CIS990Presentation0420030226

1
Presentation 5
Logic-based Knowledge Representation

• Wednesday 26 February 2003
• Pinar Ozden
• CIS990 Knowledge Based Systems and Cognitive
  Modeling
• Paper
• Logic-based Knowledge Representation
• Prof.Franz Baader
• Technical University of Dresden Germany Institute
  of Theoretical Computer Science,
• Chair of Automata Theory, main research area
  Deduction, Knowledge Representation

2
Presentation Outline

• Review
• Propositional Logic
• First Order Predicate Logic
• Possible Worlds Semantics
• Requirements for a (Logic-based) Knowledge
  Representation Formalism
• Why do we need DL, ML and Nonmonotonic Logics?
• Description Logics
• Precursors of DL Semantic Networks and Frames
• Why do we need Description Logics (DL)?
• What is DL?
• A Network Representation of DL
• Syntax and Semantics of DL
• DL language ALC, examples

3
Presentation Outline

• Description Logics (Contd.)
• Inference Problems
• Inference Algorithms
• Problems Encountered
• Connection with other logical formalisms
• Connection with Logic Programming
• DL in a nutshell
• Modal Logics
• Why do we need Logics (ML)?
• What is ML?
• Syntax and Semantics of ML, examples
• Connection with DL and Logic Programming

4
Presentation Outline

• Nonmonotonic Logics
• What is Monotonic Logics and what is Nonmonotonic
  Logics?
• Why do we need Nonmonotonic Logics?
• Approaches to Nonmonotonic Logics, examples
• Consistency-based
• Autoepistemic Logic
• Circumscription
• Nonmonotonic Inference Relation
• Connection with Logic Programming
• Summary
• Terminology
• References

5
ReviewPropositional Logic
Propositional Logic
6
ReviewPropositional Logic1

• Simple statement
• does not contain any other statement as a part,
  lower-case letters, p, q, r, ..., as symbols for
  simple statements
• p "p is true assertion
• ?p "p is false negation
• Connectives
• ?, ?, ?, ? joins simple statements into
  compounds, and joins compounds into larger
  compounds
• Compound Statement
• compound statement is one with two or more simple
  statements as parts i.e. components
• p ? q "either p is true, or q is true, or
  both disjunction
• p ? q "both p and q are true
  conjunction

7
ReviewPropositional Logic2

• p ? q if p is true, then q is true
  implication/conditional
• p ? q p and q are either both true or both
  false equivalence/biconditional
• All meaningful statements have truth values, i.e.
  p is either true or false
• A compound statement is truth-functional if its
  truth value as a whole can be determined based on
  the truth values of its components.
• e.g if we knew the truth values of p and of q,
  then we can find out the truth value of the
  compound, p ? q

8
ReviewPropositional Logic3

• Tautology
• A statement whose truth value is always true,
• e.g it rains or it does not rain
• Contradiction
• A statement whose truth value is always false,
• e.g it rains and it does not rain
• Contingency
• Statements whose truth values may be true or
  false depending on the truth values of its
  compounds
• e.g it rains and the sun shines

9
ReviewFirst Order Predicate Logic
First Order Predicate Logic
10
ReviewFirst Order Predicate Logic1

• A subject is what we make an assertion about, and
  a predicate is what we assert about the subject
• predicate(Subject)
• When the subject of the sentence is an individual
  object, then it is first order logic. When the
  subject is another predicate, then it is second
  order logic or higher order logic
• predicate(predicate(..(..(..(Subject))))).
• Individual constants specifies no individual on
  its own, i.e are short names or abbreviations for
  longer names
• e.g s for Sokrates

11
ReviewFirst Order Predicate Logic2

• Individual variables are place holders that range
  over individual objects
• e.g x for Amy, John,
• Quantifiers inform how many objects the predicate
  is asserted. Universal quantifier asserts a
  predicate of all objects. Existential quantifier
  asserts a predicate of some objects (at least
  one)
• e.g Every student walks ?xstudent(x) ?
  walks(x) Universal Quantifier
• e.g At least somebody loves everybody
  ?x?yloves(x,y)
• scope of ?y
• scope of ?x
• e.g. Some student walks ?xstudent(x) ?
  walks(x) Existential Quantifier
• e.g. Everybody is loved by somebody
  ?x?yloves(x,y)
• In first order logic, all variables range over
  individual objects all predicate letters are
  constants and all quantifiers use individual
  variables. In higher order logics there are
  predicate variables an quantification over
  predicates is allowed

