On Fuzzy Description Logics From Description Logics to Fuzzy Description Logics DLs (ALC) FOL Interpretation

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  • On Fuzzy Description Logics

    Francesc Esteva

    Institut d’Investigació en Intel·ligència Artificial-CSIC

    DIPLEAP workshop, Wien November 2010

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • Network-based structures plus logical systems

    The antecedents of DL languages are Network-based structures carrying the intuition that, by exploiting the notion of hierarchical structure, one could gain in terms of both

    ease of representation

    the efficiency of reasoning

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • Network-based structures

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • Network-based structures plus logical systems

    The basic elements of the representation are characterized as

    unary predicates, denoting sets of individuals, and

    binary predicates, denoting relationships between individuals.

    Such a characterization does not capture the constraints of network-based structures with respect to logic.

    They did not require all the machinery of FOL,

    but could be regarded as fragments of it [Brachman and Levesque, 1985].

    In addition, different features of the representation language would lead to different fragments of FOL.

    Important consequence:

    recognition that the typical forms of reasoning used in structure-based representations could be accomplished by specialized reasoning techniques,

    without necessarily requiring FOL theorem provers.

    Moreover, reasoning in different fragments of FOL leads to computational problems of differing complexity.

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • Terminological systems / Concept languages

    First name: Terminological Systems

    To emphasize that the representation language was used to establish the basic terminology adopted in the modeled domain.

    Second name: Concept Languages

    The emphasis was on the set of concept-forming constructs admitted in the language.

    Third name: Description Logics

    In more recent years, after attention was further moved towards the properties of the underlying logical systems, the term Description Logics became popular.

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • Description Logics: The language

    Terminology

    Denote a hierarchical structure built to provide an intensional representation of the domain of interest.

    DL language

    Concepts are interpreted semantically as sets and in FOL as unary predicates

    Roles are interpreted as binary relations and in FOL as binary predicates

    the Description language is built from concepts and roles by means of constructors

    Different DLs depend on constructors (more or less expressive)

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • Example: the language AL

    AL (= attributive language) introduced by [Schmidt-Schauß and Smolka, 1991] as a minimal language that is of practical interest.

    attributive language: a language to express, concisely, certain natural constructions that are commonly used to build other concepts from simpler ones.

    Given A ∈ NA, and R ∈ NR , an AL-description formula is defined in accordance with the following syntactic rules (C,D are metavariables for descriptions of concepts):

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • Description Logics: concept constructors

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • Description Logics: a basic hierarchy of AL-languages

    6

    6

    6

    6 6

    � � ���

    � � ���

    � � ���

    � � ���

    � ���XX

    XXX XXy

    XX XXX

    XXy

    XX XXX

    XXy

    XX XXX

    XXy

    XX XXy

    AL

    ALU

    ALE

    ALUE = ALC

    ALN

    ALUN

    ALEN

    ALUEN = ALCN

    r

    r r

    r

    r

    r r

    r

    +N

    +E

    +U

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • Description formulas as FOL formulas

    Description signature: D = 〈NI ,NC ,NR〉 First order signature: ΣD = 〈CD,PD〉

    CD = NI , PD = NC ∪ NR , where we read each c ∈ NI as an object constant, each A ∈ NC as a unary predicate symbol, each R ∈ NR as a binary predicate symbol.

    Instance of an ALC-description

    ⊥ := 0̄, > := 1̄

    A(t)

    (¬C)(t) :=∼ C(t)

    (C1 t C2)(t) := C1(t) Y C2(t)

    (C1 u C2)(t) := C1(t)&C2(t)

    (C1 = C2)(t) := C1(t)→ C2(t)

    (∀R.C)(t) := (∀y)(R(t, y)→ C(y))

    (∃R.C)(t) := (∃y)(R(t, y)&C(y))

    R(t1, t2)

    An interpretation for D:

    I = 〈M, (cI)c∈NI , (A I)A∈NC , (R

    I)R∈NR 〉

    is also an interpretation for ΣD.

    The following are equivalent:

    a ∈ CI

    CI(a) = 1

    ‖C(x)‖I,v [x→a] = 1

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • Description Logics: knowledge base (KB)

    Terminology

    Denote a hierarchical structure built to provide an intensional representation of the domain of interest.

