Knowledge representation

Rushdi Shams, Dept of CSE, KUET, Bangladesh 1

Knowledge Representation IIKnowledge Representation IILogicsLogics

Artificial IntelligenceArtificial IntelligenceVersion 1.0Version 1.0

There are 10 types of people in this world- who understand binary There are 10 types of people in this world- who understand binary and who do not understand binaryand who do not understand binary


Propositional LogicPropositional Logic


IntroductionNeed formal notation to represent

knowledge, allowing automated inference and problem solving.

One popular choice is use of logic.Propositional logic is the simplest.

Symbols represent facts: P, Q, etc..These are joined by logical connectives (and,

or, implication) e.g., P Λ Q; Q RGiven some statements in the logic we can

deduce new facts (e.g., from above deduce R)


Syntactic Properties of Propositional Logic

If S is a sentence, S is a sentence (negation)If S1 and S2 are sentences, S1 S2 is a sentence

(conjunction)If S1 and S2 are sentences, S1 S2 is a sentence

(disjunction)If S1 and S2 are sentences, S1 S2 is a sentence

(implication)If S1 and S2 are sentences, S1 S2 is a

sentence (bi-conditional)


Semantic Properties of Propositional Logic

S is true iff S is false

S1 S2 is true iff S1 is true and S2 is true

S1 S2 is true iff S1is true or S2 is true

S1 S2 is true iff S1 is false or S2 is true

i.e., is false iff S1 is true and S2 is false

S1 S2 is true iff S1S2 is true and

S2S1 is true


Truth Table for Connectives


Model of a FormulaIf the value of the formula X holds 1 for the

assignment A, then the assignment A is called model for formula X.

That means, all assignments for which the formula X is true are models of it.


Model of a Formula


Model of a Formula: Can you do it?


Satisfiable FormulasIf there exist at least one model of a formula

then the formula is called satisfiable.The value of the formula is true for at least

one assignment. It plays no rule how many models the formula has.


Satisfiable Formulas


Valid FormulasA formula is called valid (or tautology) if all

assignments are models of this formula.The value of the formula is true for all

assignments. If a tautology is part of a more complex formula then you could replace it by the value 1.


Valid Formulas


Unsatisfiable FormulasA formula is unsatisfiable if none of its

assignment is true in no models


Logical equivalenceTwo sentences are logically equivalent iff true in same

models: α ≡ ß iff α╞ β and β╞ α


Deduction: Rule of Inference

1.Either cat fur was found at the scene of the crime, or dog fur was found at the scene of the crime. (Premise)

C v D



2.If dog fur was found at the scene of the crime, then officer Thompson had an allergy attack. (Premise)

D → A



3.If cat fur was found at the scene of the crime, then Macavity is responsible for the crime. (Premise)

C → M



4.Officer Thompson did not have an allergy attack. (Premise)

¬ A



5.Dog fur was not found at the scene of the crime. (Follows from 2 D → A and 4. ¬ A). When is ¬ A true? When A is false- right? Now, take a look at the implication truth table. Find what is the value of D when A is false and D → A is true

¬ D


Rules for Inference: Modus TollensIf given α → β

and we know ¬β Then ¬α



6.Cat fur was found at the scene of the crime. (Follows from 1 C v D and 5 ¬ D). When is ¬ D true? When D is false- right? Now, take a look at the OR truth table. Find what is the value of C when D is false and C V D is true

C


Rules for Inference: Disjunctive SyllogismIf given α v β

and we know ¬α then β

If given α v βand we know ¬β then α



7.Macavity is responsible for the crime. (Conclusion. Follows from 3 C → M and 6 C). When is C → M true given that C is true? Take a look at the Implication truth table.

M


Rules for Inference: Modus PonensIf given α → β

and we know α Then β


Conjunctive Normal Form (CNF)A formula is in conjunctive normal form

(CNF) if it is a conjunction (AND) of clauses, where a clause is a disjunction (OR) of literals or a single literal.

It is similar to the canonical product of sums form used in circuit theory

All of the following formulas are in CNF:


Conjunctive Normal Form (CNF)The following formulae are not in CNF:

The above three formulas are respectively equivalent to the following three formulas that are in conjunctive normal form:


Conjunctive Normal Form (CNF)Eliminate implication with its equivalence.

This will turn P → Q into ¬ P V QUse de Morgan's law to move the ¬ symbol

onto atoms (not sentences), replace:

Perform the following operation:


CNF

(p ^ ~q) V (r V s) ^ (r V t)(p V r V s ) ^ (p V r V t) ^ (~q V r V s)^(~q V r V t)


Horn Clause A Horn clause is a clause with at most one positive

literal. Any Horn clause therefore belongs to one of four

categories: 1. A rule: 1 positive literal, at least 1 negative literal.