12
ReviewFirst Order Predicate Logic3

• wff.
• A string of symbols from the alphabet of the
  formal language that
• conforms to the grammar of the formal language.
  A sentence is in predicate logic, a wff with no
  free occurrences of any variable. There can also
  be wff.s with 1 free variable, 2 free variables,
  ...n free variables.
• Decidable wff.
• A wff that is either a theorem or the negation
  of a theorem
• Decidable system
• A formal system in which there is an effective
  method for determining whether any given wff is a
  theorem. A system in which the set of theorems is
  a decidable set (a set for which there is an
  effective method to determine whether any given
  object is a member). The question whether a
  system is decidable is often called the
  Entscheidungsproblem. Undecidable system is a
  system for which there is no such effective method

13
ReviewPossible Worlds Semantics
Possible Worlds Semantics
14
ReviewPossible Worlds Semantics
What is Semantics? Semantics determines the
facts in the world to which the sentences
refer. Without semantics a sentence is just an
arrangement of electrons or a collection of marks
on page. With semantics each sentence makes a
claim about the world. (R N Chapter
6) Language World Frog ?
_at_fg(, xcP9_Y ???
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ReviewPossible Worlds Semantics1

• Denotation
• The meaning of a sentence is a function between
  the expressions of a language and the world. The
  argument of a monadic (unary) predicate is the
  set of individuals and the argument of a binary
  predicate is the set of pairs i.e. the Cartesian
  product of DXD where D is the domain of
  discourse. It depicts a relationship between
  individuals

e.g. Peter sleeps is true if Peter belongs to
the set of individuals such that they sleep
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ReviewPossible Worlds Semantics2

• Model
• A world in which a sentence is true under a
  particular interpretation
• Validity
• A sentence which is true in all worlds under all
  interpretations is valid.
• e.g. tautologies
• Satisfiability
• A sentence is satisfiable when it has at least
  one model, i.e. when there is at least one world
  in which it holds

17
ReviewPossible Worlds Semantics3

• Interpretation
• It is the fact to which a sentence refers. If
  this fact occurs in the actual world, then the
  interpretation is true. An interpretation is a
  pairing of expressions and semantic values
• Possible worlds
• a higher level abstraction of states-of-affairs
• not only how the world has to be for a sentence
  to become true
• (states-of-affairs) but also how the world
  might be.
• states-of-affairs is one of all the possible
  worlds.
• e.g. a possible world might be where _at_fg(,
  xcP9_Y has a model

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Requirements for a (Logic-based) Knowledge
Representation Formalism

• An intelligent Knowledge Representation (KR)
  formalism should be able to find implicit
  consequences of its explicitly represented
  knowledge
• A KR formalism should be capable of symbolically
  representing all relevant knowledge in a given
  application domain. This requires
• declarative semantics
• the meaning of the entries in a knowledge base
  must be defined independently of the programs
  that operate on the KB
• maps symbolic expressions into the world, is
  truth-functional
• intelligent retrieval mechanism
• extract relevant knowledge
• structured representation of knowledge for
  cognitive adequacy and faster retrieval
• correlated information should be stored in
  related parts i.e. should be grouped together

19
Why do we need DL, ML and Nonmonotonic Logics?

• First Order Predicate Logic is not sufficient to
  be used as a logical KR formalism because
• there is no treatment for incomplete and
  contradictory knowledge, nor for subjective or
  time-dependent knowledge ? NML and ML deals with
• usual syntax of FOL does not support structured
  knowledge ? DL deals with
• there are no semantically adequate inference
  procedures (because all relevant inference
  problems are undecidable) ? DL deals with
• Logic Programming Languages are programming
  languages thus they are not necessarily
  appropriate as representation languages.
• e.g. PROLOG as the knowledge is not encoded
  independently of the way which it is processed
  (top-down, left-to-right, order matters)

