Basics of Reasoning in Description Logics

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Basics of Reasoning in Description Logics

• Jie Bao
• Iowa State University
• Feb 7, 2006

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An ontology of this talk
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Roadmap

• What is Description Logics (DL)
• Semantics of DL
• Basic Tableau Algorithm
• Advanced Tableau Algorithm

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Description Logics

• A formal logic-based knowledge representation
  language
• Description" about the world in terms of
  concepts (classes), roles (properties,
  relationships) and individuals (instances)
• Decidable fragments of FOL
• Widely used in database (e.g., DL CLASSIC) and
  semantic web (e.g., OWL language)

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A Family Knowledge Base

• Person include Man(Male) and Woman(Female),
• A Man is not a Woman
• A Father is a Man who has Child
• A Mother is a Woman who has Child
• Both Father and Mother are Parent
• Grandmother is a Mother of a Parent
• A Wife is a Woman and has a Husband( which as
  Man)
• A Mother Without Daughter is a Mother whose all
  Child(ren) are not Women

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DL for Family KB
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DL Basics

• Concepts (unary predicates/formulae with one free
  variable)
• E.g., Person, Father, Mother
• Roles (binary predicates/formulae with two free
  variables)
• E.g., hasChild, hasHudband
• Individual names (constants)
• E.g., Alice, Bob, Cindy
• Subsumption (relations between concepts)
• E.g. Female ? Person
• Operators (for forming concepts and roles)
• And(?) , Or(U), Not ()
• Universal qualifier (?), Existent qualifier(?)
• Number restiction ?, ?,
• Inverse role (-), transitive role (), Role
  hierarchy

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More for Family Ontology

• (Inverse Role) hasParent hasChild-
• hasParent(Bob,Alice) -gt hasChild(Alice, Bob)
• (Transitive Role)hasBrother
• hasBrother(Bob,David), hasBrother(David, Mack)
  -gt hasBrother(Bob,Mack)
• (Role Hierarchy) hasMother ? hasParent
• hasMother(Bob,Alice) -gt hasParent(Bob, Alice)
• HappyFather ? Father ? ?1 hasChild.Woman ? ?1
  hasChild.Man

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DL Architecture
Knowledge Base
Tbox (schema)
HappyFather ? Person ? ?1 hasChild.Woman ? ?1
hasChild.Man
Interface
Inference System
Abox (data)
Happy-Father(Bob)
(Example from Ian Horrocks, U Manchester, UK)
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DL Representives

• ALC the smallest DL that is propositionally
  closed
• Constructors include booleans (and, or, not),
• Restrictions on role successors
• SHOIQ OWL DL
• SALCR ALC with transitive role
• H role hierarchy
• O nomial .e.g WeekEnd Saturday, Sunday
• I Inverse role
• Q qulified number restriction e.g. gt1
  hasChild.Man
• N number restriction e.g. gt1 hasChild

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Roadmap

• What is Description Logic (DL)
• Semantics of DL
• Basic Tableau Algorithm
• Advanced Tableau Algorithm

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Interpretations

• DL Ontology is a set of terms and their
  relations
• Interpretation of a DL Ontology A possible world
  ("model") that materalizes the ontology

Ontology Student ? People Student ?
?Present.Topic KR ? Topic DL ? KR
Interpretation
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DL Semantics

• DL semantics defined by interpretations I (DI,
  .I), where
• DI is the domain (a non-empty set)
• .I is an interpretation function that maps
• Concept (class) name A -gt subset AI of DI
• Role (property) name R -gt binary relation RI over
  DI
• Individual name i -gt iI element of DI
• Interpretation function .I tells us how to
  interpret atomic concepts, properties and
  individuals.
• The semantics of concept forming operators is
  given by extending the interpretation function in
  an obvious way.

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DL Semantics example

• I (DI, .I)
• DI Jie_Bao, DL_Reasoning
• PeopleIStudentIJie_Bao
• TopicIKRIDLIDL_Reasoning
• PresentI(Jie_Bao, DL_Reasoning)

An interpretation that satisifies all axioms in
an DL ontology is also called a model of the
ontology.
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Source Description Logics Tutorial, Ian Horrocks
and Ulrike Sattler, ECAI-2002,
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Source Description Logics Tutorial, Ian Horrocks
and Ulrike Sattler, ECAI-2002,
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Roadmap

• What is Description Logic (DL)
• Semantics of DL
• Basic Tableau Algorithm
• Advanced Tableau Algorithm

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What is Reasoning?

