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Title: 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 ???
15
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
16
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

18
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
21
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

22
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!)
23
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
26
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

28
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

29
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
31
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