A rule has the form "~P1 V ~P2 V ... V ~Pk V Q".

2. A fact or unit: 1 positive literal, 0 negative literals. 3. A negated goal : 0 positive literals, at least 1

negative literal. 4. The null clause: 0 positive and 0 negative literals.

Appears only as the end of a resolution proof.


First Order Logic orFirst Order Logic orFirst Order Predicate Logic orFirst Order Predicate Logic or

Predicate LogicPredicate Logic


IntroductionPropositional logic is declarativePropositional logic allows partial/disjunctive/negated

information(unlike most data structures and databases)

Meaning in propositional logic is context-independent(unlike natural language, where meaning depends on context)

Propositional logic has very limited expressive power(unlike natural language)E.g., cannot say “if any student sits an exam they either pass or fail”.

Propositional logic is compositional(meaning of B ^ P is derived from meaning of B and of P)


Introduction You see that we can convert the sentences

into propositional logic but it is difficultThus, we will use the foundation of

propositional logic and build a more expressive logic


IntroductionWhereas propositional logic assumes the

world contains facts,first-order logic (like natural language)

assumes the world containsObjects: people, houses, numbers, colors,

baseball games, wars, …Relations: red, round, prime, brother of, bigger

than, part of, comes between, …Functions: father of, best friend, one more

than, plus, …


Syntax of FOL: Basic ElementsConstants KingJohn, 2, NUS,... Predicates Brother, >,...Functions Sqrt, LeftLegOf,...Variables x, y, a, b,...Connectives , , , , Equality = Quantifiers ,


ExamplesKing John and Richard the Lion heart are

brothersBrother(KingJohn,RichardTheLionheart)

The length of left leg of Richard is greater than the length of left leg of King John> (Length(LeftLegOf(Richard)),Length(LeftLegOf(KingJohn)))


Atomic Sentences


Atomic Sentences


Complex SentencesComplex sentences are made from atomic sentences

using connectives:S, S1 S2, S1 S2, S1 S2, S1 S2,

Example

Sibling(KingJohn,Richard) Sibling(Richard,KingJohn)


Complex Sentences


FOL illustrated Five objects-1. Richard the Lionheart2. Evil King John3. Left leg of Richard4. Left leg of John5. The crown


FOL illustrated Objects are related with

Relations For example, King John

and Richard are related with Brother relationship

This relationship can be denoted by(Richard,John),(John,Richard)


FOL illustrated Again, the crown and

King John are related with OnHead Relationship-OnHead (Crown,John)

Brother and OnHead are binary relations as they relate couple of objects.


FOL illustrated Properties are relations

that are unary. In this case, Person can

be such property acting upon both Richard and JohnPerson (Richard)Person (John)

Again, king can be acted only upon JohnKing (John)


FOL illustrated Certain relationships

are best performed when expressed as functions.

Means one object is related with exactly one object.Richard -> Richard’s left legJohn -> John’s left leg


Universal quantification<variables> <sentence>

Everyone studies at KUET is smart:

x Studies (x,KUET) Smart (x)

x P is true in a model m iff P is true with x being each possible object in the model

Roughly speaking, equivalent to the conjunction of instantiations of P


Universal quantification Remember, we had five

objects, let us replace them with a variable x-

1. x ―›Richard the Lionheart

2. x ―› Evil King John3. x ―› Left leg of Richard4. x ―› Left leg of John5. x ―› The crown


Universal quantification Now, for the quantified

sentencex King (x) Person (x)

Richard is king Richard is Person

John is king John is personRichard’s left leg is king

Richard’s left leg is personJohn’s left leg is king John’s

left leg is personThe crown is king the crown is

person


Universal quantificationRichard is king Richard is

PersonJohn is king John is personRichard’s left leg is king

Richard’s left leg is person

John’s left leg is king John’s left leg is person

The crown is king the crown is person

Only the second sentence is correct, the rest is incorrect


A common mistake to avoidTypically, is the main connective with

Common mistake: using as the main connective with :


means “Everyone Studies at KUET and everyone is smart”


Existential Quantification<variables> <sentence>

Someone studies at KUET is smart:x Studies (x,KUET) Smart (x)

x P is true in a model m iff P is true with x being some possible object in the model

Roughly speaking, equivalent to the disjunction of instantiations of P


Another common mistake to avoidTypically, is the main connective with

Common mistake: using as the main connective with :


means some guys, if they study in KUET, then they are smart


Properties of quantifiers

x y is the same as y x

x y is the same as y x

x y is not the same as y x


Properties of quantifiersx y Loves(x,y)

“There is a person who loves everyone in the world”

y x Loves(x,y)

“Everyone in the world is loved by at least one person”


