Description Logic

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Description Logic 2
DL-2
Description Logic 2
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History

• Description Logic Handbook Chapter 1
• Early knowledge representation work from 60s
• Representing classes of objects
• Abstraction Hierarchy
• Properties of those objects
• Constraints on the properties
• Led to object oriented programming ideas
• CommonLisp Object System (CLOS)
• SmallTalk, C, Java
• Networks of knowledge
• Semantic Networks (Quillian 1967)
• Frames (Minsky 1981)

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Example of Network KR

• Person, Female, etc are concepts
• hasChild is a property of Person
• hasChild relates Parent to Person
• Nil means infinity. A Parent is a Person with
  between 1 and infinity children
• Large arrows are IS-A links
• A Mother is a (specialization of a) Parent

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Networks and Logic systems were defined

• First order logic can express most of the network
• Predicates can be arity 1 or 2 (unary and binary)
• Rules can be used for inheritance and some
  constraints
• Mother ISA Parent
• ?X Mother(X) ?Parent(X)
• All children of Parents are Persons
• ?X,Y Parent(X) ? hasChild(X, Y) ? Person(Y)
• Expect some inferences to be made
• Inheritance All properties of a superclass
  should also be properties of its subclass
• So all children of Mothers should be Persons
• ?X,Y Mother(X) ? hasChild(X, Y) ? Person(Y)

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Network and Logic Systems have advantages/disadvan
tages

• Networks considered more appealing and more
  practical than Logic
• Some Network reasoners were ad hoc
• Different reasoners give different results
• Logic engines all produce the same conclusions
• Clearly defined semantics
• Logic engines at the time were very general
• Not much known about special reasoners for these
  formulas
• General resolution-based theorem provers used
• Generality meant reasoning procedures had high
  complexity

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Description Logic systems

• Brachman and Levesque 1984 there is a tradeoff
  between the expressiveness of a representation
  language and the difficulty of reasoning over the
  representations built using that language.
• The more expressive the language, the harder the
  reasoning.
• Schmidt-Schauss and Smolka 1991 specialized
  classical settings for deductive reasoning to the
  DL subsets of first-order logics,
• Using tableau calculus
• Best of both worlds
• Unary predicates for names of classes
• Binary predicates for properties
• Conclusions based on inheritance
• Derived from Logic but driven by an understanding
  of how much reasoning power is needed.

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Description Logic

• This lecture from DL Handbook Chapter 2
• Description Logics
• overcome the ambiguities of early semantic
  networks and frames
• first realized in the system KL-One Brachman and
  Schmolze, 1985
• Well-studied and decidable (most DL languages)
• Tight coupling between theory and practice

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TBox and ABox

• TBox terminology
• the vocabulary of an application domain
• Concepts sets of individuals
• Roles binary relationships between individuals.
• Examples
• Concepts Person, Female, Mother
• Role hasChild, meaning that some person is the
  child of some other
• ABox assertions
• about named individuals in terms of this
  vocabulary
• Example
• Elizabeth and Charles are Persons. We write this
  as Person(Elizabeth), and Person(Charles).
• Individuals, like myCar, have attributes, like
  color, and those attributes have values, like
  red. When this happens we say that red is the
  colorOf attribute of myCar.We write this as
  colorOf(myCar, red).

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Architecture of a DL System
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Formulas

• Building blocks that allow complex descriptions
  of concepts and roles.
• Example (well look at the syntax in more detail
  soon.)
• A Woman is a Female Person
• Woman Person u Female
• A Mother is a Woman and she has a child
• Mother Woman u 9 hasChild.T
• The TBox can be used to assign names to complex
  descriptions. We will use the terms description
  and concept interchangably.

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Reasoning Tasks

• TBox reasoning Determine
• whether a description is satisfiable (i.e.,
  non-contradictory)
• whether description A is more general than
  description B
• A subsumes B if every individual of concept B is
  also of concept A
• With Subsumption tests one can organize the
  concepts of a terminology into a hierarchy
  according to their generality.
• ABox reasoning Determine
• whether its set of assertions is consistent,
• whether a particular individual is an instance of
  a given concept description.
• A concept description can be a query, describing
  a set of objects
• With instance tests one can retrieve the
  individuals that satisfy the query.
• We will focus on the reasoning problem of TBox
  satisifiability

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Model Theory

• Statements in the TBox and in the ABox can be
  identified with formulae in first-order logic and
  the logical semantics is defined by those formulas

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Wrapping your mind around model theory

• Recall that a model provides a set of domain
  elements and a way to interpret each piece of
  syntax.
• It just tells you what is meant by what you wrote
  down.
• Example
• you can interpret Father(Bruce) and
  hasChild(Bruce, Mary) when I tell you who Bruce
  is and who Mary is and what Father means (a set
  of individuals) and what hasChild means (a role,
  a binary relation)
• Whether or not the statements are true depends on
  whether Bruce is a Father and on whether Bruces
  child is Mary in the world.
• In model theory the weird thing we do is fix the
  formulas and let the interpretations vary.
• We dont always have a specific world in mind
• When a set of formulas is true is some world we
  say that the formulas are a model of that world.
  
