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Title: Logics for Data and Knowledge Representation


1
Logics for Data and KnowledgeRepresentation
  • Description Logics

2
Outline
  • Overview
  • Syntax
  • Semantics
  • Terminology
  • Reasoning with Terminologies
  • World Description
  • Reasoning with World Descriptions
  • Tableau Algorithm

3
Overview
  • Description Logics (DLs) is a family of KR
    formalisms that represent the knowledge of an
    application domain (the world) by
  • defining the relevant concepts of the domain
    (i.e., its terminology), and then
  • using these concepts to specify the properties of
    objects in the domain (i.e., the worlds
    description).

4
Some History on DLs
  • Descended from the structured inheritance
    networks (Brachman, 1977).
  • Introduced to overcome the ambiguities of early
    semantic networks and frames.
  • First realized in the system Kl-One by Brachman
    and Schmolze (1985).
  • First DL presented in the B Ss paper.
  • Some nicknames for DLs
  • Terminological knowledge representation
    languages,
  • Concept languages,
  • Term subsumption languages,
  • Kl-One-based knowledge representation languages.

5
Three Basic Features
Extended from ClassL
  1. The basic syntactic building blocks are atomic
    concepts, atomic roles, individuals.
  2. The expressive power of DLs is restricted to a
    rather small set of constructors for building
    complex concepts and relations.
  3. Implicit knowledge about concepts and individuals
    can be inferred automatically with the help of
    specific reasoning services.

Syntax
Extended from ClassL to Complest Role
Logical
6
Architecture of a KR system on DL
TBox
Description Language
Reasoning
ABox
Application Programs
Rules
7
Language (Syntax)
  • The first step in setting up a formal language
    (viz. a descriptive language) is to list the
    symbols, that is, the alphabet of symbols.
  • We denote a generic alphabet of a descriptive (or
    description) language dS.
  • Similarly to any logical language we can divide
    symbols in dS in descriptive (nonlogical) and
    non-descriptive (logical).

8
Outline
  • Overview
  • Syntax
  • Semantics
  • Terminology
  • Reasoning with Terminologies
  • World Description
  • Reasoning with World Descriptions
  • Tableau Algorithm

9
Syntax of DL
  • Descriptive dS consists of concept names (set C),
    which denote sets of individuals, role names (set
    R), which denote binary relations between
    individuals, and individual names, (set I), which
    denote individuals.
  • e.g.concept names Room, Person, Fruitrole
    names likeSkiing, hasChild, partOf,
    isA,..individual names I, you, apple, Fido, ...

10
DL Language and Previous Languages
  • concept names are propositional variables
  • (PL/ClassL)
  • role names are binary predicate symbols
  • (FOL)
  • individual names are constants
  • (FOL)

11
Language (Syntax)
  • Non-descriptive dS provides concept constructors
    to build complex formulas, called concept
    descriptions and role descriptions, from atomic
    formulas.
  • e.g. ?(negation),
  • ?(conjunction),?(for all),
  • ?(there exists)

12
AL-family Languages
  • We shall now discuss various descriptive
    languages from the family of AL-languages.
  • An AL-language ( Attributive Languages) is a
    minimal DL language of practical interest.
  • More expressive descriptive languages are usually
    extensions of some AL-language.
  • AL-languages do not deal with individuals.

13
AL Logical Symbols
  • Universal concept symbol ?.
  • Bottom concept symbol ?.
  • Parentheses (auxiliary symbols) (, )
  • Logical constants (concept constructors)?
    (atomic negation), ? (conjunction)
  • ?R (for all atomic roles)?R (there exists an
    atomic role)

The SAME as in ClassL
New!
14
AL Non-logical Symbols
The SAME as in ClassL
  • Atomic concept names A, B, ...
  • Concept names C, D, ...
  • Atomic role names R (generic)
  • NOTE There is no logical symbol in AL for
    logical implication (as ? in PL and in FOL).
    For this, we will use the subsumption symbol ?
    instead (as in classL).

New!
15
Defined Symbols
  • Similarly to ClassL, ? and ? can be defined
  • For all concept names C,
  • ? df C ? ? C
  • ? df ? ? or also ? df U for U be a special
    coincept name denoting the Universal Concept.
  • We prefer to consider ? and ? AL symbols.