Properties of quantifiersQuantifier duality: each can be expressed using

the other

x Likes(x,IceCream) is equivalent to x Likes(x,IceCream)

x Likes(x,Broccoli) is equivalent tox Likes(x,Broccoli)


Properties of quantifiers Equivalences-1. x P is equivalent to x P2. x P is equivalent to x P3. x P is equivalent to x P4. x P is equivalent to x P


Equalityterm1 = term2 is true under a given interpretation if

and only if term1 and term2 refer to the same object

E.g., definition of Sibling in terms of Parent:x,y Sibling(x,y) [(x = y) m,f (m = f) Parent(m,x)

Parent(f,x) Parent(m,y) Parent(f,y)]


Example knowledge baseThe law says that it is a crime for an

American to sell weapons to hostile nations. The country Nono, an enemy of America, has some missiles, and all of its missiles were sold to it by Colonel West, who is American.

Prove that Col. West is a criminal


Example knowledge base... it is a crime for an American to sell weapons to hostile nations:

American(x) Weapon(y) Sells(x,y,z) Hostile(z) Criminal(x)Nono … has some missiles,

Owns(Nono,x)Missile(x)

… all of its missiles were sold to it by Colonel WestMissile(x) Owns(Nono,x) Sells(West,x,Nono)

Missiles are weapons:Missile(x) Weapon(x)

An enemy of America counts as "hostile“:Enemy(x,America) Hostile(x)

West, who is American …American(West)

The country Nono, an enemy of America …Enemy(Nono,America)


Forward ChainingAmerican(x) Weapon(y) Sells(x,y,z) Hostile(z) Criminal(x)

Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(x,America) Hostile(x)American(West)Enemy(Nono,America)


Forward ChainingAmerican(x) Weapon(y) Sells(x,y,z) Hostile(z) Criminal(x)



Forward ChainingAmerican(West) Weapon(y) Sells(x,y,z)

Hostile(z) Criminal(x)



Forward ChainingAmerican(West) Weapon(y) Sells(West,y,z)










Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(Nono,America) Hostile(x)American(West)Enemy(Nono,America)




Owns(Nono,x)Missile(x)Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missile(x) Weapon(x)Enemy(Nono,America) Hostile(Nono)American(West)Enemy(Nono,America)



Hostile(Nono) Criminal(x)



Backward ChainingAmerican(West) Weapon(y) Sells(West,y,z)




















Backward ChainingAmerican(West) Weapon(y) Sells(West,y,Nono)




Backward ChainingAmerican(West) Weapon(y) Sells(West,y,Nono)




…& the InferenceAmerican(West) Weapon(y) Sells(West,y,Nono)

Hostile(Nono) Criminal(West)



Probability: Logic for Probability: Logic for UncertaintyUncertainty


Conditional ProbabilityDefinition of conditional probability:

P(a | b) = P(a b) / P(b) if P(b) > 0Product rule gives an alternative formulation:

P(a b) = P(a | b) P(b) = P(b | a) P(a)


Inference with Probability


Inference in ProbabilityP(toothache) =

0.108 + 0.012 + 0.016 + 0.064 = 0.2


Inference in ProbabilityP(cavity V toothache) =

0.108 + 0.012 + 0.072 + .008 + 0.016 + 0.064 = 0.28


Inference in ProbabilityCan also compute conditional probabilities:


Inference in ProbabilityCan also compute conditional probabilities:


Baye’s RuleProduct rule gives an alternative formulation:

P(a b) = P(a | b) P(b) = P(b | a) P(a)

Joining them together, we can find-P(a | b) = P(b | a) P(a)

P(b)


Application of Bayes’ RuleA doctor knows that the disease meningitis causes the patient to have a stiff neck is 50%Means probability of stiff neck given the probability of having meningitis P(s | m) = 0.5He also knows that in every 50000 patients, 1 may have meningitisMeans probability that a patient has meningitisP (m) = 1/50000He also knows that in every 20 patients, 1 may have stiff neckMeans probability that a patient has meningitisP (m) = 1/20Then, from Bayes’ ruleP(m | s) = P(s | m) P(m)

P(s)


Application of Bayes’ RuleP(m | s) = P(s | m) P(m)

P(s)= 0.5 X (1/50000)

1/20= 0.0002

Means he can expect only 1 in 5000 patients with a stiff neck

to have meningitis


ReferencesArtificial Intelligence: A Modern Approach

(2nd Edition)by Russell and NorvigChapter 7, 8, 9, 13

http://www.iep.utm.edu/p/prop-log.htm#H5 http://www.cs.yale.edu/homes/cc392/node5.h

tml

http://www.cs.nyu.edu/courses/spring03/G22.2560-001/horn.html


Acknowledgement Dr. Adel Elsayed

Research Leader, M3C Lab, University of Bolton, UK

Weiqiang WeiPhD Student, University of Bolton, UK

Education

Knowledge representation