Approximate Reasoning for the Semantic Web Part II OWL Semantics - - PDF document

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van Harmelen, Hitzler, Wache ESSLLI 2006 Malaga, Spain August 2006 AIFB Approximate Reasoning for the Semantic Web Part II OWL Semantics and Tableau Reasoning Frank van Harmelen Pascal Hitzler Holger Wache ESSLLI 2006 Summer

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Approximate Reasoning for the Semantic Web Part II OWL Semantics and Tableau Reasoning

Frank van Harmelen Pascal Hitzler Holger Wache ESSLLI 2006 Summer School Malaga, Spain, August 2006

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 2

AIFB

Introducing the speaker

  • 1998 Diplom (Master) in Mathematics

– Uni Tübingen (Helmut Salzmann)

  • 1999-2001 PhD in Mathematics

– Cork, Irland (Tony Seda) – Formal Aspects of Knowledge Representation

  • 2001-2004 Postdoc

– TU Dresden, Artificial Intelligence (Steffen Hölldobler)

  • since 2004 Assistant Professor

– AIFB Univ. Karlsruhe, Semantic Web (Rudi Studer)

  • 2005 Habilitation in Computer Science

Main Interests: Semantic Web (Knowledge Representation/Logic) Neural-symbolic Integration Mathematical Foundations of Artificial Intelligence

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Karlsruhe: Location for Semantic Technologies and Applications Semantic Karlsruhe

Knowledge Management B2B, EAI Business Intelligence Electronic Markets eGovernment Semantic Web Infrastructure Ontology Management Data, Web & Text Mining Peer-to-Peer, Semantic Grid Semantic Web Services Application-oriented Research Know-how Transfer Realizing new Scenarios Application-oriented Research Product Development Innovative Solutions Basic Research Application-oriented Research

AIFB

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 4

AIFB Who are we? ... Semantic Web Research Group

FZI AIFB

Rudi Studer

Valentin Zacharias Max Völkel Nenad Stojanovic York Sure Andreas Abecker Ljiljana Stojanovic Johanna Völker Stephan Bloehdorn Sudhir Agarwal Jens Hartmann Philipp Cimiano Mark Hefke Stephan Grimm Peter Haase Steffen Lamparter Saartje Brockmans Pascal Hitzler Denny Vrandecic Christoph Tempich Markus Krötzsch Anupriya Ankolekar Hans-Jörg Happel Heiko Haller Holger Lewen Sebastian Blohm Yimin Wang Julien Tane Knowledge Management Semantic Web Intelligent WWW-Applications Business Intelligence eGovernment Ontology Engineering Data/Text Mining Ontology Learning Peer-to-Peer Web Services Simone Braun

& ~40 people at Ontoprise

Sebastian Rudolph Guilin Qi Tuvshintur Tserendorj Thanh Tran Duc

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Partners and Projects

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Semantic Web Layer Cake

O W L + + now

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

What is Semantics?

Syntax: strings without meaning Semantics: meaning of syntax Why logic is so successful: Semantics can be captured syntactically!

syntax meaning e.g. „the world“ IF cond(A,B) THEN display(_354)

show pixel set „_354“ on screen if „A“ is of type "B".

assignment of meaning

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

What is semantics? Example programming language

FUNCTION f(n:natural):natural; BEGIN IF n=0 THEN f:=1 ELSE f:=n*f(n-1); END;

syntax intended semantics formal semantics procedural semantics computing factorial What the program does when exectued

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB Semantics of logics/knowledge representation languages

∀ X (p(X) → q(X))

syntax intended semantics model theoretic semantics proof theoretic semantics all humans are mortal

` ²

logical consequence deducible in a calculus

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Part II contents

1. OWL Model-theoretic Semantics a. Description Logics: ALC b. OWL as SHOIN(D) c. OWL Examples 2. Proof Theory a. Reasoning as Satisfiability checking b. Tableaux Reasoning

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Description Logics, DLs

  • FOL (First-Order Logic) fragments
  • usually decidable
  • "expressive"
  • come from semantic networks and frame systems
  • close relation with multi-modal logics
  • W3C Standard OWL DL is the description logic

SHOIN(D)

  • We first talk about the simpler ALC

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

General DL Architecture

Knowledge Base

Tbox (schema) Abox (data)

Man ≡ Hum an u Male Happy-Father ≡ Man u ∃ has-child.Fem ale u … Happy-Father( John) has-child( John, Mary)

I nference System I nterface

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

DLs – general remarks

  • DLs are a family of logic-based KR formalisms
  • DLs characterised by:

– Different constructors for generating complex class expressions. – Axioms for describing properties for roles.