16
Formation Rules for AL
  • Atomic Concepts
  • 1. A, B,..., ?, ?.
  • Concepts (concept descriptions) 2. All the
    atomic concepts3. ?A for A (atomic concept
    negation)4. C?D (intersection) 5. ?R.C (value
    restriction)6. ?R.?(limited existential
    quantification)
  • Resulting language attributive language (AL).

17
Examples
  • Atomic concepts Person, Female, Room, ...Atomic
    roles hasChild, partOf, isIn, isA,...
  • Concepts Person ? Female,Person ? ?hasChild.? (
    ?hasChild )Person ? ?hasChild.?
    (Not?hasChild.?) Person ? ?hasChild.Female
  • Question What is the intended meaning?

18
Solution
  • Person ? Female
  • persons that are female
  • Person ? ?hasChild. ? (all those) persons that
    have a child
  • Person ? ?hasChild. ? (all those) persons
    without a child
  • Person ? ?hasChild.Female persons all of whose
    children are female

19
ALs Extensions - ALU
  • Extended Alphabet Logical constants (concept
    constructors) ? (disjunction).
  • Extended concepts (descriptions) C ? D (union)
  • The resulting new language (i.e. AL plus the new
    set of concepts) usually denoted ALU.
  • E.g. Mother?Father describes the extension of
    parent.

20
ALs Extensions - ALE
  • Extended Alphabet Logical constants (concept
    constructors)?R (there exists an arbitrary
    role)
  • Extended concepts (descriptions) ?R.C (full
    existential quantification)
  • The resulting new language (i.e. AL plus the new
    set of concepts) usually denoted ALE.
  • E.g. ?hasChild.Female describes the extension of
    those parents who has at least a daughter.

21
ALs Extensions - ALN
  • Extended Alphabet Logical constants (concept
    constructors)n, n for all n ? N
    (at-least/at-most n)
  • Extended concepts (descriptions) nR (at-least
    number restriction)nR (at-most number
    restriction)
  • The resulting new language (i.e. AL plus the new
    set of concepts) usually denoted ALN.
  • E.g. 2 hasChild describes the extension those
    parents who has at least 2 children.

22
ALs Extensions - ALC
  • Extended Alphabet Logical constants (concept
    constructors) ? (general negation)
  • Extended concepts (descriptions) ? C (full
    concept negation)
  • The resulting new language (i.e. AL plus the new
    set of concepts) usually denoted ALC. (C stands
    for Complement).
  • E.g. ?(2 hasChild ) describes the extension
    those parents who has at most 1 children.

23
ALs Extensions (Summary)
  • Extending AL by any subsets of the above
    constructors yields a particular DL language.
  • Each language is denoted by a string of the form
    ALUENC, where a letter in the name stands
    for the presence of the corresponding
    constructor. Notation AL.
  • ALC as the most important in many aspects.(Well
    see that ALU ? ALC and ALE ? ALC.)

24
ALs Sub-languages
  • By eliminating some of the syntactical symbols,
    we get some sub-languages of AL.
  • The most important sub-language obtained by
    elimination in the AL family is ClassL.
  • Historically, another important sub-language is
    named as the Frame Language FL0.

25
From AL to ClassL
  • Elimination (w.r.t. ALUEC!)
  • Name Symbols atomic role names R
  • Concept constructors ?R, ?R
  • Descriptions ?R.C, ?R.C
  • The new language is a description language
    without roles which is ClassL.
  • NOTE So far, we are considering DL without
    individuals and ABox.

26
ALs Contractions FL-
  • Elimination (w.r.t ALs alphabet)
  • Symbols Universal and bottom symbols ?, ?
  • Concept constructors (atomic negation)
  • Descriptions ?, ? , ? A (atomic negation)
  • The resulting new language (i.e. AL without
    negation) is denoted FL-.

27
ALs Contractions FL0
  • Elimination(w.r.t FL-s alphabet)
  • Concept constructors?R (there exists an atomic
    role)
  • Descriptions ?R.?(limited existential
    quantification)
  • The resulting new language (i.e. FL- without the
    limited existential quantification) is denoted
    FL0.
  • FL Frame Language (for historical reasons)

28
Summary
  • DLs are a family of logic-based KR formalisms to
    describe a domain in terms of - concepts - roles
    - individuals (grounding).
  • Strictly speaking we do not need DLs to represent
    concepts and roles, but the variable-free syntax
    of DLs is much more concise! (Thats good for
    automation!)
  • ClassL is ALC without Role.

