Knowledge Representation and Reasoning

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Knowledge Representation and Reasoning
Master of Science in Artificial Intelligence,
2009-2011

• University "Politehnica" of Bucharest
• Department of Computer Science
• Fall 2009
• Adina Magda Florea
• http//turing.cs.pub.ro/krr_09
• curs.cs.pub.ro

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Lecture 5

• Description Logic
• Lecture outline
• About DL
• DL language
• Terminologies
• World descriptions
• Reasoning services
• Reasoning algorithms

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1. About Description Logics

• Description Logics (DLs) - knowledge
  representation formalisms that
• represent the knowledge of an application domain
  (the world)
• by de?ning the relevant concepts of the domain
  (its terminology)
• and then using these concepts to specify
  properties of individuals occurring in the domain
  (the world description).
• Inferences classi?cation of
• concepts - subconcept/superconcept relationships
  (subsumption)
• individuals - whether a given individual is
  always an instance of a concept

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Description Logics

• Semantic networks IS-A, AKO
• KL-ONE formal semantic
• Subsumption
• Unlike IS-A links, subsumption relationships and
  instance relationships are inferred from the
  de?nition of the concepts and the properties of
  the individuals
• DLs are based on a formal, logic-based semantics

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A DL system
TBox the terminology, i.e., the vocabulary of
an application domain concepts (set of
individuals) roles ABox assertions about
named individuals in terms of this vocabulary
TBox
Description Language
Reasoning
ABox
KB
Rules
Application Programs
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Inference services

• Typical reasoning tasks
• Whether a TBox description is satis?able (i.e.,
  non-contradictory)
• Whether one description subsumes another one in a
  TBox - organize the concepts of a terminology
  into a hierarchy according to their generality
• Find out whether the set of assertions in a ABox
  is consistent (has a model)
• Whether the assertions in the ABox entail that a
  particular individual is an instance of a given
  concept description
• A concept description can also be conceived as a
  query - retrieve the individuals that satisfy the
  query.

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2. DL Language

• Elementary descriptions
• atomic concepts (A,B)
• atomic roles (R)
• Complex descriptions of concepts (C,D) concept
  constructors, or of roles role constructors
• Basic language Attribute Language AL
• Description languages are distinguished by the
  constructors they provide (syntactic rules of
  concept formation)
• Different extensions of AL

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Syntax
ALUENC

• C, D ? A (atomic concept)
• T (universal concept)
• ? (bottom concept)
• A (atomic negation)
• C D (intersection)
• ?R.C (value restriction)
• ?R.T (limited existential
  quantification)
• C D (union)
• ?R.C (full existential
  quantification)
• C (negation of concepts)
• ? n R (number restrictions)
• ? n R

AL
ALU
ALE
ALC
ALN
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Examples

• Atomic concepts Pers, Fem
• Atomic role hasChild
• Pers Fem Pers Fem
• Pers ?hasChild.T Pers ?hasChild.Fem
• Pers ?hasChild. ?
• Pers Fem ?hasChild.T ?hasChild.Fem
• Pers Fem ? 3 hasChild

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Semantics

• Interpretation I
• The domain of the interpretation ?I
• An interpretation function
• assigns to every atomic concept A a set AI ? ?I
• assigns to every atomic role R a binary relation
• RI ? ?I x ?I
• I (?I , I)

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Semantics

• The interpretation function is extended to
  concept definitions by the following inductive
  definitions

Pers ?hasChild.Fem
Pers ?hasChild.T
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Semantics
Pers ?hasChild.Fem
Fem ? 3 hasChild
Pers Fem
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Link with FOPL

• Atomic concepts - unary predicates
• Atomic roles binary predicates
• Concept C ? pc(x), for every I the set of
  elements that satisfies ?I is CI

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3. Terminologies (TBox)

• Terminological axioms - make statements about how
  concepts or roles are related to each other
• C D (R S) - inclusion axiom
• C ? D (R ? S) - equality axiom
• Semantics of axioms
• C D if CI ? DI
• C ? D if CI ? DI
• If T is a set of axioms, then I satis?es T i? I
  satis?es each element of T I is a model of T