  • ALC is the smalles DL which is propositionally closed

– Conjunction, disjunction, negation are constructors, written as u, t, ¬. – Quantifiers used only together with roles:

Man u ∃hasChild.Female u u ∃hasChild.Male u u ∀hasChild.(Rich t Happy)

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Other DL language components

  • E.g.

– Number restrictions (cardinality constraints) for Roles: ≥3 hasChild, ·1hasMother – Qualified number restrictions: ≥2 hasChild.Female, ·1 hasParent.Male – Nominals (definition by extension): {Italy, France, Spain} – Concrete domains (datatypes): hasAge.(≥21) – Inverse roles: hasChild– ≡ hasParent – Transitive roles: hasAncestor v+ hasAncestor – Role composition: hasParent.hasBrother(uncle)

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

ALC: basic language elements

  • basic language components:

– classes – roles – individuals

  • Professor(RudiStuder)

– Individual RudiStuder is in class Professor

  • affiliation(RudiStuder,AIFB)

– RudiStuder has affiliation AIFB

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

ALC: subclass relation

  • Professor v Faculty

– translates to (∀x)(Professor(x) → Faculty(x)) – corresponds to owl:subClassOf

  • Professor ≡ Faculty

– translates to (∀x)(Professor(x) ↔ Faculty(x)) – corresponds to owl:equivalentClass

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

ALC: complex class descriptions

  • conjunction u
  • disjunction t
  • negation ¬
  • Professor v (Person u Faculty)

t (Person u ¬PhDStudent) (∀x)(Professor(x) → ((Person(x) ∧ Faculty(x)) ∨ (Person(x) ∧ ¬PhDStudent(x)))

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

ALC: Quantifiers

  • Exam v ∀hasExaminer.Professor

(∀x)(Exam(x) → (∀y)(hasExaminer(x,y) → Professor(y))) – corresponds to owl:allValuesFrom

  • Exam v ∃hasExaminer.Person

(∀x)(Exam(x) → (∃y)(hasExaminer(x,y) ∧ Person(y))) – corresponds to owl:someValuesFrom

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Modelling in ALC

  • owl:nothing:

⊥ ≡ C u ¬C

  • owl:thing:

> ≡ C t ¬C

  • owl:disjointWith:

C u D ≡ ⊥ equivalently: C v ¬D

  • rdfs:range:

> v ∀R.C

  • rdfs:domain:

∃R.> v C

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

ALC: Syntax

  • The following rules generate classes in ALC, where A is

an atomic (named) class and R is a role. C,D → A | > | ⊥ | ¬C | C u D | C t D | ∀R.C | ∃R.C

  • An ALC TBox consists of assertions (axioms) of the form

C v D and C ≡ D, where C,D are classes.

  • An ALC ABox consists of assertions of the form C(a) and

R(a,b), where C is a complex class, R is a role and a,b are individuals.

  • An ALC-knowledge base consists of an ABox and a TBox.
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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

ALC: Semantics

Defined by translating TBox axioms into FOL via the mapping π (shown to the right). Here, C,D are complex classes, R is a role and A is an atomic class.

A(y)

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

DL knowledge bases

  • DL knowledge bases consist of two parts:

– TBox: Axioms containing schema knowledge:

  • HappyFather ≡ Man u ∃hasChild.Female u …
  • Elephant v Animal u Large u Grey
  • transitive(hasAncestor)

– Abox: Axioms describing data:

  • HappyFather(John)
  • hasChild(John, Mary)
  • Distinction between ABox and TBox has no logical

significance whatsoever …but it makes some things easier to talk about.

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Simple example

Terminological knowledge (TBox): Human v ∃parentOf.Human Orphan ≡ Human u ¬∃hasParent.Alive Data (ABox): Orphan(harrypotter) hasParent(harrypotter,jamespotter) Semantics and logical consequences are understood via translation to FOL.