20
Description Logics
Description Logics (DL) Formalism for
Representing Terminological Knowledge
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Precursors of DL Frames and Semantic Networks

• Frames
• are introduced my Minski and they are
  record-like data structures which represent
  situations and objects. The main objective is to
  collect all the information necessary to treat a
  situation in one place
• Semantic Networks
• are developed by Quillian and they represent
  objects and concepts as nodes in a graph. They
  have two types of edges, the property edges and
  IS-A edges. Property edges assign properties to
  concepts and objects and IS-A-edges depict
  hierarchical relationships among concepts and
  instance relationships between objects and
  concepts. Properties are inherited along
  IS-A-edges

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Semantic Networks
Property color Property edge assigns color to
concept green and object Kermit Concept frog
IS-A-edge Kermit is a treefrog, a tree frog is
a frog, a frog is an animal Object
Kermit Inheritance tree frogs, thus Kermit
inherit the property green grass frogs are
brown (not green!)
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Why do we need DL?

• Usual syntax of FOL does not support structured
  knowledge
• there are no semantically adequate inference
  procedures (because all relevant inference
  problems are undecidable)
• Frames and Semantic Networks lack a formal
  semantics. The meaning of a frame or a semantic
  network is left to the intuition of
  users/programmers which results in ambiguities.
• e.g. Figure 1 has two interpretations
• Green is the only possible color for frogs
• Any frog has at least the color green but may
  have other colors too
• To solve the first two problems Description
  Logics introduces a non-standard syntax and
  restricts the expressive power. Value
  Restrictions attempt to solve the third problem

24
What is DL?

• Description Logics (DL) is a class of KR
  formalisms with inference procedures for
  representing terminological knowledge
• are descended from so-called structured
  inheritance networks. The system KL-ONE is the
  first realization
• Main idea start with atomic concepts (unary
  predicates) and roles (binary predicates) and use
  a rather small set of epistemologically adequate
  constructors to build complex concepts and roles.
  Restrict expressive power
• complex concepts concept terms
• complex roles role terms

25
A Network Representation of DL 1
Representation of knowledge about parents,
persons, children, etc. in terms of concepts
w.r.t generality/specificity
Value restriction
Example of a network with DL modification
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A Network Representation of DL 2

• IS-A relationship
• Description Logics has the ability to represent
  other kinds of relationships that can hold
  between concepts, beyond IS-A relationships.
• the concept of Parent has a property that is the
  role labeled hasChild. The role has a value
  restriction v/r and number restriction
• value restriction a limitation on the range
  of types of objects that can fill that role.
• A parent is a person having at least one child,
  and all of his/her children are persons.
• Inheritance of relations from concepts to their
  subconcepts IS-A relationship
• e.g the concept Mother, i.e., a female parent,
  is a more specific descendant of concepts Female
  and Parent thus inherits the restriction on its
  hasChild role from Parent.

27
Syntax and Semantics of DL1

• the concept Frog from Figure 1
• atomic concept
• Animal ? ?color.Green
• atomic role
• the concept definition Frog from Figure 1
• Frog ? Animal ? ?color.Green
• abbreviation
• Interpretations I consist of
• non-empty set 4I (the domain of interpretation)
• an interpretation function assigns
• to every atomic concept A a set AI ? 4I
• to every atomic role R a binary relation RI ? 4I
  X 4I
• every element aI ? 4I to individual names a

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Syntax and Semantics of DL2

• Translation into FOL
• Frog Animal ? ?color.Green gt
• Animal(x) ??ycolor(x,y)?Green(y))
• Terminology (Tbox)
• Consists of a finite set of role definitions of
  the form A?C and P?R where A is a concept name, P
  is a role name , C is a concept term and R is a
  role term
• Definitions are unique (any name may occur at
  most once as a left-hand side definition) and
  acyclic ( the definition of a name must not,
  directly or indirectly refer to this name)
• An interpretation I is a model of a TBox iff it
  satisfies all the definitions
• A?C and P?R in the TBox, i.e AI CI and PI
  RI