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TBox

• De?nition an equality whose left-hand side is
  an atomic concept
• Introduces symbolic names for complex
  descriptions
• Name symbols NT (defined concepts) - occur on the
  left-hand side of some axiom
• Base symbols BT (primitive concepts) - occur only
  on the right-hand side of axioms

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Example of TBox

• Woman Person Female
• Man Person Woman
• Mother Woman ?hasChild.Person
• Father Man ?hasChild.Person
• Parent Father Mother
• Grandmother Mother ?hasChild.Parent
• MotherWithManyChildren Mother ? 3 hasChild
• MotherWithoutDaughter Mother ?hasChild.
  Woman
• Wife Woman ?hasHusband.Man

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TBox

• A base interpretation J for T is an
  interpretation that interprets only the base
  symbols.
• An interpretation I that interprets also the name
  symbols is an extension of J if it has the same
  domain as J , i.e., ?I ?J , and if it agrees
  with J for the base symbols.
• T is de?nitorial if every base interpretation has
  exactly one extension that is a model of T.
• If we know what the base symbols stand for, and T
  is de?nitorial, then the meaning of the name
  symbols is completely determined.

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TBox

• The question whether a terminology is de?nitorial
  or not is related to the question whether or not
  its de?nitions are cyclic.
• Human Animal ?hasParent.Human
• Let A, B be atomic concepts occurring in T
• A directly uses B in T - B appears on the
  right-hand side of the de?nition of A
• A uses B in T - the transitive closure of the
  relation directly uses.
• T contains a cycle i? there exists an atomic
  concept in T that uses itself.
• Otherwise, T is acyclic

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TBox

• If a terminology T is acyclic, then it is
  de?nitorial
• T' the expansion of T (all axioms of the form
  AC', C' contains only base symbols)
• Woman Person Female
• Mother Woman ?hasChild.Person
• MotherWithManyChildren Mother ? 3 hasChild
• MotherWithManyChildren ((Person Female)
  ?hasChild.Person) ?3 hasChild

only one extension
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4. World descriptions (ABox)

• ABox introduce individuals, by giving them names,
  and asserts properties of these individuals
• Individuals a, b, c
• C(a) - concept assertions - a belongs to (the
  interpretation of) C,
• R(b,c) - role assertions - c is a ?ller of the
  role R for b.
• MotherWithoutDaughter(mary)
• Father(peter)
• hasChild(mary, peter)
• hasChild(peter, harry)
• hasChild(mary, paul)

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ABox Semantics

• Semantics of ABoxes extend interpretations to
  individual names
• An interpretation I (?I , I) not only maps
  atomic concepts and roles to sets and relations,
  but in addition maps each individual name a to an
  element aI ? ?I.
• The mapping has to respect the unique name
  assumption (UNA) - that distinct individual names
  denote distinct objects, i.e., if a?b then aI?bI

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ABox Semantics

• An interpretation I satisfies C(a) if aI ? CI
• An interpretation I satisfies R(a,b) if (aI ,bI)
  ? RI
• An interpretation I satisfies an ABox if it
  satisfies each assertion in the ABox I is a
  model of A
• An interpretation I satisfies an ABox A with
  respect to a TBox T if it is a model of T and a
  model of A

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Individual names in TBox

• Sometimes, it is convenient to allow individual
  names (also called nominals) not only in the
  ABox, but also in the description language
• The most basic concept constructors employing
  individuals is the set (or one-of ) constructor
• a1,...,an, where a1,...,an are individual
  names.
• The semantics of a set concept is defined as
• a1,...,anI a1I ,...,anI

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Individual names in TBox

• Another constructor involving individual names is
  the ?lls constructor
• R a, for a role R
• The semantics of this constructor is de?ned as
• (R a)I x ? ?I (x, aI) ? RI
• R a stands for the set of those objects that
  have a as a ?ller of the role R.