This will be discussed later.
29
Outline
  • Overview
  • Syntax
  • Semantics
  • Terminology
  • Reasoning with Terminologies
  • World Description
  • Reasoning with World Descriptions
  • Tableau Algorithm

30
Semantics
  • The elements of the description languages in
    AL-family (AL) are plain strings of symbols
  • The meaning which is intended to be attached to
    concept, role, and individual names form an
    informal interpretation of the given AL
    languages expressions.

No (formal) meaning.
31
Recall Interpretation (?, I)
  • Def. An interpretation of a language L is a pair
    I (?,I), where- ? (domain) is a non-empty set
    of objects - I(interpretation function) is a
    mapping from L to ?.
  • NOTE sometimes, the mapping I is written as
    I but of the same function.
  • E.g. I(Person) pp is a person is written as
  • PersonI pp is a person.

32
AL Interpretation (?,I)
  • I(?) Ø and I(?) ? (full domain, Universe)
  • For every concept name A of L, I(A) ? ?.
  • I(C) ? \ I(C).
  • I(C?D) I(C) n I(D) I(C ? D) I(C) ? I(D).
  • For every role name R of L, I(R) ? ? ?.
  • I(?R.C) a ? ? for all b, if (a,b)?I(R) then
    b?I(C).
  • I(?R. ?) a ? ? exists b s.t. (a,b) ? I(R).
  • I(?R.C) a ? ? exists b s.t. (a,b) ? I(R), b
    ? I(C).
  • I(nR) a ? ? b (a, b) ? I(R) n.
  • I(nR) a ? ? b (a, b) ? I(R) n.

The SAME as in ClassL
33
Interpretation of Existential Quantifier
  • I(?R.C) a ? ? exists b s.t. (a,b) ? I(R), b
    ? I(C)
  • Those that have some value in C with role R.
  • Remark a ? ?

34
Example
  • Those who have a daughter.
  • I(?R.C) I(?hasChild.Female) a ? ? exists b
    s.t. (a,b) ? I(hasChild), b ? I(Female)

hasChild
Anna has a child betty and betty is a female,
then Anna is an individual of the concept
?hasChild.Female.
35
Interpretation of Value Restriction
  • I(?R.C) a ? ? for all b, if (a,b)?I(R) then
    b?I(C)
  • Those that have only value in C with role R.
  • Remark a ? ?

36
Example
  • Those who have only daughter(s)
  • I(?R.C) I(?hasChild.Female) a ? ? for all
    b, if (a,b)?I(R) then b?I(C)

hasChild
37
Interpretation of Number Restriction
  • I(nR) a?? b (a, b) ? I(R) n
  • Those that have relation R to at least n
    individuals.

b (a, b) ? I(R) n
38
Example
  • Those who has at least 2 children
  • I(nR) I(2hasChild)a ? ? b (a, b) ?
    I(R)n

?
hasChild
Annas children betty, bianca, babara,... are
2. Anna is one individual of those who has at
least 2 children.
39
Interpretation of Number Restriction Cont.
  • I(nR) a ? ? b (a, b) ? I(R) n
  • Those that have relation R to at most n
    individuals.

?
b
b'

a
b (a,b) ? I(R) n
40
Example
  • Those who as at most 2 children
  • I(nR) I(2hasChild)a?? b (a, b) ?
    I(R)n

?
hasChild
Anna has only one child Betty, so Anna belongs to
the concept of those who has at most 2 children.
41
Simple Exercises
  • Verify the following equivalences hold for all
    interpretations (?,I)
  • I((C ? D)) I(C ? D)
  • I((C ? D)) I(C ? D)
  • I(?R.C) I(?R.C)
  • I(?R.C) I(?R.C)
  • E.g.
  • I(?R.C) a ? ? not exists b s.t. (a,b) ?
    I(R), b ? I(C) a ? ? ?b. R(a,b)?C(b)
  • a ? ? ?b. (R(a,b)?C(b))
  • a ? ? ?b. R(a,b)?C(b)
  • I(?R.C)

42
Outline
  • Overview
  • Syntax
  • Semantics
  • Terminology
  • Reasoning with Terminologies
  • World Description
  • Reasoning with World Descriptions
  • Tableau Algorithm

43
Terminological Axioms
  • Inclusion Axiom
  • C?D
  • e.g.
  • Master ? 18Pass.Exam?1Finish.Project,
  • Professor??Give.Lecture??Organize.(Exam?Experiment
    )
  • Equivalence (Equality) Axiom
  • CD
  • e.g.
  • PhD Postgraduate?3Publish.Paper,
  • Parent Man??hasChild.?
  • Is the TBox definition the SAME here as in ClassL?

