-standard reasoning

• ontology management

• semantic extensions


Chapter 2

A Basic Description Logic

A Basic Description Logic

ALCattributive language with complement

Naming Schema• basic language AL• additional constructors are denoted by appending a letter

• C stands for complement

ALC is obtained by adding ¬ to AL


Structure of Description Logic Systems

TBox

terminology ofapplication domain

ABox

facts about a world

LanguageLanguage Reasoning

◦ constructors forbuilding complexconcepts

◦ formal logic-basedsemantics

◦ implicit knowledge

◦ practical algorithms

Ontology


Section 2.1

The Description Language

Syntax of ALC

Definition 2.1 (Syntax of ALC)Let NC and NR two disjoint sets of concept names and role names, respectively.

ALC (complex) concepts are defined by induction:

• if A ∈ NC , then A is an ALC concept

• if C, D are ALC concepts and r ∈ NR , then the following are ALC concepts:– C u D (conjunction)

– C t D (disjunction)

– ¬C (negation)

– ∃r.C (existential restriction)

– ∀r.C (value restriction)

Abbreviations• > := A t ¬A (top)

• ⊥ := A u ¬A (bottom)


Notation

• concept names are also called atomic concepts

• all other concepts are called complex

• instead of ALC concept, we often say concept

• A, B stand for concept names

• C, D for (complex) concepts

• r, s for role names


Examples of Concepts

Hero u Female

∀fights.Mutant

Rich t ¬Human

∃fights.¬Human

Mutant u ∃fights.(¬Human t ∀sidekick.Female)


Semantics of ALC

Definition 2.2 (Semantics of ALC)An interpretation I = (∆I , ·I) consists of:

• a non-empty domain ∆I , and

• an extension mapping ·I (also called interpretation function):– AI ⊆ ∆I for all A ∈ NC (concepts interpreted as sets)– rI ⊆ ∆I ×∆I for all r ∈ NR (roles interpreted as binary relations)

The extension mapping is extended to concept descriptions as follows:

(C u D)I := CI ∩ DI

(C t D)I := CI ∪ DI

(¬C)I := ∆I \ CI

(∃r.C)I := {d ∈ ∆I | there is e ∈ ∆I with (d, e) ∈ rI and e ∈ CI}(∀r.C)I := {d ∈ ∆I | for all e ∈ ∆I : (d, e) ∈ rI implies e ∈ CI}


Interpretation Example

R

Hero

B

Hero

S

Hero

W

Hero,Female

Rich,Human,FemaleRich,Human,Female HumanHuman Female Rich,HumanRich,Human

helps

helps

fights fights fights fights fights

( Hero u ∃fights.Human )I = {B,S}( Hero u ∀fights.(Rich t ¬Human) )I = {R,S,W}( ∀helps.Human )I = ∆I \ {R}


DL vs FOL

ALC can be seen as a fragment of First-order Logic

• concept names are unary predicates

• role names are binary predicates

Interpretations can obviously be seen as first-order interpretations for this signature

concepts then correspond to FOL formulae with one free variable

Let φ(x) be such a formula with free variable x, and I an interpretation. The extension of φw.r.t. I is given by

φI := {d ∈ ∆I | I |= φ(d)}

Goaltranslate ALC concepts C into FOL formulae τx(C) such that their extensions coincide


Translation to FOLSyntactic translation C τx(C)

• τx(A) := A(x) for A ∈ NC

• τx(C u D) := τx(C) ∧ τx(D)

• τx(C t D) := τx(C) ∨ τx(D)

• τx(¬C) := ¬τx(C)

• τx(∃r.C) := ∃y.(r(x, y) ∧ τy C) y new variable different from x• τx(∀r.C) := ∀y.(r(x, y)→ τy C)

Exampleτx(∀r.(A u ∃s.B)) = ∀y.(r(x, y)→ τy (A u ∃s.B))

= ∀y.(r(x, y)→ (A(y) ∧ ∃z.(s(y, z) ∧ B(z))))

Lemma 2.3C and τx(C) have the same extension; that is,

CI = {d ∈ ∆I | I |= τx(C)(d)}.

