Knowledge Representation for the Semantic Web Lecture 5...

Satisfaction and Entailment Other Reasoning Problems Algorithmic Approaches to DL Reasoning

Knowledge Representation for the Semantic WebLecture 5: Description Logics IV

Daria Stepanovaslides based on Reasoning Web 2011 tutorial “Foundations of Description Logics and OWL” by S. Rudolph

Max Planck Institute for InformaticsD5: Databases and Information Systems group

WS 2017/18

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Unit Outline

Satisfaction and Entailment

Other Reasoning Problems

Algorithmic Approaches to DL Reasoning

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Satisfaction and Satisfiability

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Satisfaction and Satisfiability of Knowledge Bases

Satisfaction of a KB by an interpretation

An interpretation I satisfies (or is a model of) a knowledge baseK = 〈R, T ,A〉, if I satisfies every axiom of K, i.e., I |= α for α ∈ K.

KB (un)satisfiability / (in)consistency

A KB K is satisfiable (also: consistent), if it has some model; otherwise itis unsatisfiable (also: inconsistent or contradictory).

� unsatisfiability of a KB hints at a design bug

� unsatisfiable axioms carry no information:

α is unsatisfiable ⇐⇒ ¬α is tautologic (if negation is applicable),i.e., I |= ¬α for every interpretation I

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Example: KB Satisfiability

RBox Rowns v caresFor

”If somebody owns something, s/he cares for it.”

TBox THealthy v ¬Dead

”Healthy beings are not dead.”

Cat v Dead tAlive”Every cat is dead or alive.”

HappyCatOwner v ∃owns.Cat u ∀caresFor .Healthy”A happy cat owner owns a cat andall beings he cares for are healthy.”

ABox AHappyCatOwner(schroedinger)”Schrodinger is a happy cat owner.”

Is K = 〈R, T ,A〉 satisfiable?

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Example: KB Satisfiability








Is K = 〈R, T ,A〉 satisfiable? Yes!

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Example: KB SatisfiabilityTBox T

Deer v Mammal

”Deers are mammals.”

Mammal u Flies v Bat

”Mammals, who fly are bats.”

Bat v ∀worksFor .{batman}”Bats work only for Batman”

ABox ADeer u ∃hasNose.Red(rudolph)

”Rudolph is a deer with a red nose.”

∀worksFor−.(¬Deer t Flies)(santa)

”Only non-deers or fliers work for Santa.”

worksFor(rudolph, santa)

”Rudolph works for Santa.”

santa 6≈ batman

“Santa is different from Batman.”

Is K satisfiable?6 / 40


Example: KB SatisfiabilityTBox T

Deer v Mammal

”Deers are mammals.”

Mammal u Flies v Bat

”Mammals, who fly are bats.”

Bat v ∀worksFor .{batman}”Bats work only for Batman”

ABox ADeer u ∃hasNose.Red(rudolph)

”Rudolph is a deer with a red nose.”

∀worksFor−.(¬Deer t Flies)(santa)

”Only non-deers or fliers work for Santa.”

worksFor(rudolph, santa)

”Rudolph works for Santa.”

santa 6≈ batman

“Santa is different from Batman.”

Is K satisfiable? No!6 / 40


Entailment of Axioms

Entailment checking

A knowledge base K entails an axiom α (in symbols, K |= α), if everymodel I of K satisfies α.

models of K

interpretations satisfying α

� Informally, K |= α elicits implicit knowledge

� If α occurs in K, then trivially K |= α

� If K is unsatisfiable, then K |= α for every axiom α

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Example: EntailmentRBox Rowns v caresFor







� K |= ∃caresFor .(Cat uAlive)(schroedinger)

� K |= ∀owns.¬Cat v ¬HappyCatOwner

� K |= Cat v Healthy

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� K |= ∃caresFor .(Cat uAlive)(schroedinger)?



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� K |= ∀owns.¬Cat v ¬HappyCatOwner?


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� K |= Cat v Healthy?

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� K 6|= Cat v Healthy

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Decidability of DLs

DLs are decidable, i.e., there exists an algorithm that

Given: a KB and an axiom α,

Output: “yes” iff KB |= α and no otherwise.

