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Basics of Reasoning in Description Logics

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Title: Basics of Reasoning in Description Logics


1
Basics of Reasoning in Description Logics
  • Jie Bao
  • Iowa State University
  • Feb 7, 2006

2
An ontology of this talk
3
Roadmap
  • What is Description Logics (DL)
  • Semantics of DL
  • Basic Tableau Algorithm
  • Advanced Tableau Algorithm

4
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)

5
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

6
DL for Family KB
7
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

8
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

9
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)
10
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

11
Roadmap
  • What is Description Logic (DL)
  • Semantics of DL
  • Basic Tableau Algorithm
  • Advanced Tableau Algorithm

12
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
13
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.

14
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.
15
Source Description Logics Tutorial, Ian Horrocks
and Ulrike Sattler, ECAI-2002,
16
Source Description Logics Tutorial, Ian Horrocks
and Ulrike Sattler, ECAI-2002,
17
Roadmap
  • What is Description Logic (DL)
  • Semantics of DL
  • Basic Tableau Algorithm
  • Advanced Tableau Algorithm

18
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

19
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

20
Reasoning Tasks(2)
  • Many inference tasks can be reduced to
    subsumption reasoning
  • Subsumption can be reduced to satisfiability

21
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"

22
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.

23
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.

24
NNF
25
Tableaux Expansion(Selected)
Clash
26
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.

27
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)

28
A Expansion Example
Search
29
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.

30
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.

31
Roadmap
  • What is Description Logic (DL)
  • Semantics of DL
  • Basic Tableau Algorithm
  • Advanced Tableau Algorithm

32
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

33
SHIQ Expansion Rules
34
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.
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