    DL knowledge base (KB)

    TBox: definitions and hierarchies of the relevant domain concepts

    ABox: specifications of properties of the domain instances

    Issues

    The statements in the KB can be identified with formulas in FOL

    Tools from FOL can be used to obtain implicit knowledge from the explicit knowledge in the KB by means of deductive reasoning

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • The Robots dataset

    r1 r5r4r3r2

    . . . . . . . . . .

    r6

    . .

    r9

    . .. .

    r7

    . .

    r8

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • Classical TBox

    Friendly ≡ Robot & (∃hasObject.FriendlyObject)& (Happy Y Homogeneous) Unfriendly ≡ Robot & ∼ Friendly

    FriendlyObject v Object UnfriendlyObject ≡ Object & ∼ FriendlyObject

    Robot & Object v 0̄ 1̄ v Robot Y Object

    Homogeneous v Robot Happy v Robot

    WearsTie v Robot

    Flower v FriendlyObject Balloon v FriendlyObject

    Flag v FriendlyObject Sword v UnfriendlyObject

    Ax v UnfriendlyObject

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • Classical ABox

    For each i, 1 ≤ i ≤ 9, Robot(ri ), hasObject(ri , oi )

    Homogeneous(r1),Balloon(o1),Happy(r1),WearsTie(r1)

    Homogeneous(r2),Flag(o2),Happy(r2),WearsTie(r2)

    ∼ Homogeneous(r3),Sword(o3),Happy(r3),WearsTie(r3)

    ∼ Homogeneous(r4),Flower(o4),∼ Happy(r4),∼ WearsTie(r4)

    ∼ Homogeneous(r5),Sword(o5),∼ Happy(r5),∼ WearsTie(r5)

    ∼ Homogeneous(r6),Flag(o6),∼ Happy(r6),∼ WearsTie(r6)

    Homogeneous(r7),Ax(o7),∼ Happy(r7),WearsTie(r7)

    ∼ Homogeneous(r8),Ax(o8),∼ Happy(r8),WearsTie(r8)

    Homogeneous(r9),Balloon(o9),∼ Happy(r9),∼ WearsTie(r9)

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • Knowledge bases: TBox and ABox

    Knowledge base = Terminological Box + Assertional Box

    TBox (T ): definitions and hierarchies of the relevant domain concepts ABox (A): specifications of properties of the domain instances

    K = 〈T ,A〉

    TBox: finite set of (general) concept inclusion axioms

    C v D : correspond to sentences of the form (∀x)(C(x)→ D(x)) Given I = 〈M, (.)I〉, I |= C v D iff CI ⊆ DI

    C ≡ D: abbreviation for the two axioms C v D and D v C

    ABox: finite set of assertion axioms

    a : C or (a, b) : R : correspond to sentences of the form C(a) or R(a, b)

    Given I = 〈M, (.)I〉,

    I |= a : C iff aI ∈ CI ; I |= (a, b) : R iff 〈aI , bI〉 ∈ RI

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • Reasoning tasks for concepts

    Reasoning algorithms: Structural subsumption algorithms

    In order to obtain decision procedures for any of the four inferences: it is sufficient to develop algorithms that decide the satisfiability of concepts,

    provided the language supports conjunction as well as negation of arbitrary concepts.

    Reasoning algorithms: specialized TABLEAU CALCULI

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • Description Logics

    Classical Description Logics

    From logical point of view: They are decidable fragments of FOL Relation with modal logic

    Development of DLs

    More complex languages (related to constructors and expresivity) OWL Efficient algorithms for satisfiability (validity and subsumption) A good balance between expressivity and efficiency. Application to ontologies and semantic web

    Francesc Esteva DIPLEAP workshop, Wien, November 2010

  • From Description Logics to Fuzzy Description Logics

    Fuzzy Description Logic

    Gradual concepts and roles

    Take the same language and constructors

    Interpret concepts by Fuzzy Sets and roles as fuzzy relations

    Patient with high fever Person living near Paris

    First papers on Fuzzy Description Logic

    Generalizing term subsumption languages to fuzzy logic, John Yen. Proceedings of the IJCAI’91.

    A Description Logic for Vague Knowledge, Christopher B. Tres