• "Machine Understanding"
• Find facts that are implicit in the ontology
  given explicitly stated facts
• Find what you know, but you don't know you know
  it - yet.
• Example
• A is father of B, B is father of C, then A is
  ancestor of C.
• D is mother of B, then D is female

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Reasoning Tasks

• Knowledge is correct (captures intuitions)
• C subsumes D w.r.t. K iff for every model I of K,
  CI µ DI
• Knowledge is minimally redundant (no unintended
  synonyms)
• C is equivallent to D w.r.t. K iff for every
  model I of K, CI DI
• Knowledge is meaningful (classes can have
  instances)
• C is satisfiable w.r.t. K iff there exists some
  model I of K s.t. CI ? ?
• Querying knowledge
• x is an instance of C w.r.t. K iff for every
  model I of K, xI ? CI
• hx,yi is an instance of R w.r.t. K iff for,
  every model I of K, (xI,yI) ? RI
• Knowledge base consistency
• A KB K is consistent iff there exists some model
  I of K

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Reasoning Tasks(2)

• Many inference tasks can be reduced to
  subsumption reasoning
• Subsumption can be reduced to satisfiability

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Tableau Algorithm

• Tableau Algorithm is the de facto standard
  reasoning algorithm used in DL
• Basic intuitions
• Reduces a reasoning problem to concept
  satisfiability problem
• Finds an interpretation that satisfies concepts
  in question.
• The interpretation is incrementally constructed
  as a "Tableau"

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Short Example

• given Wife? Woman, Woman? Person question if
  Wife? Person
• Reasoning process
• Test if there is a individual that is a Woman but
  not a Person, i.e. test the satisfiability of
  concept C0(Wife?Person)
• C0(x) -gt Wife(x), (Person)(x)
• Wife(x)-gtWoman(x)
• Woman(x) -gtPerson(x)
• Conflict!
• C0 is unsatisfiable, therefore Wife? Person is
  true with the given ontology.

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General Process

• Transform C into negation normal form(NNF), i.e.
  negation occurs only in front of concept names.
• Denote the transformed expression as C0, the
  algorithm starts with an ABox A0 C0(x0), and
  apply consistency-preserving transformation rules
  (tableaux expansion) to the ABox as far as
  possible.
• If one possible ABox is found, C0 is satisfiable.
• If not ABox is found under all search pathes, C0
  is unsatisfiable.

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NNF
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Tableaux Expansion(Selected)
Clash
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Termination Rules

• An ABox is called complete if none of the
  expansion rules applies to it.
• An ABox is called consistent if no logic clash is
  found.
• If any complete and consistent ABox is found, the
  initial ABox A0 is satisfiable
• The expansion terminates, either when finds a
  complete and consistent ABox, or try all search
  pathes ending with complete but inconsistent
  ABoxes.

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Internalisation

• Embed the TBox in the initial ABox concept
• C?D is equivalent T? C U D (T is the "top"
  concept. It imeans C U D is the super concept
  for ANY concepts)
• E.g.
• Given ontology Mother ? Woman ? Parent, Woman ?
  Person
• Query Mother ? Person
• The intitial ABox is Mother U(Woman ? Parent)
  ? (Woman U Person) ? (Mother ? Person)

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A Expansion Example
Search
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Tree Model

• Another explanation of tableaux algorithm is that
  it works on a finite completion tree whose
• individuals in the tableau correspond to nodes
• and whose interpretation of roles is taken from
  the edge labels.

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Requirments for Tab. Alg.

• Similar tableaux expansions can be designed for
  more expressive DL languages.
• A tableau algorithm has to meet three
  requirements
• Soundness if a complete and clash-free ABox is
  found by the algorithm, the ABox must satisfies
  the initial concept C0.
• Completeness if the initial concept C0 is
  satisfiable, the algorithm can always find an
  complete and clash-free ABox
• Termination the algorithm can terminate in
  finite steps with specific result.

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Roadmap

• What is Description Logic (DL)
• Semantics of DL
• Basic Tableau Algorithm
• Advanced Tableau Algorithm

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Advanced Tableau Alg.

• Rich literatures in the past decade.
• Advanced techniques
• Blocking (Subset Blocking,Pair Locking, Dynamic
  Blocking)
• For more expressive languages number
  restriction, transitive role, inverse role,
  nomial, data type
• Detailed analysis of complexities.
• Refer to references at the end of this
  presentation for details

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SHIQ Expansion Rules
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References

• F. Baader, W. Nutt. Basic Description Logics. In
  the Description Logic Handbook, edited by F.
  Baader, D. Calvanese, D.L. McGuinness, D. Nardi,
  P.F. Patel-Schneider, Cambridge University Press,
  2002, pages 47-100.
• Ian Horrocks and Ulrike Sattler. Description
  Logics Tutorial, ECAI-2002, Lyon, France, July
  23rd, 2002.
• Ian Horrocks and Ulrike Sattler. A tableaux
  decision procedure for SHOIQ. In Proc. of the
  19th Int. Joint Conf. on Artificial Intelligence
  (IJCAI 2005), 2005.
• I. Horrocks and U. Sattler. A description logic
  with transitive and inverse roles and role
  hierarchies. Journal of Logic and Computation,
  9(3)385-410, 1999.