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Syntax and Semantics of DL3

• Assertional Component (ABox)
• Introduces individuals by giving them names and
  asserts properties of these individuals
• Let a,b be names for individuals, C be a concept
  term and R be a role term. Then
• C(a) and R(a,b) are assertions
• ABox is a finite set of such assertions
• An Interpretation I is a model of these
  assertions iff aI?CI and (aI,bI)?RI
• e.g. Frog(KERMIT) a concept assertion
• color(KERMIT,Color07) a role assertion
• A Knowledge Base consists of a TBox and an ABox.

30
Syntax and Semantics of DL4
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DL Language ALC
Natural Language Notation
DL Syntax
Semantic Interpretation
32
Inference Problems1

• Objective
• Draw inferences from the explicit knowledge to
  retrieve the implicit knowledge in the KB.
• Satisfiability
• Is the concept description C non-contradictory?
• C is satisfiable iff there is an I such that CI
  ? Ø.
• Consistency
• Is the ABox A non-contradictory?
• A is consistent iff it has a model

33
Inference Problems2

• Subsumption Problem
• Is a concept a subconcept of another concept?
• Concept term C is subsumed by concept term D
  w.r.t. TBox T
• (C vT D) iff CI ? DI holds in all models of I
  of T
• Instance Problem
• Is a an instance of C w.r.t. both T and A?
• the individual a is an instance of the concept
  term C w.r.t. T iff
• a? ? C? holds in all interpretations of ? that
  are models of both T and A
• e.g. if the TBox contains the definition of the
  concept Frog and the ABox contains the assertions
  for KERMIT, then color07 is an instance of Green
  w.r.t TBox and ABox

34
Inference Algorithms1

• Structural Algorithms
• Knowledge Base is viewed as a directed graph
• efficient, sound but incomplete
• Tableaux-based algorithms
• the Tableaux Calculus is a decision procedure for
  solving the problem of satisfiability.
• 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 looks at all the possibilities, so
  that it can eventually prove that no model could
  be found for unsatisfiable formulas.

35
Inference Algorithms2

• Tableaux-based algorithm for ALC

36
Problems Encountered

• Main problem in DL decidability of subsumption
  problem
• No subsumption algorithms both complete and
  polynomial
• Expansion of TBox definitions may lead to an
  exponential blow up
• Instance problem can be harder than subsumption
  problem

37
Connection with Other Logical Formalisms1

• General first order theorem provers, when applied
  to reasoning in DL yield semidecision procedures
  for DL inference problems like subsumption (how?)
• General purpose resolution provers can be applied
  to ALC by appropriate translation techniques
• L2 is a two variable fragment of FOL and is
  decidable. ALC can be translated into FOL thus
  ALC becomes decidable
• ?R.(?R.A) translates to ?yR(x,y) ? ?z
  (R(y,z) ? A(z)
• subformula ?z (R(y,z) ? A(z) does not contain
  x, x can be re-used
• rename the bound variable z into x translates to
• ?yR(x,y) ? ?x (R(y,x) ? A(x)

38
Connection with Other Logical Formalisms2

• Quantifiers in DL are always guarded by role
  expressions
• ?R.C translates to ?yR(x,y) ? C(y)
• thus formula belongs to the guarded fragment of
  FOL.
• Satisfiability of formulae in guarded fragment of
  FOL is decidable
• therefore satisfiability of formulae in ALC is
  also satisfiable (???)