Proof by induction on structure of C


Decidable Fragments of FOL

ALC can be seen as a fragment of FOL:each concept C yields a formula τx(C) with one free variable

DecidabilityThese formulae belong to known decidable subclasses of FOL:

• two variable fragment

• guarded fragment

τx(∀r.(A u ∃s.B)) = ∀y.(r(x, y)→ τy (A u ∃s.B))

= ∀y.(r(x, y)→ (A(y) ∧ ∃x.(s(y, x) ∧ B(x))))


DL vs Modal Logic

ALC is a syntactic variant of multimodal K

• concept names are propositional variables

• role names are transition relations

multimodal K: several pairs ofboxes and diamonds

Concept descriptions C yield modal formulas Θ(C)

• Θ(A) := A for A ∈ NC

• Θ(C u D) := Θ(C) ∧Θ(D)

• Θ(C t D) := Θ(C) ∨Θ(D)

• Θ(¬C) := ¬Θ(C)

• Θ(∃r.C) := [r ]Θ(C)

• Θ(∀r.C) := 〈r〉Θ(C)

C and Θ(C) have the same semantics:I is a Kripke structureCI is the set of worlds that make Θ(C) true in I.


More Expressivity

ALC is only one example of many description logics that have been studied

many other constructors exist and can be used for KR

Qualified Number Restrictions (Q)• (≥ n r.C)I := {d ∈ ∆I | |{e | (d, e) ∈ rI , e ∈ CI}| ≥ n}

“at least n r -successors that belong to C”

• (≤ n r.C)I := {d ∈ ∆I | |{e | (d, e) ∈ rI , e ∈ CI}| ≤ n}“at most n r -successors that belong to C”

A hero that fights at least two villains, of which at most one is a sidekick

Hero u (≥ 2 fights.Villain) u (≤ 1 fights.(∃helps.Villain))


More Expressivity



Number Restrictions (N )• (≥ n r)I := (≥ n r.>)I = {d ∈ ∆I | |{e | (d, e) ∈ rI , e ∈ CI}| ≥ n}

“at least n r -successors”

• (≤ n r)I := (≤ n r.>)I = {d ∈ ∆I | |{e | (d, e) ∈ rI , e ∈ CI}| ≤ n}“at most n r -successors”


More Expressivity



Inverse Roles (I)• for a role name r , r−1 denotes the inverse:

(r−1)I := {(e, d) | (d, e) ∈ rI}

A hero that only fights villains with female sidekicks

Hero u ∀fights.(Villain u ∃helps−1.Female)



TBox


TBox


ABox

facts about a world

Language Reasoning





Ontology


Section 2.2

Terminological Knowledge

GCIs and TBoxes

Definition 2.4 (GCIs and TBoxes)• A general concept inclusion (GCI) is of the form C v D, where C, D are concepts

• A TBox is a finite set of GCIs

• The interpretation I satisfies the GCI C v D iff CI ⊆ DI

• I is a model of the TBox T iff it satisfies all GCIs in T

Note: this definition is not specific of ALC; applies to any description language

ExampleHero u Villain v ⊥

Hero u ∀hasPower.⊥ v Rich t ∃hasSidekick−1.Rich

Two TBoxes are equivalent if they have the same models


Concept Definitions

Definition 2.5A concept definition is of the form A ≡ C where

• A is a concept name

• C is a concept description

The interpretation I satisfies the concept definition A ≡ C if AI = CI

A ≡ C abbreviates the two GCIsA v C, C v A


Acyclic TBoxes

Definition 2.5 (continued)An acyclic TBox is a finite set of concept definitions that

• does not contain multiple definitions

A ≡ CA ≡ D

• does not contain cyclic definitions (directly or indirectly)

A ≡ ∃r.BB ≡ C

C ≡ ∀s.A

The interpretation I is a model of the acyclic TBox T is it satisfies all concept definitions in T

A concept name A occurring in T is a

• defined concept iff there is C such that A ≡ C ∈ T ;

• primitive concept otherwise


Example

Heroine ≡ Hero u Female

Sidekick ≡ ∃helps.>

Criminal ≡ ∃fights.Hero

MutantCriminal ≡ Criminal u ∀fights.Mutant

Superhero ≡ Hero u (Rich t ¬Human t ∃hasPower.SuperPower)

Overlord ≡ (≥ 3 helps−1.Criminal) u ∀fights.Superhero


ABox Expansion

Proposition 2.6For every acyclic TBox T , we can effectively construct an equivalent acyclic TBox T such thatthe right-hand sides of concept definitions in T contain only primitive concepts.