• They say something accurate about it, but dont
  tell you everything.
• What makes this useful is that when you do some
  syntactic manipulation to generate new formulas
  from the model, we expect that the new thing we
  found is also true in the world

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DL Semantics

• Defined by standard Tarski-style interpretations
  
• I (DI, I), where
• DI is the domain (a non-empty set)
• I is an interpretation function that maps
• Concept (class) name A ! subset AI of DI
• Role (property) name R ! binary relation RI over
  DI
• Individual name i ! iI element of DI

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The Basic Description Language AL
Negation can only be applied to atomic concepts
Syntax Semantic
gtI ?I (universal concept)
?I (bottom concept)
( A)I ?I \ AI (atomic negation)
(C u D)I CI u DI (intersection)
(8 R.C)I a 2 ?I 8 b.(a,b) 2 RI ! b 2 CI (value restriction)
(9 R.gt)I a 2 ?I 9 b.(a,b) 2 RI (limited exists quantification)
R RI µ ?I ?I (R is an atomic role)
A AI µ ?I (A is an atomic concept)
Only the top concept is allowed in the scope of
an existential quantification over a role
?The sublanguage FL is obtained by disallowing
atomic negation and FL0 is obtained by
disallowing limited existential quantification.
In predicate logic, concept C can be translated
into , C(x) with a free variable x.
,8hasChild.Female(x) 8y.hasChild(x,y) !
Female(y)
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What does 8 R.C and 9 R.C mean?

• A DogLover is someone whose pets are all dogs, in
  this case C
• DogLover 8 hasPet.Dog
• p 8 a, (p, a) 2 hasPet ! a 2 Dog
• Also writen more simply as
• p hasPet(p, a) ! Dog(a)
• A DogLiker is someone who owns a dog, , in this
  case A, C
• DogLiker 9 hasPet.Dog
• p hasPet(p, a) Æ Dog(a)

hasPet hasPet
A Fido
A Fluffy
B Tabby
C Rover
C Flip
Cat
Fluffy
Tabby
Dog
Fido
Rover
Flip
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The Family of AL-Languages

• ALUENC
• U - union
• (C t D) I CI DI
• E - full existential quantification
• (9 R.C)I a 2 ?I 9 b.(a,b) 2 RI Æ b 2 CI
• N - number restrictions
• at least (gt n R)I a 2 ?I b(a,b)2RI gt
  n
• at most (6 n R)I a 2 ?I b(a,b)2RI 6 n
  
• Person u (6 1 hasChild t (gt 3 hasChild u 9
  hasChild.Female))
• denotes those persons that have either not more
  than one child or at least three children, one of
  which is female.
• C - full negation
• CI ?I \ CI

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DL as Fragments of Predicate Logic

• Any concept D as unary predicate with one free
  variable
• Any role R as primitive binary predicate
• 9 R.C corresponds to 9 y.R(x,y) Æ C(y)
• 8 R.C corresponds to 8 y.R(x,y) gt C(y)
• nR corresponds to
• 9 y1,...,yn. R(x,y1) Æ... Æ R(x,yn) Æ 8 iltj.
  yi?yj
• nR corresponds to
• 8 y1,...,yn1. R(x, y1) Æ... Æ R(x,yn1) gt 9
  iltj. yiyj
• Last two examples demonstrate advantage of
  variable free syntax

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Knowledge Base in DL

• A knowledge base K hT , Ai comprise two
  components
• TBOX T introduces terminology (vocabulary of an
  application domain)
• ABOX A contains assertions about named
  individuals in terms of this vocabulary
• The vocabulary consists of concepts and roles
• concepts denote sets of individuals
• roles denote binary relationships between
  individuals

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TBox and ABox

• T (TBox) is a set of axioms of the form
• C v D (concept inclusion)
• C D (concept equivalence)
• R v S (role inclusion)
• - has_parent v has_ancestor
• R S (role equivalence)
• R v R (role transitivity)
• - has_mother v has_ancestor
• - 8has_ancestor.human applies to all successors
  of has_ancestor
• A (ABox) is a set of axioms of the form
• x 2 D (concept instantiation)
• hx, yi 2 R (role instantiation)

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An Example about Family Relationships
value restriction(v/r) (1, NIL)
a role defines a property of Parent
hasChild.Person
IS-A relationship defines a hierarchy over
concepts
Examples in ABox hasChild(MARY,
PETER) Father(PETER)
Examples in TBox Woman Person u Female Mother
Woman u 9 hasChild.Person
These are concepts
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Name Symbols vs. Base Symbols

• atomic concepts occurring in a TBox T can be
  divided into two sets, name symbols NT (or
  defined concepts) and base symbols BT (or
  primitive concepts, occur only on the right-hand
  side)
• a base interpretation for T only interprets the
  base symbols.