� Likewise, there is a similar algorithm that decides whether an inputKB is satisfiable

� Just ask KB |= > v ⊥: if the answer is “yes”, then KB isunsatisfiable, otherwise it is satisfiable.

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Standard Reasoning Problems

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Standard DL Reasoning Problems

� KB Satisfiability: verify whether the KB is satisfiable

� Entailment: verify whether the KB entails a certain axiome.g., K |= CatOwner(Schroedinger)

� Concept Satisfiability: verify whether a given concept is(un)satisfiable, e.g., K |= Dead uAlive v ⊥

� Coherence: verify whether none of the concepts in the KB isunsatisfiable

� Classification: compute the subsumption hierarchy of all atomicconcepts, e.g. K |= Healthy v ¬Dead , etc.

� Instance Retrieval: retrieve all the individuals known to beinstances of a certain concept, e.g., find all a, s.t. ∃caresFor(a)

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Deciding KB Satisfiability

� deciding KB satisfiability is a basic inference task(the “mother” of all standard reasoning tasks)

� directly needed in the process of KB engineering� detect severe modeling errors

� other reasoning tasks can be reduced to checking KB(un)satisfiability (and vice versa)

Theorem 1: Reducing reasoning problems to KB satisfiability

Let K be a KB and a an individual name not in K. Then

2. C is satisfiable w.r.t. K iff K ∪ C(a) is satisfiable;

3. K is coherent iff, for each concept name C, K ∪ C(a) is satisfiable;

4. K |= A v B iff K ∪ (A u ¬B)(a) is unsatisfiable;

5. K |= B(b) iff K ∪ ¬B(b) is unsatisfiable.

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Entailment Checking

� used in the KB modeling process to check, whether the specifiedknowledge has the intended consequences

� used for querying the KB if certain propositions are necessarily true

Reduction of entailment problem K |= α to checking KB inconsistency(see Th.1) follows the idea of proof by contradiction:

� negate the axiom α

� add the negated axiom ¬α to K� check for inconsistency of the resulting KB K ∪ {¬α}

If an axiom cannot be negated directly, its negation can be emulated({¬α} Aα).

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Entailment Checking, cont’dAxiom sets Aα such that K |= α exactly if K ∪Aα is unsatisfiable:

α Aαr1 ◦ . . . ◦ rn v r {¬r(c0, cn), r1(c0, c1), . . . , rn(cn−1, cn)}Dis(r, r′) {r(c1, c2), r′(c1, c2)}C v D {(C u ¬D)(c)} or {> v ∃u(C u ¬D)}C(a) {¬C(a)}¬C(a) {C(a)}r(a, b) {¬r(a, b)}¬r(a, b) {r(a, b)}a ≈ b {a 6≈ b}a 6≈ b {a ≈ b}

� Individual names c with possible subscripts are supposed to be fresh1.

� For GCIs (third line), the first variant is normally employed; the second islogical equivalent instead of just emulating.

1Fresh individuals are those not appearing in the given KB K.14 / 40


Concept Satisfiability

Concept Satisfiability

A concept expression C is called satisfiable with respect to a knowledgebase K, if there exists a model I of K such that CI 6= ∅.

� Unsatisfiable atomic concepts normally indicate KB modeling errors.

� Concept satisfiability can be reduced to KB consistency (Th. 1) andnon-entailment resp.:

� C is satisfiable wrt. K ⇐⇒ K ∪ {C(a)} is consistent, where a is afresh individual name

� C is satisfiable wrt. K ⇐⇒ K 6|= C v ⊥

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Concept Satisfiability, cont’d

Entailment of general concept inclusions C v D and equivalences C ≡ Dcan be reduced to both concept (un)satisfiability and KB(un)satisfiability.

�

C v D ⇐⇒ C u ¬D is unsatisfiable

⇐⇒ KB {C(a), ¬D(a)} is unsatisfiable

�

C ≡ D ⇐⇒ (C u ¬D) t (D u ¬C) is unsatisfiable

⇐⇒ both C u ¬D and D u ¬C are unsatisfiable

⇐⇒ both KBs {C(a), ¬D(a)} and{D(a), ¬C(a)} are unsatisfiable

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Classification

KB Classification

Classification of a knowledge base K is to determine for any two conceptnames A,B , whether K |= A v B holds.