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Part II contents

1. OWL Model-theoretic Semantics a. Description Logics: ALC b. OWL as SHOIN(D) c. OWL Examples 2. Proof Theory a. Reasoning as Satisfiability checking b. Tableaux Reasoning

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

OWL and ALC

The following OWL DL primitives are expressible in ALC:

  • classes, roles, individuals
  • class membership, role instances
  • > and ⊥
  • class inclusion, equivalence, and disjointness
  • u, t
  • ¬
  • role restrictions
  • range and domain

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

OWL as SHOIN(D): Individuals

  • owl:sameAs

– equality of individuals – DL: a=b – FOL: need extension with equality predicate

  • owl:differentFrom

– inequality of individuals – DL: a≠b – FOL: ¬(a=b)

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

OWL as SHOIN(D): nominals

Nominals

  • owl:oneOf

– closed class (definition by extension) – DL: C ≡ {a,b,c} – FOL: (∀x) (C(x) ↔ (x=a ∨ x=b ∨ x=c))

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

OWL as SHOIN(D): number restrictions

number restrictions need equality predicate

<owl:Class rdf:about="#Exam"> <rdfs:subClassOf> <owl:Restriction> <owl:onProperty rdf:resource="#hasExaminer"/> <owl:maxCardinality rdf:datatype="&xsd;nonNegativeInteger">2</owl:maxcardinality> </owl:Restriction> </rdfs:subClassOf> </owl:Class>

An exam may have at most two examiners.

  • DL: Exam v ≤2 hasExaminer
  • In FOL: (E… Exam, h…hasExaminer)

(∀x)(E(x) → ¬(∃x1)(∃x2)(∃x3)(x1 ≠ x2 ∧ x2 ≠ x3 ∧ x1 ≠ x3 ∧ h(x,x1) ∧ h(x,x2) ∧ h(x,x3))) Similarly for the other number restrictions.

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

OWL as SHOIN(D): role constructors

  • Rdfs:subPropertyOf

– DL: R v S – FOL: (∀x)(∀y)(R(x,y) → S(x,y))

  • similarly for role equivalence
  • Inverse Roles: R ≡ S-

– FOL: (∀x)(∀y)(R(x,y) ↔ S(y,x))

  • Transitive Roles: R+ v R

– FOL: (∀x)(∀y)(∀z)(R(x,y) ∧ R(y,z) → R(x,z))

  • Symmetry: R ≡ R-
  • Functionality: > v ≤1 R
  • Inverse Functionality: > v ≤1 R-

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

datatypes

  • Allow usage of datatypes in the second argument of

concrete roles in the ABox.

  • Furthermore, a nominal (closed class) can consist of

a set of datatype elements.

  • It is not possible to express datatypes directly in FOL.

But FOL syntax/semantics can be extended to encompass datatypes.

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

OWL DL as SHOIN(D): overview

Allowed are:

  • ALC
  • Equality and inequality between individuals
  • Nominals
  • Number restrictions
  • Subroles and role equivalence
  • Inverse and transitive roles
  • datatypes

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Naming conventions for DLs

  • ALC: Attribute Language with Complement
  • S: ALC + role transitivity
  • H: subrole relations
  • O: nominals
  • I: inverse roles
  • N: number restrictions ≤n R etc.

– Q: Qualified number restrictions ≤n R.C etc.

  • (D): Datatypes
  • F: Functional roles
  • OWL DL is SHOIN(D)
  • OWL Lite is SHIF(D)
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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB Ontology (=Knowledge Base)

Overview syntax for DLs (without datatypes)

{i1,…,in} Nominal C u D And ≤n R.C (≤n R) At most ≥n R.C (≥n R) At least ∀ R.C For all ∃ R.C Exists C t D Or ¬C Not A, B Atomic

Concepts Roles

R- Inverse R Atomic

ALC Q (N) I Concept Axioms (TBox)

C ≡ D Equivalent C v D Subclass

Role Axioms (RBox) Assertional Axioms (ABox)

a ≠ b Different a = b Same R(a,b) Role C(a) Instance

H O S

Trans(S) Transitivity R v S Subrole

S = ALC + Transitivity OWL DL = SHOIN(D) (D: concrete domain)