Proofblackboard

We call T the expanded version of T


Primitive Interpretations

Given an acyclic TBox T , a primitive interpretation J for T consists of a non-empty set ∆J

together with an extension mapping ·J that maps

• primitive concepts P to sets PJ ⊆ ∆J

• role names r to binary relations rJ ⊆ ∆J ×∆J

The interpretation I is an extension of the primitive interpretation J iff

• ∆J = ∆I ,

• PJ = PI for all primitive concepts P

• rJ = rI for all role names r

Corollary 2.7Let T be an acyclic TBox. Any primitive interpretation J has a unique extension to a model of T

Proofblackboard


Translation to FOL

Any ALC-TBox can be translated into FOL:

τ(T ) :=∧

CvD∈T∀x.(τx(C)→ τx(D))

Lemma 2.8Let T a TBox and τ(T ) its translation into FOL.Then T and τ(T ) have the same models

Proofblackboard



TBox


ABox

facts about a world

ABox

facts about a world

Language Reasoning





Ontology


Section 2.3

Assertional Knowledge

Assertions and ABoxes

Definition 2.9 (Assertions and ABoxes)An assertion is of the form

C(a) (concept assertion) or r(a, b) (role assertion)

where C is a concept, r a role, and a, b are individual names from a set NI (disjoint with NC ,NR )

An ABox is a finite set of assertions

Interpretations I are extended to assign elements aI ∈ ∆I to individual names a ∈ NI

I is a model of the ABox A if it satisfies all its assertions:

• aI ∈ CI for all C(a) ∈ A• (aI , bI) ∈ rI for all r(a, b) ∈ A


Examples of Assertions

Rich(batman)

¬Human(superman)

fights(superman,bizarro)

helps(robin,batman)


Translation to FOL

Any ALC-ABox can be translated into FOL:

τ(A) :=∧

C(a)∈Aτx(C)(a) ∧

∧r(a,b)∈A

r(a, b)

(individual names are viewed as constants)

Lemma 2.10Let A an ABox and τ(A) its translation into FOL.Then A and τ(A) have the same models

Proofeasy (Exercise!)


Ontologies

Definition 2.11An ontology O = (T ,A) consists of a TBox T and an ABox A

The interpretation I is a model of the ontology O = (T ,A) iffit is a model of T and a model of A

FOL translation: τ(O) := τ(T ) ∧ τ(A)

Lemma 2.12Let O be an ontology and τ(O) its FOL translation. O and τ(O) have the same models

ProofImmediate from Lemmas 2.8 and 2.10


Nominals

We can increase the expressive power of the description language by using individual names asconcept constructors

They are interpreted as singleton sets containing only the extension of the individual name

Nominals (O)Syntax: {a} for a ∈ NI

Semantics: {a}I := {aI}

Using nominals, ABox assertions can be expressed through GCIs:

C(a) is expressed by {a} v Cr(a, b) is expressed by {a} v ∃r.{b}



TBox


ABox

facts about a world

Language ReasoningReasoning





Ontology


Section 2.4

Reasoning Problems and Services

Reasoning ProblemsGoal: to make implicitly represented knowledge explicit

Definition 2.13 (Terminological Reasoning)Let T be a TBox. Terminological reasoning refers to deciding the following problems

Satisfiability:C is satisfiable w.r.t. T iff CI 6= ∅ for some model I of T

Subsumption:C is subsumed by D w.r.t. T (C vT D) iff

CI ⊆ DI for all models I of T

Equivalence:C is equivalent to D w.r.t. T (C ≡T D) iff

CI = DI for all models I of T

If T = ∅ we simply remove the “w.r.t T ” from the nameDresden, WS 2013-2014 Description Logics 36

Examples

• A u ¬A and ∀r.A u ∃r.¬A are not satisfiable (unsatisfiable)

• A u ¬A and ∀r.A u ∃r.¬A are equivalent

• A u B is subsumed by A and by B

• ∃r.(A u B) is subsumed by ∃r.A and by ∃r.B

• ∀r.(A u B) is equivalent to ∀r.A u ∀r.B

• ∃r.A u ∀r.B is subsumed by ∃r.(A u B)


Properties of Subsumption

Lemma 2.141. The subsumption relation vT is a pre-order on concepts:

– C vT C (reflexivity)– if C vT D and D vT E, then C vT E (transitivity)

2. Existential restrictions and value restrictions are monotonic w.r.t. subsumption– if C vT D, then

∃r.C vT ∃r.D and ∀r.C vT ∀r.D

Proofblackboard

NotevT is not a partial order since it is not antisymmetric:

C vT D and D vT C does not imply that C = D

(no syntactic equivalence, just semantical)


Reasoning Problems

Definition 2.15 (Assertional Reasoning)Let O = (T ,A) be an ontology. The following are assertional reasoning problems

Consistency:O is consistent iff there exists a model of O

Instance:a is an instance of C w.r.t. O iff aI ∈ CI for all models I of O

Lemma 2.16Let O = (T ,A) be an ontology.If a is an instance of C w.r.t. O and C vT D, then a is an instance of D w.r.t. O

Exercise!


Reductions Between Reasoning Problems

The following polynomial time reductions between the reasoning problems hold

equivalence subsumption satisfiability

instance consistency

1.

2.

3.

4.

6.

7.5.

holds for any DL that hasconjunction and negation


Reductions

Theorem 2.17Let O = (T ,A) be an ontology, C, D concepts and a ∈ NI .

1. C ≡T D iff C vT D and D vT C

2. C vT D iff C ≡T C u D

3. C vT D iff C u ¬D is unsatisfiable w.r.t. T

4. C is satisfiable w.r.t. T iff C 6vT ⊥

5. C is satisfiable w.r.t. T iff (T , {C(a)}) is consistent

6. a is an instance of C w.r.t. O iff (T ,A ∪ {¬C(a)}) is inconsistent

7. O is consistent iff a is not an instance of ⊥ w.r.t. O

Proofblackboard


Expansions

Let O = (T ,A) be an ontology where T is acyclic, and C a concept

The expanded versions C and A of C and A w.r.t. T are obtained by

replacing all defined concepts occurring in C and A by their defintions from T

T :

Woman ≡ Person u Female

Criminal ≡ ∃fights.Hero

Minion ≡ Person u ∃helps.Criminal

C: Woman uMinion

C = Person u Female u Person u ∃helps.∃fights.Hero


Expansions

Let O = (T ,A) be an ontology where T is acyclic, and C a concept

The expanded versions C and A of C and A w.r.t. T are obtained by

replacing all defined concepts occurring in C and A by their defintions from T

Proposition 2.18

1. C is satisfiable w.r.t. T iff C is satisfiable

2. O = (T ,A) is consistent iff (∅, A) is consistent

Proofblackboard

similar reductions exist for other reasoning problems


Not a Polynomial Reduction

The expansion of concepts and ABoxes is in general not polynomial

The acyclic TBox T

A0 ≡ ∀r.A1 u ∀s.A1

A1 ≡ ∀r.A2 u ∀s.A2

...

An−1 ≡ ∀r.An u ∀s.An

has n axioms, all of the same size; i.e., the size of T is linear in n

The expanded version A0 of A0 contains the name An 2n times!

induction on n


Relationship with FOL

We can translate ALC reasoning into FOL reasoning

Lemma 2.19Let O = (T ,A) be an ontology, C, D be concepts, and a ∈ NI

1. C vT D iff τ(T ) |= ∀x.(τx(C)(x)→ τx(D)(x))

2. O is consistent iff τ(O) is consistent

3. a is an instance of C w.r.t. O iff τ(O) |= τx(C)(a)

Proofblackboard


ClassificationComputing the subsumption relations between all concept names in T


MaleHero ≡ Hero u ¬Female

MutantHeroine ≡ Heroine uMutant

Elite ≡ Rich t ¬Human

Superhero ≡ Hero u Elite

HeroElite

Human

Female

Heroin

Mutant

MaleHeroSuperHeroRich

MutantHeroin


Realization

Computing the most specific concept names to which an individual belongs


MaleHero ≡ Hero u ¬Female

MutantHeroine ≡ Heroine uMutant

Elite ≡ Rich t ¬Human

Superhero ≡ Hero u Elite

Hero(Superman)

Superman is an instance of

Hero, MaleHero, Elite, Superhero


phase 1 - lehrstuhl für automatentheories for role names dresden, ws 2013-2014 description logics 9...

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