� This is useful at KB design time for checking the inferred concepthierarchy. Also, computing this hierarchy once and storing it canspeed up further queries.

� Classification can be reduced to checking entailment of GCIs.

� While this requires quadratically many checks, one can often domuch better in practice by applying optimizations and exploitingthat subsumption is a preorder.

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Instance Retrieval

Instance Retrieval

Instance retrieval task is to find all named individuals that are known tobe in a certain concept (role).

� retrieve(C,K) = {a ∈ NI | aI ∈ CI for every model I of K }

� retrieve(r,K) = {(a, b) ∈ NI2 | (aI , bI) ∈ rI for every model I of K }

� It can be reduced to checking entailment of as many individualassertions as there are named individuals in K i.e., test K |= C(a)for each a occurring in K (excluding pathologic cases).

� Depending on the system used and the inference algorithm, this canbe done in a much more efficient way (e.g. by a transformation intoa database query, in SQL or Datalog).

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Novel Reasoning Problems

In recent years, other reasoning tasks have been gaining attention

� Conjunctive Query Answering

conjunctive queries allow to join pieces of informationmore expressive: union of CQAs (akin to SQL), rules

� Inconsistency Handling

repair inconsistent KB or avoid that K |= α for every α (‘knowledgeexplosion’) if K is inconsistent by introducing, e.g., new semantics

� Entailment Explanation

identify axioms in the knowledge base that support a conclusionK |= α, typically a smallest K′ ⊆ K such that K′ |= α(’axiom pinpointing’)

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Conjunctive Query Answering

Generalize Instance Retrieval by allowing joins and projections:

q(Z) = ∃Y ∃X.childOf (Z, Y ) ∧ childOf (Z,X) ∧marriedWith(Y,X)

� In databases:

� just one model (the DB itself) by Closed World Assumption(R. Reiter, 1978: if atom A is not provable from DB, ¬A is true).

� this is rather easy

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Conjunctive Query Answering, cont’d

Generalize Instance Retrieval by allowing joins and projections:q(Z) = ∃Y ∃X.childOf (Z, Y ) ∧ childOf (Z,X) ∧marriedWith(Y,X)

� In Description Logics:

� one knowledge base, many models (Open World Assumption)

� not so easy

� the ∃-variables must be suitably mapped in every model

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Algorithmic Approaches to DL Reasoning

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Types of Reasoning Procedures

� Roughly, DL inference algorithms can be separated into two groups:

model-based algorithms: show satisfiability by constructing a model(or a representation of it).

Examples: tableau, automata, type elimination algorithms

proof-based algorithms: apply deduction rules to the KB to infernew axioms.

Examples: resolution, consequence-based algorithms

Note:

� both strategies are known from first-order logic theorem proving

� additional care is needed to ensure decidability (in particularcompleteness and termination of algorithms).

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Tableau Algorithm for DLs

Tableau-based techniques

They try to decide the satisfiability of a formula (or theory) by using rulesto construct (a representation of) a model.

� Tableau-based techniques have been used in FOL and modal logicsfor many years.

� For DLs, they have been extensively explored since the late 1990s[Smolka, 1990], [Baader and Sattler, 2001].

� They are considered well-suited for implementation.

� In fact, many of the most successful DL reasoners implementtableau techniques or variations of them,e.g.: RACER, FaCT++, Pellet, Hermit, etc.

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Tableau Algorithm for Deciding KB Satisfiability

� Perform a ”bottom-up” construction of a model:

� Initialize an interpretation by all explicitly known (i.e., named)individuals and their known properties.

� Most probably, this ”model draft” will violate some of the axioms.

� Iteratively ”repair” it by adding new information about concept orrole memberships and/or introducing new (i.e., anonymous)individuals, this may require case distinction and backtracking.

� If we arrive at an interpretation satisfying all axioms, satisfiability hasbeen shown.

� If every repairing attempt eventually results in an overt inconsistency,unsatisfiability has been shown.

Note: as the finite model property does not hold in general, not a fullmodel is constructed but a finite representation of it (cf. “blocking”).

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Tableau Overview Example: Happy Cat Owner








Is K satisfiable?