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

OWL DL as DL: overview class constructors

nesting is allowed: Person u ∀hasChild.(Doctor t ∃hasChild.Doctor)

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

OWL DL as DL: axioms

  • General Class Inclusion (v) suffices:

C ≡ D gdw ( C v D und D v C )

  • Equivalences

C ≡ D ⇔ (∀x) ( C(x) ↔ D(x) ) C v D ⇔ (∀x) ( C(x) → D(x) )

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Computational complexity (worst-case)

Exptime Exptime NExptime undecidable combined complexity NP OWL Lite NP (IJCAI 2005) OWL DL without nominals unknown OWL DL undecidable OWL Full data complexity OWL variant data complexity: complexity w.r.t. ABox size combined complexity: complexity w.r.t. ABox and TBox size

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Part II contents

1. OWL Model-theoretic Semantics a. Description Logics: ALC b. OWL as SHOIN(D) c. OWL Examples 2. Proof Theory a. Reasoning as Satisfiability checking b. Tableaux Reasoning

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Examples (abstract syntax)

Class(a:bus_driver complete intersectionOf(a:person restriction(a:drives someValuesFrom (a:bus)))) Class(a:driver complete intersectionOf(a:person restriction(a:drives someValuesFrom (a:vehicle)))) Class(a:bus partial a:vehicle)

  • A bus driver is a person that drives a bus.
  • A bus is a vehicle.
  • A bus driver drives a vehicle, so must be a driver.

The subclass is inferred due to subclasses being used in existential quantification.

bus_driver ≡ person u ∃drives.bus bus v vehicle driver ≡ person u ∃drives.vehicle

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Examples

Class(a:driver complete intersectionOf(a:person restriction(a:drives someValuesFrom (a:vehicle)))) Class(a:driver partial a:adult) Class(a:grownup complete intersectionOf(a:adult a:person))

  • Drivers are defined as persons that drive cars (complete

definition)

  • We also know that drivers are adults (partial definition)
  • So all drivers must be adult persons (i.e. grownups)

An example of axioms being used to assert additional necessary information about a class. We do not need to know that a driver is an adult in order to recognize one, but once we have recognized a driver, we know that they must be adult. driver ≡ person u ∃drives.vehicle driver v adult grownup ≡ adult u person

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 40

AIFB

Part II contents

1. OWL Model-theoretic Semantics a. Description Logics: ALC b. OWL as SHOIN(D) c. OWL Examples 2. Proof Theory a. Reasoning as Satisfiability checking b. Tableaux Reasoning

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Important inference problems

  • global consistency of knowledge base

KB ² false? – Is knowledge base reasonable?

  • class consistency

C ≡ ⊥? – Is class c well-modeled?

  • subsumption

C v D? – structuring the knowledge base

  • class equivalence

C ≡ D? – Are they actually the same?

  • class disjointness

C u D = ⊥? – Do they have common instances?

  • class membership

C(a)? – Does the instance belong to the class?

  • instance retrieval

„find all X with C(X)“ – Give me all instances with some given properties.

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Decidability of OWL DL reasoning

  • Decidability: For every inference problem there's a

terminating decision procedure.

  • OWL DL is FOL fragment. In principle, one could

attempt to use standard FOL algorithms

  • But they don't always terminate!
  • Problem: Find terminating algorithms.
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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Reasoning via satisfiability checking

  • We will modify standard tableaux algorithms.

– We will cover ALC only.

  • Tableaux algorithms work by showing unsatisfiability.

→ Reduce reasoning to finding inconsistencies in some knowledge base, i.e. we show unsatisfiability of the knowledge base!

