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					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

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

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 CIS 990: Knowledge-Based Systems and Cognitive Modeling
Kansas State University Department of Computing and Information Sciences

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

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

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

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Review Propositional Logic

Propositional Logic

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Review Propositional Logic[1]
• Simple statement
– does not contain any other statement as a part, lower-case letters, p, q, r, ..., as symbols for simple statements

– p – p

"p is true“ "p is false“

assertion 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
– pq

"either p is true, or q is true, or both“
"both p and q are true“

disjunction
conjunction

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Review Propositional Logic[2]
– 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

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Review Propositional Logic[3]
• 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”

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Review First Order Predicate Logic

First Order Predicate Logic

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Review First Order Predicate Logic[1]
• 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‟

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Review First Order Predicate Logic[2]
• 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‟ = x[student(x)  walks(x)] = Universal Quantifier – e.g „At least somebody loves everybody‟ = x[y[loves(x,y)]]
scope of y scope of x

– e.g. „Some student walks‟ = x[student(x)  walks(x)] = Existential Quantifier – e.g. „Everybody is loved by somebody‟ = x[y[loves(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
Kansas State University Department of Computing and Information Sciences

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Review First Order Predicate Logic[3]
• 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

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Review Possible Worlds Semantics

Possible Worlds Semantics

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Review Possible 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 Frog  World

@#fg(,|| xc&}P%9_Y

???

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Review Possible Worlds Semantics[1]
• 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 CIS 990: Knowledge-Based Systems and Cognitive Modeling
Kansas State University Department of Computing and Information Sciences

Review Possible Worlds Semantics[2]
• 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

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Review Possible Worlds Semantics[3]
• 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 @#fg(,|| xc&}P%9_Y has a model

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

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
Kansas State University Department of Computing and Information Sciences

CIS 990: Knowledge-Based Systems and Cognitive Modeling

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)

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Description Logics

Description Logics (DL)
Formalism for Representing Terminological Knowledge

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

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

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Semantic Networks

Property: color Concept: frog

Property edge: assigns color to concept green and object Kermit 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!)

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

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 nonstandard syntax and restricts the expressive power. Value Restrictions attempt to solve the third problem
Kansas State University Department of Computing and Information Sciences

CIS 990: Knowledge-Based Systems and Cognitive Modeling

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

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

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

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

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.

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Syntax and Semantics of DL [1]
• 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
Kansas State University Department of Computing and Information Sciences

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Syntax and Semantics of DL [2]
• Translation into FOL:
Frog : Animal  color.Green => Animal(x) y[color(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

•

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Syntax and Semantics of DL [3]
• 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) color(KERMIT,Color07) a concept assertion a role assertion

•

A Knowledge Base consists of a TBox and an ABox.

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Syntax and Semantics of DL [4]

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

DL Language ALC

Natural DL Syntax Language Notation

Semantic Interpretation

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Inference Problems [1]
• 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

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Inference Problems [2]
• 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

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Inference Algorithms [1]
• 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.

–

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Inference Algorithms [2]
Tableaux-based algorithm for ALC

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

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

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Connection with Other Logical Formalisms [1]
• 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 y[R(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 y[R(x,y)  x[ (R(y,x)  A(x)]

•

•

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Connection with Other Logical Formalisms [2]
• Quantifiers in DL are always “guarded” by role expressions:
– – – R.C translates to y[R(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 (???)

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

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 Bill‟s 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 don‟t represent that the union of a set with its complement is the whole universe

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

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 [Brachman&Schmolze 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, ...)

• • •

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Modal Logics

Modal Logics (ML)
Formalism for Representing Time-Dependent or Subjective Knowledge

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

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”

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

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”
“Robot 1 believes that  holds”

box operator [future] 
[robi1] 

“Robot 1 believes that  is possible” robi1 
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CIS 990: Knowledge-Based Systems and Cognitive Modeling

Syntax and Semantics of ML [1]
• 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.

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Syntax and Semantics of ML [2]
– – 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

–

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Connection with DL and Logic Programming [1]
– – –


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

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Connection with DL and Logic Programming [2]

–

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”
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–

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Connection with DL and Logic Programming [3]
• 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. Farin~as del Cerro. Molog: A system that extends Prolog with modal logic. New Generation Computing, 4:35--50, 1986 M. Abadi and Z. Manna. Temporal logic programming. Journal of Symbolic Computation, 8:277--295, 1989

–

–

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Nonmonotonic Logics

Nonmonotonic Logics
Formalism for Representing Incomplete Knowledge

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

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 Therefore Sokrates is mortal Birds typically fly Tweety is a bird Therefore Tweety presumably flies NONMONOTONIC LOGICS STANDARD LOGICS

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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 the rule is applied as long as no contradictory information is found “Kermit is a frog, therefore Kermit is green” 

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

CIS 990: Knowledge-Based Systems and Cognitive Modeling

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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

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Approaches to nonmonotonic logics [1]
• Consistency-based approaches:
– –


Reiter‟s Default Logic “ A normally implies B” Deals with the question of how to resolve conflicts between different rules
e.g. Frogs are normally green
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 Frogs are normally green  Default rule does not apply!  Default rule

 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
Kansas State University Department of Computing and Information Sciences

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Approaches to nonmonotonic logics [2]
• 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. agent‟s 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 don‟t believe that x cannot fly, then I will conclude that x flies

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Approaches to nonmonotonic logics [3]
• 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”

–

–

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Approaches to nonmonotonic logics [4]
 e.g. Frogs are normally green 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  Default rule

–

To achieve the circumscription of a theory, add a second order axiom that limits the extension of certain predicates to a set of axioms

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Approaches to nonmonotonic logics?
• Nonmonotonic Inference Relation |~ :
Inference rules for nonmonotonic reasoning to generate new nonmonotonic consequences

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

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”

•

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

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
Kansas State University Department of Computing and Information Sciences

•

•

•

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CIS 990: Knowledge-Based Systems and Cognitive Modeling

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

• •

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

Terminology
• Propositional Logic, First Order Predicate Logic
– Validity/Satisfiability,Domain of Discourse, Denotation, Tautology, Contradiction, Contingence, Truth Value, Truth-functional, Model, Interpretation, Possible Worlds

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Description Logics
– Semantic Networks, Frames,Concept, Interpretation,Subsumption, Instantiation, Declarative Semantics, Tableaux Calculus,TBox, ABox

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Modal Logics
– Subjective, Beliefs, Time-dependent, Kripke Structure, Relation, Axiomatization

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Nonmonotonic Logics
– Closed World Assumption, Circumscription, Autoepistemological Logics, Default Logics, Preferential Semantics, Nonmonotonic Inference Relation

CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences

References
• • • • • •
•

1st ed. (Chapter 6), Russell and Norvig Talks of Prof. Franz Baader
http://lat.inf.tu-dresden.de/~baader/Talks/Tableaux2000.pdf

Introduction Seminar to Semantics, Horst Lohnstein, Uni. Cologne,
http://www.uni-koeln.de/phil-fak/idsl/dozenten/lohnstein

“Nonmonotonic Logic”, Leora Morgenstern,
http://www-formal.stanford.edu/leora/krcourse/nonmon.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/loghome.htm Course on Description Logics, Enrico Franconi, http://www.cs.man.ac.uk/~franconi/dl/course/slides/prop-DL/propositional-dl.pdf “Introduction to Montague Semantics” (Chapter 1), D.R. Dowty

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CIS 990: Knowledge-Based Systems and Cognitive Modeling

Kansas State University Department of Computing and Information Sciences