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Tableau Overview Example: Happy Cat OwnerRBox Rowns v caresFor


Cat v Dead tAliveHappyCatOwner v ∃owns.Cat u ∀caresFor .Healthy

ABox AHappyCatOwner(s)

Is K satisfiable?

s

c

L(s) = {HappyCatOwner}

L(c) = {Cat}

owns caresFor

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Is K satisfiable?

s

c


L(c) = {Cat}

owns

caresFor

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Is K satisfiable?

s

c


L(c) = {Cat}

owns caresFor

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Is K satisfiable?

s

c


L(c) = {Cat , Healthy}

owns caresFor

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Is K satisfiable?

s

c


L(c) = {Cat , Healthy,¬Dead}

owns caresFor

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Is K satisfiable?

s

c


L(c) = {Cat , Healthy,¬Dead,Alive}

owns caresFor

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Is K satisfiable? Yes!

s

c


L(c) = {Cat , Healthy,¬Dead,Alive}

owns caresFor

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Naive Tableau Algorithm for ALCGiven a KB in NNF we construct a tableau, which for ALC KBs consists of

� a set of nodes, labeled with individual names or variable names

� directed edges between some pairs of nodes

� for each node labeled x, a set L(x) of class expressions and

� for each pair of nodes x and y, a set L(x, y) of role names.

Provisos:

� omit edges which are labeled with the empty set

� assume > is contained in L(x) for any x.

� concept expressions should be in negation normal form

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Recall ALC Syntax

Construct Syntax Example Semantics

atomic concept A Doctor AI ⊆ ∆I

atomic role r hasChild rI ⊆ ∆I ×∆I

conjunction C uD Human uMale CI ∩DI

unqual. exist. res.2 ∃r ∃hasChild {o | ∃o′.(o, o′) ∈ rI}value res. ∀r.C ∀hasChild .Male {o | ∀o′.(o, o′) ∈ rI → o′ ∈ CI}full negation ¬C ¬∀hasChild .Male ∆I\CI

2Unqualified existential restriction29 / 40


Negation Normal Form

Negation normal form (NNF)

A concept expression C is in negation normal form, if negation occurs inC only in front of atomic concepts, nominal concepts and self-restrictions.

Given a KB K to construct nnf (K) we need to� replace every C ≡ D by C v D and D v C;� replace every C v D by ¬C tD;� recursively translate every C into nnf (C ):

nnf(C) C if C ∈ {A,¬A, {a1 . . . an},¬{a1 . . . an},∃r.Self,¬∃r.Self,>,⊥}nnf(¬¬C) nnf(C)nnf(C uD) nnf(C) u nnf(D)nnf(C tD) nnf(C) t nnf(D)

nnf(¬(C tD)) nnf(¬C u ¬D)nnf(¬(C uD)) nnf(¬C t ¬D)

nnf(∀r.C) ∀r.nnf(C)nnf(∃r.C) ∃r.nnf(C)

nnf(¬∀r.C) ∃r.nnf(¬C)nnf(¬∃r.C) ∀r.nnf(¬C)nnf(≤ k r.C) ≤ k r.nnf(C)nnf(≥ k r.C) ≥ k r.nnf(C)

nnf(¬ ≤ k r.C) ≥ (k + 1) r.nnf(C)nnf(¬ ≥ k r.C) ≤ (k − 1) r.nnf(C)

nnf(¬>) ⊥

C and nnf(C) are logically equivalent, i.e., CI = nnf(C)I , for every interpre-tation I, and the translation process terminates in linear time.

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Given a KB K to construct nnf (K) we need to� replace every C ≡ D by C v D and D v C;� replace every C v D by ¬C tD;

� recursively translate every C into nnf (C ):






nnf(¬>) ⊥


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nnf(¬>) ⊥


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nnf(¬>) ⊥

C and nnf(C) are logically equivalent, i.e., CI = nnf(C)I , for every interpre-tation I, and the translation process terminates in linear time. 30 / 40


Example: Negation Normal Form


FilmActor v (∃actedIn uArtist) t ¬(¬∃actedIn t ∃playsIn.Theater)