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 44

AIFB

Reasoning by satisfiability checking

  • class consistency

C ≡ ⊥? – KB ∪ {C(a)} unsatisfiable (a new)

  • subsumption

C v D? – KB ∪ {C u ¬D(a)} unsatisfiable (a new)

  • class equivalence

C ≡ D? – C v D and D v C

  • class disjointness

C u D = ⊥? – KB ∪ {(C u D)(a)} unsatisfiable (a new)

  • class membership

C(a)? – KB ∪ {¬C(a)} unsatisfiable (a new)

  • instance retrieval

„find all X with C(X)“ – Check membership for all known individuals

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 45

AIFB

Part II contents

1. OWL Model-theoretic Semantics a. Description Logics: ALC b. OWL as SHOIN(D) c. OWL Examples 2. Proof Theory a. Reasoning as Satisfiability checking b. Tableaux Reasoning

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 46

AIFB

ALC Tableaux: contents

  • Transformation to negation normal form
  • Naive tableau algorithm
  • Tableau algorithm with Blocking
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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Transformation to negation normal form

Let W be a knowledge base

  • replace C ≡ D by C v D and D v C.
  • replace C v D by ¬C t D.
  • Apply rules on next slides exhaustively.

resulting knowledge base: NNF(W) negation normal form of W. Negation only occurs directly in front of atomic classes.

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 48

AIFB

NNF(C) = C, if C atomic NNF(¬C) = ¬C, if C atomic NNF(¬¬C) = NNF(C) NNF(C t D) = NNF(C) t NNF(D) NNF(C u D) = NNF(C) u NNF(D) NNF(¬(C t D)) = NNF(¬C) u NNF(¬D) NNF(¬(C u D)) = NNF(¬C) t NNF(¬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) W and NNF(W) are logically equivalent.

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

negation normal form: example

P v (E u U) t ¬(∀R.E t D). In negation normal form: ¬P t (E u U) t (∃R.¬E u ¬D).

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

ALC Tableaux: contents

  • Transformation to negation normal form
  • Naive tableau algorithm
  • Tableau algorithm with Blocking
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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Naive tableau algorithm

reduce to finding an inconsistency Idea:

  • Given a knowledge base W.
  • Generate consequences of the form C(a) and ¬C(a),

until a contradiction is found.

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 52

AIFB

Simple example

C(a) (¬C u D)(a) ¬C(a) is logical consequence: 2nd formula in FOL: ¬C(a)∧ D(a) hence ¬C(a) Contradiction has been found.

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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Another example

C(a) ¬C t D ¬D(a) Derive logical consequences: C(a) ¬D(a) (¬C t D)(a) Now we consider two cases

  • 1. ¬C(a)

contradiction

  • 2. D(a)

contradiction

Splitting of the tableau into two brances

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

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AIFB

Tableau: definitions

  • Tableau branch:

Finite set of statements of the form C(a), ¬C(a), R(a,b).

  • Tableau: Finite set of tableau branches.
  • A tableau branch is closed if it contains a pair C(a)

and ¬C(a) of contradictory statements.

  • Tableau is closed if every branch of it is closed.
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van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 55

AIFB

Generating a tableau

If the resulting tableau is closed, then the original knowledge base is unsatisfiable. Always select only such elments which add new elements to the tableau. If this is impossible, then terminate the algorithm – then the knowledge base is satisfiable.

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 56

AIFB

Example

  • P … Professor

E … Person U … University member D … PhD student

  • knowledge base: P v (E u U) t (E u ¬D)

Is P v E a logical consequence?

  • Knowledge base (with query) in NNF:

{¬P t (E u U) t (E u ¬D), (P u ¬E)(a)}

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SLIDE 29

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 57

AIFB

TBox: ¬P t (E u U) t (E u ¬D) Tableau: (P u ¬E)(a) (from knowledge base) P(a) ¬E(a) (¬P t (E u U) t (E u ¬D))(a) ¬P(a) ((E u U) t (E u ¬D))(a) (E u ¬D)(a) E(a) ¬D(a) (E u U)(a) E(a) U(a)

Example (ctd)

Knowledge base is unsatisfiable, i.e. P v E.

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 58

AIFB

The termination problem

  • Single Axiom: ¬Person t ∃hasParent.Person

we want to show: ¬Person(Bill) Person(Bill) hasParent(Bill,x1) Person(x1) ∃ ¬Person t ∃hasParent.Person(Bill) ¬Person(Bill) ∃hasParent.Person(Bill) t ¬Person(x1) ∃hasParent.Person(x1) t hasParent(x1,x2) Person(x2) ∃ ¬Person(x2) ∃hasParent.Person(x2) t ¬Person t ∃hasParent.Person(x1) ¬Person t ∃hasParent.Person(x2)

Problem occurs due to existential quantification (and minCardinality) etc.