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5. Reasoning services in DL

• Reasoning tasks for TBox
• Checking satisfiability of a concept
• Checking subsumption of a concept by another
• Checking equivalence of 2 concepts
• Checking disjointness of 2 concepts

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Reasoning tasks for TBox

• Satis?ability A concept C is satis?able with
  respect to T if there exists a model I of T such
  that CI is nonempty (I is a model of C)
• Subsumption A concept C is subsumed by a concept
  D with respect to T if CI ?DI for every model I
  of T
• C D or T C D.
• Equivalence Two concepts C and D are equivalent
  with respect to T if CI DI for every model I of
  T
• C T D or T C D
• Disjointness Two concepts C and D are disjoint
  with respect to T if CI n DI Ø for every model
  I of T

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Properties

• Reduction to Subsumption
• C is unsatis?able ? C is subsumed by ?
• C and D are equivalent ? C is subsumed by D and D
  is subsumed by C
• C and D are disjoint ? C D is subsumed by ?
• Reduction to Unsatis?ability
• C is subsumed by D ? C D is unsatis?able
• C and D are equivalent ? both (C D) and
• (C D) are unsatis?able
• C and D are disjoint ? C D is unsatis?able

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Reasoning tasks for ABox

• Consistency of an ABox
• Instance check
• Realization
• Retrieval
• An ABox A is consistent with respect to a TBox T,
  if ?I that is a model of both A and T
• A is consistent if it is consistent with respect
  to the empty TBox.
• Mother(mary), Father(mary)

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Reasoning tasks for ABox

• The expansion of A with respect to T - the ABox
  A' that is obtained from A by replacing each
  concept assertion C(a) in A with the assertion
  C'(a), where C ' is the expansion of C with
  respect to T
• An ABox A is consistent with respect to a TBox
  iff the expansion A' is consistent

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Reasoning tasks for ABox

• Instance check
• An assertion C(a) is entailed by A if ?I that
  satis?es A also satis?es C(a) (A C(a))
• A C(a) i? A ? C(a) is inconsistent
• Retrieval problem
• Given an ABox A and a concept C, ?nd all
  individuals a such that A C(a)
• Realization problem
• Given an individual a and a set of concepts, ?nd
  the most speci?c concepts C from the set such
  that A C(a)
• The most speci?c concepts are those that are
  minimal with respect to the subsumption ordering

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Assumptions

• Closed-world vs. open-world semantics

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6. Reasoning algorithms

• Structural subsumption algorithms algorithms
  that compare the syntactic structure of concept
  descriptions
• Very efficient
• Complete for only rather simple languages
• For ALNU subsumption algorithms are not complete
• Tableau-based algorithms satisfiability of
  concepts, TBox, etc.

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Tableau algorithms

• Negation normal form of a concept negation
  occurs only in front of concept names
• Example of tableau algorithm to check subsumption
  (reduced to unsatisfiability)

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Tableau algorithms

• Test satisfiability of C0
• Let C0 a concept (ALCN) in negation normal form
• A0C0(x0)
• Apply transformation rules (preserve consistency
  if any) until no more rule apply or a clash is
  obtained
• ABox transformed in many new ABox' by some
  rules
• ABox is consistent iff one of the ABox' is
  consistent

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Transformation rules for satisfiability algorithms
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Transformation rules for satisfiability algorithms
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Tableau algorithms

• An ABox is complete iff none of the
  transformation rules apply to it
• Consistency of a set of complete ABoxes can be
  decided by looking for clashes (contradictions).
• An ABox A contains a contradiction if
• ?(x) ? A for some individual x
• A(x), A(x) ? A for some individual x and some
  concept A
• (?n R)(x) ? R(x,yi) 1?i?n1 ? yi ?yj
  1?iltj?n1 ? A for individuals x, y1, , yn1, a
  nonnegative integer n, and a role R

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Tableau algorithms

• Tr. rule
• ABox set of complete ABoxes
• if one ABox is clash-free, then C0(x0) is
  consistent and C0 is satisfiable
• if all ABoxes contain a clash, then C0(x0) is
  inconsistent and C0 is unsatisfiable

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Theorems

• T1 (termination)
• Let C0 be an ALCN concept description in negation
  normal form. There cannot be an infinite sequence
  of rule applications
• C0(x0) ? S1 ? S2 ?
• T2 (decidability)
• It is decidable whether or not an ALCN-concept is
  satisfiable

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Credits

• Slides based on the book
• The Description Logic Handbook Theory,
  Implementation, and Applications, Edited by Franz
  Baader, Diego Calvanese, Deborah L. McGuinness,
  Daniele Nardi, Peter F. Patel-Schneider,
  Cambridge University Press, 2007, 2nd Edition