1. ¬FilmActor t (∃actedIn uArtist) t ¬(¬∃actedIn t ∃playsIn.Theater)

2. ¬FilmActor t (∃actedIn uArtist) t (∃actedIn u ¬∃playsIn.Theater)

3. ¬FilmActor t (∃actedIn uArtist) t (∃actedIn u ∀playsIn.¬Theater)

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Initial Tableau

For an ALC knowledge base K in negation normal form, the initial tableau isdefined as follows:

1. For each individual a occurring in K, create a node labeled a and setL(a) = ∅.

2. For all pairs a, b of individuals, set L(a, b) = ∅.

3. For each ABox statement C(a) in K, set L(a)← C.

4. For each ABox statement r(a, b) in K set L(a, b)← r.

Note: “←” means “update with”, “add to”

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Example: Initial Tableau

K = {A(a), (∃r.B)(a), r(a, b), r(a, c), s(b, b), (A tB)(c),¬A t (∀s.B)}

b

a

c

L(b) = ∅

L(a) = {A, ∃r.B}

L(c) = {A tB}

r

s

s

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b

a

c

L(b) = ∅

L(a) = {A,

∃r.B}

L(c) = {A tB}

r

s

s

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b

a

c

L(b) = ∅

L(a) = {A, ∃r.B}

L(c) = {A tB}

r

s

s

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Expansion Rules for the Naive Tableau

� u-rule: If C uD ∈ L(x) and {C,D} 6⊆ L(x), then setL(x)← {C,D}.

� t-rule: If C tD ∈ L(x) and {C,D} ∩ L(x) = ∅, then setL(x)← C or L(x)← D.

� ∃-rule: If ∃r.C ∈ L(x) and there is no y with r ∈ L(x, y) andC ∈ L(y), then

1. add a new node with label y (where y is a new node label),

2. set L(x, y) = {r}, and

3. set L(y) = {C}.

� ∀-rule: If ∀r.C ∈ L(x) and there is a node y with r ∈ L(x, y) andC 6∈ L(y), then set L(y)← C.

� TBox-rule: If C is a (rewritten and normalized) TBox statement andC 6∈ L(x), then set L(x)← C.

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Naive Tableau AlgorithmInput: ALC knowledge base K in negation normal form

Output: “yes” if K is satisfiable

1. Initialize the tableau;

2. While some expansion rule is applicable:

2.1. nondeterministically apply an applicable rule;2.2. if for some node x, there exists some C ∈ L(x) such that ¬C ∈ L(x),

output “no” and terminate;

3. Output “yes”.

A nondeterministic run of the algorithm terminates, if either

� for some node x, L(x) contains a contradiction (“no”; attempt to find a modelfor K was unsuccessful), or

� no expansion rule is applicable (“yes”; attempt was successful)

� K is satisfiable if some run outputs “yes”, andunsatisfiable if every run outputs “no’

� only the t-rule creates true branching regarding yes/no output

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Example: Expansion Rules

K = {C(a), ¬C t ∃r.D, ¬D t E, ∀r.¬E(a)}

a

L(a) = {C, ∀r.¬E,¬C t ∃r.D, ∃r.D}

x

r

L(x) = {D,¬D t E, E, ¬E}

Clash is obtained, KB is unsatisfiable!

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K = {C(a), ¬C t ∃r.D, ¬D t E, ∀r.¬E(a)}

a L(a) = {C,

∀r.¬E,¬C t ∃r.D, ∃r.D}

x

r

L(x) = {D,¬D t E, E, ¬E}


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K = {C(a), ¬C t ∃r.D, ¬D t E, ∀r.¬E(a)}

a L(a) = {C, ∀r.¬E,

¬C t ∃r.D, ∃r.D}

x

r

L(x) = {D,¬D t E, E, ¬E}


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K = {C(a), ¬C t ∃r.D, ¬D t E, ∀r.¬E(a)}

a L(a) = {C, ∀r.¬E,¬C t ∃r.D,

∃r.D}

x

r

L(x) = {D,¬D t E, E, ¬E}


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K = {C(a), ¬C t ∃r.D, ¬D t E, ∀r.¬E(a)}