R?S
RS
44
Terminology (TBox)
  • A terminology (or TBox) is a set of a
    (terminological) axioms.
  • e.g. T is
  • PhD Postgraduate?3Publish.Paper,
  • Parent Person??hasChild.Person,
  • hasGrandChild?hasChild
  • NOTE the punctuations mean logical conjunctions.
  • .

45
Outline
  • Overview
  • Syntax
  • Semantics
  • Terminology
  • Reasoning with Terminologies
  • World Description
  • Reasoning with World Descriptions
  • Tableau Algorithm

46
TBox Reasoning
  • Let T be a TBox
  • Satisfiability(with respect to T)
  • T satisfies P?
  • Subsumption (with respect to T)
  • T P ? Q?
  • Equivalence (with respect to T)
  • (T P Q) T P ? Q and T P ? Q?
  • Disjointness (with respect to T)
  • T P ? Q ? ??
  • NOTE they are the SAME as in ClassL with more
    concepts built by the role constructors.

47
Outline
  • Overview
  • Syntax
  • Semantics
  • Terminology
  • Reasoning with Terminologies
  • World Description
  • Reasoning with World Descriptions
  • Tableau Algorithm

48
ABox
  • In a ABox, one introduces individuals, by giving
  • them names, and one asserts properties about
  • these individuals.
  • We denote individual names as a, b, c,
  • An assertion with Role R is called Role assertion
    (in contrast to Concept assertion C(a) in ClassL)
    as
  • R(a, b), R(b, c),
  • e.g.
  • Study(Tin, ldkr)
  • NOTE the other part of the ABox is the Concept
    assertions the SAME as in Ground ClassL.

The SAME as in ClassL
49
Semantics of the ABox
  • We give a semantics to ABox by extending
    interpretations to individual names.
  • An interpretation I (?, I) not only maps atomic
    concepts to sets, but in addition maps each
    individual name a to an element aI ??., namely
  • I (C(a)) aI ?CI,
  • I(R(a, b))(aI, bI)?RI
  • We still assume unique name assumption as in
    ClassL.
  • set constructor is the SAME as in ClassL, which
    allows individual description in TBox.

The SAME as in ClassL
50
Outline
  • Overview
  • Syntax
  • Semantics
  • Terminology
  • Reasoning with Terminologies
  • World Description
  • Reasoning with World Descriptions
  • Tableau Algorithm

51
Consistency
  • Consistency An ABox A is consistent with respect
    to a TBox T if there is an interpretation I which
    is a model of both A and T.
  • e.g. TParent?1hasChild.Person
  • AhasChild(mary, bob), hasChild(mary, cate)
  • is consistent but Not consistent with respect
    the above TBox.
  • NOTE it is the SAME as in ClassL, with respect
    to more complex TBox (Role axioms).

52
Other ABox Reasoning
  • Instance Checking Checking whether an assertion
    C(a) or R(a, b) is entailed by an ABox A (and
    TBox via expansion)
  • A ? C(a) or A ? R(a, b)
  • Instance retrieval Given an ABox A and a concept
    C retrieve all instance ai which satisfies C.
  • A ? C(ai)
  • Concept Realization Given an ABox A, a set of
    concepts and an individual a find the most
    specific concepts C in the set such that A ? C(a)
  • NOTE the last two services are the SAME as in
    ClassL, with respect to more complex TBox (Role
    axioms).

53
Outline
  • Overview
  • Syntax
  • Semantics
  • Terminology
  • Reasoning with Terminologies
  • World Description
  • Reasoning with World Descriptions
  • Tableau Algorithm

54
Tableaux Calculus
  • The Tableaux calculus is a decision procedure
    solving the problem of satisfiability.
  • If a formula is satisfiable, the procedure will
    constructively exhibit a model of the formula.
  • The basic idea is to incrementally build the
    model by looking at the formula, by decomposing
    it in a top-down fashion. The procedure
    exhaustively tries all the possibilities so that
    it can prove eventually that no model could be
    found for an unsatisfiable formula.