39
Connection with Logic Programming

• Several of the DL constructors cannot be
  expressed in LP languages. Disjunction and
  existential restrictions allow for incompletely
  specified knowledge
• e.g.
• ?pet.(Dog ? Cat)) (BILL)
• which ABox individual is Bills pet?
• Is it a cat or a dog?
• to overcome this deficit extensions of Logic
  Programming languages by disjunction and
  classical negation have been introduced but still
  not sufficient because
• they represent that a set and its complement is
  disjoint but
• they dont represent that the union of a set with
  its complement is the whole universe

40
DL In A Nutshell

• Tried to overcome ambiguities in semantic
  networks and frames
• Restriction to a small set of concept definitions
  for defining concepts
• Well-defined basic inference procedures such as
  subsumption and instance problem
• First realization system KL-ONE
  BrachmanSchmolze 85
• Many successor systems (Classic, Crack, Fact,
  Flex, Kris, Loom, ...)
• First application natural language processing
  now also other domains (configuration of
  technical systems, databases, chemical
  engineering, medical terminology, ...)

41
Modal Logics
Modal Logics (ML) Formalism for Representing
Time-Dependent or Subjective Knowledge
42
Why do we need ML?

• We want to represent time-dependent (temporal)
  knowledge
• e.g. Sometime in future ? holds
• e.g. Always in future ? holds
• We want to represent knowledge about the beliefs
  (modal)
• e.g. Robot believes that ? holds
• e.g. Robot believes that ? is possible

43
What is ML?

• The Modal Logic extends FOL with modal operators
  believes and knows which take sentences as
  arguments instead of terms.
• A world is possible for an agent if it is
  consistent with everything the agent knows
  (notion of theory of possible worlds)
• The propositional multi-modal logic Kn extends
  propositional logic by n pairs of unary operators
  which are called box and diamond operators
• K stands for the basic modal logic, multi-modal
  means there are more than one pair of box and
  diamond operators
• Sometime in future ? holds diamond operator
  ?future? ?
• Always in future ? holds box operator
  future ?
• Robot 1 believes that ? holds robi1 ?
• Robot 1 believes that ? is possible ?robi1? ?

44
Syntax and Semantics of ML1

• Formulae are built from atomic propositions p and
  n.
• The propositional multi-modal logic Kn extends
  propositional logic by n different modal
  parameters m, using Boolean connectives ?,?,? and
  the modal operators m and ?m?
• e.g. robi1 ?future? (p ? ?robi2? ?p)
• translates to Robot 1 believes that sometime in
  the future p will hold while at the same time
  Robot 2 will believe that ? p is possible
• p is an atomic proposition, robi1, robi2, and
  future are modal parameters
• Semantics of Kn
• Kripke Structures K(W,R) a set of possible
  worlds W and a set R of transition relations.

45
Syntax and Semantics of ML2

• The set R contains for every modal parameter m a
  transition relation
• Rm ? W XW each possible world I?W corresponds to
  an interpretation of propositional logic, i.e.
  assigns a truth value pI?0,1 to every atomic
  proposition p
• Validity of a Kn formula ? in the world of I of a
  Kripke structure K. Kn formula ? is valid iff K,I
  ? holds for all Kripke Structures and all
  worlds I in K

46
Connection with DL and Logic Programming1

• Concept terms C of ALC can directly be translated
  into formulae ?c of Kn
• Boolean connectives of ALC to Boolean connectives
  of Kn
• Universal role restrictions (value restrictions)
  are replaced box operator, existential role
  restrictions by diamond operator
• e.g. ?R.A ? ?S.?A translates to R A ?
  ?S? ?A
• Axiomatizations
• If we want to assign modal operators a special
  meaning like the knowledge of an intelligent
  agent or in the future then axiomatizations
  are necessary

47
Connection with DL and Logic Programming2

• A formula ? of Kn is valid (i.e. holds in all
  worlds of all Kripke structures) iff it can be
  derived from instances of Taut and K using modus
  ponens and necessitation
• Knowledge of intelligent agents
• m? translates to agent m knows ?
• thus T translates to An intelligent agent does
  not have an incorrect knowledge i.e if agent m
  knows ? in a situation then ? holds in this
  situation
• 4 translates to an intelligent agent knows what
  it knows
• 5 translates to an intelligent agent knows what
  it does not know

48
Connection with DL and Logic Programming3

• Work has been done to integrate Modal Logic into
  Logic Programming, there are several modal logic
  programming languages
• M. Gelfond. Logic programming and reasoning with
  incomplete information. Annals of Mathematics and
  Artificial Intelligence, 12, 1994
• L. Farinas del Cerro. Molog A system that
  extends Prolog with modal logic. New Generation
  Computing, 435--50, 1986
• M. Abadi and Z. Manna. Temporal logic
  programming. Journal of Symbolic Computation,
  8277--295, 1989

49
Nonmonotonic Logics
Nonmonotonic Logics Formalism for Representing
Incomplete Knowledge
50
What is monotonic nonmonotonic Logics?