Base symbols
Name Symbol
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Unfolding Name Symbols

• Acyclic TBox can be unfolded or expanded by
  eliminating all defined name symbols from the
  right-hand-side with only base symbols

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Reasoning Services

• Satisfiability whether the assertions in an TBox
  and ABox has a model
• Subsumption whether one description is more
  general than another one
• Equivalence whether two classes denote same set
• Instantiation check if an individual is an
  instance of class C
• Retrieval retrieve a set of individuals that
  instantiate C
• ? Problems all reducible to satisfiability.

subsumes(C,D) ? sat ( C u D)
equiv(C,D) ? subsumes(C, D) Æ subsumes(D, C)
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Concept Satisfiability

• The concepts woman, mother, parent are
  satisfiable
• how about woman u mother?
• The conjunct woman u mother can never be
  satisfied

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ABox Satisfiability

• All other inference services can be reduced to
  asat(A) where A is the axioms in TBox and ABox
• instance checking
• instance?(a, C, A) asat (A a C)
• concept satisfiability
• sat(C) asataC
• concept subsumption
• subsumes(C, D) sat(CuD) asat(aCuD)
• Open world assumption
• A andrewmale, (charles, andrew) has_child
• Does instance?(charles, 8has_child.male, A) hold?
  

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Open World Assumption

• Can we prove that instance?(charles,
  8has_child.male, A) holds?
• No!
• Although the ABox contains only knowledge about
  one male child, it is always assumed that
  represented information is incomplete.
• To make this statement hold, we could add
• charles8has_child.male or
• assert that information about a 2nd child will
  not be added in the future charles 9 6
  1has_child (not possible in ALC as we need number
  restrictions)

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Reasoner The Tableau Algorithm

• contains a set of completion rules operating on
  constraint sets or tableaux
• clash triggers
• proof procedure
• transform unfold the TBox
• Transform all concepts into negation normal form
  (i.e. negation only occurs only in front of
  concept names)
• (C u D) ! C t D
• 9R.C ! 8 R.C
• apply completion rules in arbitrary order as long
  as possible
• stops when a clash is found
• terminates if no completion rule is applicable
  anymore
• satisfiable iff a clash-free tableau can be
  derived

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Completion Rules for the Logic ALC
Clash Trigger a A, a A µ A
Role Exists Restriction Rule if 1. a 9R.C 2 A,
and 2. 9b 2 O (a,b)R, bC µ A then A A
(a,b)R, bC with b fresh in A
Conjunction Rule if 1. a C u D 2 A, and 2.
aC, aD A then A A aC, aD
Role Value Restriction Rule if 1. a 8 R.C 2 A,
and 2. 9b 2 O (a,b)R 2 A and 3. bC
?A then A A bC
Disjunction Rule if 1. a C t D 2 A, and 2.
aC, aD Å A ? then A A aC or
A A aD
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Proof for Concept Satisfiability

• Does the concept woman subsume mother?
• Is the concept woman u mother unsatisfiable?
• Application of completion rules

• woman u mother is unsatisfiable
• ) the concept woman subsumes the concept mother

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DL for the Semantic Web

• Web Ontology Language (OWL) W3C Recommendation
  on 10 Feb 2004
• builds on RDF and RDF Schema and adds more
  vocabulary for describing properties and
  classesExtends existing Web standards
• has three increasingly-expressive sublanguages
  OWL Lite (based on DL SHIF (D)) , OWL DL (based
  on DL SHOIN(D)), and OWL Full (OWL DL RDF)
• benefits from many years of DL research
• Well defined semantics
• Formal properties well understood (complexity,
  decidability)
• Known reasoning algorithms
• Implemented systems (highly optimised)
• Example Ontology of Books

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OWL Class Constructor
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OWL Axioms

• Axioms (mostly) reducible to inclusion (v)
• e.g. C D iff both C v D and D v C

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Concrete Abstract Syntax

• Easier to read than OWL, much closer to logic
• ALCAS.pl
• Prolog implementation of ALC reasoner using CAS
• Currently only does TBox consistency
• Working on this
• Shows actions at various steps.
• See http//owl.man.ac.uk/2003/concrete/latest/