55
The Tableau Algorithm
  • Construct a model for the input concept
    description C0.
  • Model is represented by tree form.
  • The formula has been translated into Negation
    Normal Form (NNM).
  • To decide satisfiability of the concept C0 ,
    start with the initial tree (root node).
  • Repeatedly apply expansion rules until find
    contradiction or expansion completed.
  • Satisfiable iff a complete and contradiction-free
    tree is found

56
Rules for Tableau Algorithm
57
?
  • When there are not both C1(x) and C2(x) in the
    ABox.
  • Add to the ABox C1(x) and C2(x) .
  • ABoxABox?C1(x), C2(x)

58
? cont.
  • TBoxMother Female ??hasChild.Person
  • ABoxMother(Anna)
  • Is (?hasChild.Person?? hasParent.Person)
    satisfiable?
  • Mother(Anna)
  • (Female ??hasChild.Person)(Anna)
  • Female(Anna)
  • (?hasChild.Person)(Anna)
  • (?hasChild.Person?? hasParent.Person)(Anna)
  • (?hasChild.Person)(Anna)
  • ( ? hasParent.Person)(Anna)

59
?
  • When there are neither C1(x) nor C2(x) in the
    ABox.
  • Split into multiple branches
  • Add in branch 1 to the ABox C1(x)
  • and add in branch 2 to the Abox C2(x) .
  • ABoxABox?C1(x)
  • ABoxABox?C2(x)

60
? cont.
  • TBoxParent?hasChild.Female??hasChild.Male, Pers
    onMale?Female,
  • MotherParent ?Female
  • ABoxMother(Anna)
  • Is (?hasChild.Person) satisfiable?
  • Mother(Anna)
  • Parent(Anna)
  • (?hasChild.Female??hasChild.Male)(Anna)
  • (?hasChild.Female)(Anna) (?hasChild.Male)(Anna
    )

61
?
  • When there are no individual name z such that
    both C(z) and R(x,z) in the ABox.
  • Add to the ABox C(z) and R(x,z) .
  • ABoxABox?C(z), R(x,z)

62
? cont.
  • TBoxParent?hasChild.Female??hasChild.Male,
    PersonMale?Female,
  • MotherParent ?Female
  • ABoxMother(Anna), hasChild(Anna,Bob),
    Female(Bob)
  • Is (?hasChild.Person) satisfiable?
  • Mother(Anna)
  • Parent(Anna)
  • (?hasChild.Female??hasChild.Male)(Anna)
  • (?hasChild.Male)(Anna)
  • hasChild(Anna,Bob)
  • Male(Bob)

63
?
  • When there is R(x,z) but not C(z) in the ABox.
  • Add to the ABox C(z) .
  • ABoxABox?C(z)

64
? cont.
  • TBoxDaughterParent?hasChild.Female,
    Male?Female??
  • ABoxhasSibling(Anna,Bob), Female(Bob)
  • Is DaughterMather satisfiable?
  • DaughterMather(x)
  • ?hasChild.Female(x)
  • hasSibling(Anna,Bob)
  • Female(Bob)
  • Female(Bob)

?
65
Example of Tableau Reasoning
  • Is ?hasChild.Male??hasChild.Male satisfiable?
  • (?hasChild.Male??hasChild.Male)(x)
  • (?hasChild.Male)(x)
  • (?hasChild.Male)(x)
  • hasChild(x,y)
  • Male(y)
  • Male(y)

?
?
?
66
Additional Rules
67
Exercise
  • Suppose we have the following knowledge base
  • TBox
  • Female?Male?Human
  • Mother?Female
  • Father?Male
  • Child?has.Mother? ?has.Father
  • Male?Female??
  • ABox
  • Mother(Anna)
  • Father(Bob)
  • has (Cate,Anna)
  • has (Cate,Bob)

Prove Human(Anna) Female(Bob) Child(Cate) Father
?Mother??
68
Exercise
  • Prove with Tableau algorithm the satisfiability
    of the following formula.

69
Complexity of Tableau Algorithms
  • The satisifiability algorithm of ALC may need
    exponential time and space. It is
    PSPACE-complete.
  • A modified algorithm needs only polynomial space
    as it assumes a depth-first search and store only
    the correct path.
  • The consistency and instance checking problem for
    ALC are also PSPACE-complete.
  • The complexity results for other Description
    Logics varies according to corresponding
    constructors.
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