• Monotonicity
• A logic is monotonic if when we add some new
  sentences to the KB all the sentences entailed in
  the original KB are still entailed by the new
  larger KB.
• Advantage inferences need not to revised when
  new information is added in the KB
• Disadvantage if new knowledge is contradictory
  with the KB , inconsistency occurs
• Nonmonotonic Logics
• Are used to formalize plausible reasoning
  allowing more general reasoning than standard
  logics to deal with incomplete knowledge
• e.g. All men are mortal
• Sokrates is a man STANDARD
  LOGICS
• Therefore Sokrates is mortal
• Birds typically fly
• Tweety is a bird
  NONMONOTONIC LOGICS
• Therefore Tweety presumably flies

51
Why do we need nonmonotonic logics?1

• Default Rules
• Default rules apply to most individuals but not
  to all. i.e. a proposition P should be treated as
  true until additional evidence is found to prove
  that P is false.
• e.g former Frog example in Semantic Networks
• Frogs are normally green ? DEFAULT RULE
• Kermit is a frog, therefore Kermit is green ?
  the rule is applied as long as no
  contradictory information is found
• not applied to grass frogs since they are not
  green but brown!
• Closed World Assumption
• Assumes that by default available information is
  complete. If an assertion cannot be derived, then
  its negation is deduced
• e.g. if a train connection is not contained in a
  connection timetable, we conclude the connection
  does not exist. If we later learn there is a
  connection, we must withdraw the previous
  connection

52
Why do we need nonmonotonic logics?2

• Frame Problem
• By the application of an action, we need to know
  which properties have changed and which
  properties remained the same
• e.g. sending a letter changes its location but
  not its content
• nonmonotonic inference rule all aspects of the
  world that are not explicitly changed by the
  action remain invariant under its application

53
Approaches to nonmonotonic logics1

• Consistency-based approaches
• Reiters Default Logic A normally implies B
• Deals with the question of how to resolve
  conflicts between different rules
• e.g. Frogs are normally green ? Default rule
• Grass frogs are brown
• An individual cannot be both brown and green
• Grass frogs are frogs
• Kermit is a frog
• Scooter is a grass frog ? Default rule does
  not apply!
• Frogs are normally green ? Default rule
• Grass frogs are normally brown ? Default rule
• An individual cannot be both brown and green
• Grass frogs are frogs
• Kermit is a frog
• Scooter is a grass frog ?Both Default rules
  are applicable!
• We need to be able to decide which default rule
  to apply and/or not to apply one when the other
  has already been applied

54
Approaches to nonmonotonic logics2

• Autoepistemic Logic
• Moore (1985)
• Formalizes nonmonotonicity using sentences of a
  Modal Logic of belief with belief operator L.
• Focuses on stable sets of sentences which can be
  viewed as the beliefs of a rational agent i.e.
  agents reflection on its own states of knowledge
• e.g.
• If an agent does not believe in a particular
  fact, then he believes that he does not believe
  it
• L(Bird(x)) ? ?L(? Fly(x))?Fly(x) If I believe
  that x is a bird and if I dont believe that
  x cannot fly, then I will conclude that x
  flies

55
Approaches to nonmonotonic logics3

• Circumscription
• McCarthy (1980,1986)
• Circumscription is an example to preferential
  semantics for the case of predicate logic.
  Preferential semantics takes as logical
  consequences all the formulae that hold in all
  preferred models whereas predicate logic defines
  logical consequence w.r.t. all models
• Formalizes nonmonotonicity within classical logic
  by circumscribing or limiting the extension of
  certain predicates. Objects in a particular class
  are limited to those that must be in the class
  i.e. an interpretation I is preferred over an
  interpretation if PI ? PJ holds given predicate
  P.
• Default rules can be expressed by the help of an
  abnormality predicate