a L(a) = {C, ∀r.¬E,¬C t ∃r.D, ∃r.D}

x

r

L(x) = {D,

¬D t E, E, ¬E}


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K = {C(a), ¬C t ∃r.D, ¬D t E, ∀r.¬E(a)}

a L(a) = {C, ∀r.¬E,¬C t ∃r.D, ∃r.D}

x

r

L(x) = {D,¬D t E,

E, ¬E}


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K = {C(a), ¬C t ∃r.D, ¬D t E, ∀r.¬E(a)}

a L(a) = {C, ∀r.¬E,¬C t ∃r.D, ∃r.D}

x

r

L(x) = {D,¬D t E, E,

¬E}


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K = {C(a), ¬C t ∃r.D, ¬D t E, ∀r.¬E(a)}

a L(a) = {C, ∀r.¬E,¬C t ∃r.D, ∃r.D}

x

r

L(x) = {D,¬D t E, E, ¬E}


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Tableau Algorithm with Blocking for ALC

� Naive tableau algorithm does not always terminateExample: K = {¬Person t ∃hasParent , Person(a1 )}.

� Modify the naive tableau algorithm to ensure termination

� Use blocking

� A node with label x is directly blocked by a node with label y if� x is a variable (i.e., not an individual),� y is an ancestor of x, and� L(x) ⊆ L(y)

� A node with label x is blocked, if it is directly blocked or one of itsancestors is blocked

� Expansion rules may only be applied if x is not blocked

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Example: Blocking

K = {B(t), ¬H t ∃P.H, H(t)}

t

L(t) = {B,H,¬H t ∃P.H,∃P.H}

x

P

L(x) = {H} ⊆ L(t)

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Example: Blocking

K = {B(t), ¬H t ∃P.H, H(t)}

t L(t) = {B,H,

¬H t ∃P.H,∃P.H}

x

P

L(x) = {H} ⊆ L(t)

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Example: Blocking

K = {B(t), ¬H t ∃P.H, H(t)}

t L(t) = {B,H,¬H t ∃P.H,

∃P.H}

x

P

L(x) = {H} ⊆ L(t)

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Example: Blocking

K = {B(t), ¬H t ∃P.H, H(t)}

t L(t) = {B,H,¬H t ∃P.H,∃P.H}

x

P

L(x) = {H}

⊆ L(t)

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Example: Blocking

K = {B(t), ¬H t ∃P.H, H(t)}

t L(t) = {B,H,¬H t ∃P.H,∃P.H}

x

P

L(x) = {H} ⊆ L(t)

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Computational Properties

� The simple tableau algorithm is expensive in general (worst case):

� the worst case complexity is double exponential

� testing KB consistency in ALC is EXPTIME-complete

� testing concept satisfiability in ALC is PSPACE-complete

� Still in practice, (optimized) tableaux algorithms work well

� other notions of blocking might be used, e.g. “cross-path blocking”(blocking node need not be an ancestor)

� single exponential time tableaux algorithms are available

� For many other description logics, also tableaux algorithms exist(e.g. SHIQ, SHOIQ, ...)Some algorithms are quite involved!

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Summary

1. Satisfaction and Entailment� Notions� Decidability

2. Reasoning Problems� Knowledge base consistency� Entailment checking� Concept satisfiability� Classification� Instance retrieval

3. Algorithmic Approaches to DL Reasoning� Types of reasoning procedures� Tableaux

4. Novel reasoning problems� Conjunctive query answering

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References I

Franz Baader, Diego Calvanese, Deborah McGuinness, Daniele Nardi, andPeter Patel-Schneider, editors.

The Description Logic Handbook: Theory, Implementation andApplications.

Cambridge University Press, 2007.

Pascal Hitzler, Markus Krotzsch, and Sebastian Rudolph.

Foundations of Semantic Web Technologies.

Chapman and Hall, 2010.

Sebastian Rudolph.

Foundations of description logics.

In Axel Polleres, Claudia d’Amato, Marcelo Arenas, Siegfried Handschuh,Paula Kroner, Sascha Ossowski, and Peter Patel-Schneider, editors,Reasoning Web. Semantic Technologies for the Web of Data, volume 6848of Lecture Notes in Computer Science, pages 76–136. Springer Berlin /Heidelberg, 2011.

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