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SLIDE 30

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 59

AIFB

ALC Tableaux: contents

  • Transformation to negation normal form
  • Naive tableau algorithm
  • Tableau algorithm with Blocking

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 60

AIFB

Solving the termination problem

  • The following happened:

Person ∃hasParent.Person Person ∃hasParent.Person Person ∃hasParent.Person

  • But why not do it this way:

Person ∃hasParent.Person Person ∃hasParent.Person hasParent hasParent hasParent hasParent hasParent

I.e. reuse old nodes! Formal proof required that this suffices! Blocking!

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SLIDE 31

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 61

AIFB

Tableau with Blocking

Person(Bill) hasParent(Bill,x1) Person(x1) ∃ ¬Person t ∃hasParent.Person(Bill) v ¬Person(Bill) ∃hasParent.Person(Bill) t ¬Person(x1) ∃hasParent.Person(x1) t ¬Person ∪ ∃ ∪ ∃hasParent.Person(x1) v σ(Bill) = { Person, ¬Person t ∃hasParent.Person, ∃hasParent.Person } σ(x1) = { Person, ¬Person t ∃hasParent.Person, ∃hasParent.Person } σ(x1) ⊆ σ ⊆ σ(Bill), so Bill blocks x1

Person ∃hasParent.Person hasParent Person ∃hasParent.Person hasParent Bill x1

  • Single Axiom: ¬Person t ∃hasParent.Person

We want to show: ¬Person(Bill)

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 62

AIFB

Blocking

The selection of (∃R.C)(a) in branch A is blocked if there is an individual b with {C | C(a) ∈ A} ⊆ {C | C(b) ∈ A}. Now two ways of terminating

  • 1. Tableau is closed.

Then knowledge base is unsatisfiable.

  • 2. No unblocked selection extends the tableau.

Then knowledge base is satisfiable.

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SLIDE 32

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 63

AIFB

Example

  • F … Women

h … hasMother V … Bird t … Tweety

  • knowledge base {F v ∃h.F, V(t)}
  • We want to show that Tweety is not a Women,

i.e. that ¬F(t) is a logical consequence.

  • It will not be possible to prove this,

i.e. Tweety can be a Woman.

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 64

AIFB

Example (ctd)

TBox: ¬Ft ∃h.F Tableau: V(t) (from knowledge base) F(t) (negated query in NNF) (¬Ft ∃h.F)(t) ¬F(t) (∃h.F)(t) h(t,s) F(s) (¬Ft ∃h.F)(s) ¬F(s) (∃h.F)(s) blocked by t s and t fall under F, ¬F t h.F, ∃h.F no other selection possible

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SLIDE 33

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 65

AIFB

Tableaux for OWL DL

  • Basic ideas are the same.
  • Need more complex blocking rules.
  • Instance generation is not very efficient.
  • Tableau with blocking is 2NExptime!

→ worse than necessary!

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 66

AIFB

Tableaux inference engines

  • Fact

– http://www.cs.man.ac.uk/~horrocks/FaCT/ – SHIQ

  • Fact++

– http://owl.man.ac.uk/factplusplus/ – SHOIQ(D)

  • Pellet

– http://www.mindswap.org/2003/pellet/index.shtml – SHOIN(D)

  • RacerPro

– http://www.sts.tu-harburg.de/~r.f.moeller/racer/ – SHIQ(D)

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SLIDE 34

van Harmelen, Hitzler, Wache ● ESSLLI 2006 ● Malaga, Spain ● August 2006

Slide 67

AIFB

Acknowledgements

For the preperation of these slides, I did not hesitate to reuse any material which I found on the web or on my

  • computer. Some of it is derived from slides by
  • last years’ ISWWW lecture / Steffen Staab et al.
  • Sean Bechhofer, Manchester
  • Ian Horrocks, Manchester
  • Boris Motik, FZI Karlsruhe
  • Alan Rector et al., Manchester (OWL Pizzas)
  • Denny Vrandecic, AIFB Karlsruhe
  • and possibly some other people for the cases where I

couldn’t trace the origin of my files …