56
Approaches to nonmonotonic logics4

• e.g.
• Frogs are normally green ? Default rule
• Frog(x) ? ?Ab(x) ?Green(x)
• Brown(x) ? Ab(x) ? introduces the exception
  to
• the Default rule, i.e Default rule
  applied unless there is an exception
• To achieve the circumscription of a theory, add a
  second order axiom that limits the extension of
  certain predicates to a set of axioms

57
Approaches to nonmonotonic logics?

• Nonmonotonic Inference Relation
• Inference rules for nonmonotonic reasoning to
  generate new nonmonotonic consequences

58
Connection with Logic Programming

• Closed World Assumption in Logic Programs and
  the corresponding treatment of Negation as
  Failure leads to a nonmonotonic behavior of
  Logic Programs.
• More recent work in the procedings of the
  conferences
• Non-Monotonic Extensions of Logic Programming
• Logic Programming and Nonmonotonic Reasoning

59
Summary

• Representing knowledge about an application
  domain is not just storing data occurring in this
  domain
• Implicitly present knowledge in the KB should be
  able to be deduced from the explicit knowledge
  present in the KB therefore an intelligent
  retrieval mechanism is necessary to extract
  relevant knowledge
• Declarative Semantics is necessary for KR
  otherwise the domain expert cannot acquire
  knowledge without the detailed knowledge of
  implementation programs that operate on the KB
• Deduction should depend on the semantics of the
  representation language not on the syntactic form
  of the entries in the KB
• (counter example PROLOG)
• Logic Programming Languages are programming
  languages therefore not necessarily appropriate
  as representation languages

60
Summary

• FOL falls short for those requirements
• Description Logics is for representing
  terminological knowledge. Supports structured
  knowledge and provides semantically adequate
  inference procedures
• Modal Logics represents subjective and
  time-dependent knowledge
• Nonmonotonic Logics provides treatment for
  incomplete and contradictory knowledge

61
Terminology

• Propositional Logic, First Order Predicate Logic
• Validity/Satisfiability,Domain of Discourse,
  Denotation, Tautology, Contradiction,
  Contingence, Truth Value, Truth-functional,
  Model, Interpretation, Possible Worlds
• Description Logics
• Semantic Networks, Frames,Concept,
  Interpretation,Subsumption, Instantiation,
  Declarative Semantics, Tableaux Calculus,TBox,
  ABox
• Modal Logics
• Subjective, Beliefs, Time-dependent, Kripke
  Structure, Relation, Axiomatization
• Nonmonotonic Logics
• Closed World Assumption, Circumscription,
  Autoepistemological Logics, Default Logics,
  Preferential Semantics, Nonmonotonic Inference
  Relation

62
References

• 1st ed. (Chapter 6), Russell and Norvig
• Talks of Prof. Franz Baader
• http//lat.inf.tu-dresden.de/baader/Talks/Tablea
  ux2000.pdf
• Introduction Seminar to Semantics, Horst
  Lohnstein, Uni. Cologne,
• http//www.uni-koeln.de/phil-fak/idsl/dozenten/lo
  hnstein
• Nonmonotonic Logic, Leora Morgenstern,
• http//www-formal.stanford.edu/leora/krcourse/non
  mon.081198.ps
• An Introduction to Description Logics, Daniele
  Nardi, Ronald J.Brachman
• http//www.cs.man.ac.uk/franconi/dl/course/dlhb/
  dlhb-01.pdf
• Symbolic Logic Course, Peter Suber, Earlham
  College
• http//www.earlham.edu/peters/courses/log/loghom
  e.htm
• Course on Description Logics, Enrico Franconi,
• http//www.cs.man.ac.uk/franconi/dl/course/slide
  s/prop-DL/propositional-dl.pdf
• Introduction to Montague Semantics (Chapter 1),
  D.R. Dowty