Knowledge representation

Rushdi Shams, Dept of CSE, KUET, Bangladesh 1
Knowledge Representation IIKnowledge Representation II
LogicsLogics
Artificial IntelligenceArtificial Intelligence
Version 1.0Version 1.0
There are 10 types of people in this world- who understand binaryThere 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

Introduction
Need 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 ⇒ R
Given 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 S2is false
S1 ⇔ S2is true iff S1⇒S2 is true and
S2⇒S1 is true

Truth Table for Connectives

Model of a Formula
If 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 Formulas
If 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 Formulas
A 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 Formulas
A formula is unsatisfiable if none of its
assignment is true in no models

Logical equivalence
Two 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 Tollens
If 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

Disjunctive Syllogism
If given α v β
and we know ¬α
then β
If given α v β
and 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

Modus Ponens
If 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:

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:

Eliminate implication with its equivalence. This will
turn P Q into ¬ P V Q→
Use 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 or
First Order Predicate Logic orFirst Order Predicate Logic or
Predicate LogicPredicate Logic

Introduction
Propositional logic is declarative
Propositional 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 difficult
Thus, we will use the foundation of propositional
logic and build a more expressive logic

Introduction
Whereas propositional logic assumes the world
contains facts,
first-order logic (like natural language) assumes the
world contains
Objects: 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 Elements
Constants KingJohn, 2, NUS,...
Predicates Brother, >,...
Functions Sqrt, LeftLegOf,...
Variables x, y, a, b,...
Connectives ¬, ⇒, ∧, ∨, ⇔
Equality =
Quantifiers ∀, ∃

Examples
King John and Richard the Lion heart are
brothers
Brother(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

Complex Sentences
Complex 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 Lionheart
2. Evil King John
3. Left leg of Richard
4. Left leg of John
5. 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 John
Person (Richard)
Person (John)
 Again, king can be acted
only upon John
King (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 leg
John -> 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

 Remember, we had five
objects, let us replace them
with a variable x-
1. x ―›Richard the Lionheart
2. x ―› Evil King John
3. x ―› Left leg of Richard
4. x ―› Left leg of John
5. x ―› The crown

 Now, for the quantified
sentence
∀x King (x) ⇒ Person (x)
Richard is king ⇒ Richard is Person
John is king ⇒ John is person
Richard’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

Richard is king ⇒ Richard is
Person
John is king ⇒ John is person
Richard’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 avoid
Typically, ⇒ is the main connective with ∀
Common mistake: using ∧ as the main connective
with ∀:
∀x Studies (x,KUET) ∧ Smart (x)
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
avoid
Typically, ∧ is the main connective with ∃
Common mistake: using ⇒ as the main connective with ∃:
∃x Studies (x,KUET) ⇒ Smart (x)
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

∃x ∀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”

Quantifier 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 to
¬∀x ¬Likes(x,Broccoli)

 Equivalences-
1. ∃x P is equivalent to ¬∀x ¬P
2. ¬∃x ¬P is equivalent to ∀x P
3. ∃x ¬P is equivalent to ¬∀x P
4. ¬∃x P is equivalent to ∀x ¬P

Equality
term1 = 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 base
The 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 West
Missile(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 Chaining
American(x) ∧ Weapon(y) ∧ Sells(x,y,z) ∧ Hostile(z) ⇒
Criminal(x)
Owns(Nono,x)
Missile(x)
American(West)
Enemy(Nono,America)

Forward Chaining
American(West) ∧ Weapon(y) ∧ Sells(x,y,z) ∧ Hostile(z)
⇒ Criminal(x)
Owns(Nono,x)
Missile(x)
American(West)
Enemy(Nono,America)

Forward Chaining
American(West) ∧ Weapon(y) ∧ Sells(West,y,z) ∧
Hostile(z) ⇒ Criminal(x)
Owns(Nono,x)
Missile(x)
American(West)
Enemy(Nono,America)

Forward Chaining
Owns(Nono,x)
Missile(x)
American(West)
Enemy(Nono,America)

Forward Chaining
Owns(Nono,x)
Missile(x)
Enemy(Nono,America) ⇒ Hostile(x)
American(West)
Enemy(Nono,America)

Forward Chaining
Owns(Nono,x)
Missile(x)
Enemy(Nono,America) ⇒ Hostile(Nono)
American(West)
Enemy(Nono,America)

Forward Chaining
Hostile(Nono) ⇒ Criminal(x)
Owns(Nono,x)
Missile(x)
American(West)
Enemy(Nono,America)

Backward Chaining
Owns(Nono,x)
Missile(x)
American(West)
Enemy(Nono,America)

Backward Chaining
Owns(Nono,x)
Missile(x)
Missile(x) ∧ Owns(Nono,x) ⇒Sells(West,x,Nono)
American(West)
Enemy(Nono,America)

Backward Chaining
Owns(Nono,x)
Missile(x)
American(West)
Enemy(Nono,America)

Backward Chaining
American(West) ∧ Weapon(y) ∧ Sells(West,y,Nono) ∧
Owns(Nono,x)
Missile(x)
American(West)
Enemy(Nono,America)

Backward Chaining
Owns(Nono,x)
Missile(x)
American(West)
Enemy(Nono,America)

…& the Inference
Hostile(Nono) ⇒ Criminal(West)
Owns(Nono,x)
Missile(x)
American(West)
Enemy(Nono,America)

Probability: Logic forProbability: Logic for
UncertaintyUncertainty

Conditional Probability
Definition of conditional probability:
P(a | b) = P(a ∧ b) / P(b) if P(b) > 0
Product rule gives an alternative formulation:
P(a ∧ b) = P(a | b) P(b) = P(b | a) P(a)


Inference with Probability

Inference in Probability
P(toothache) =
0.108 + 0.012 + 0.016 + 0.064
= 0.2

P(cavity V toothache) =
0.108 + 0.012 + 0.072 + .008 +
0.016 + 0.064 = 0.28

Can also compute conditional probabilities:

Baye’s Rule
Product 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’ Rule
A 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) = 0.5
He also knows that in every 50000 patients, 1 may have meningitis
Means probability that a patient has meningitis
P (m) = 1/50000
He also knows that in every 20 patients, 1 may have stiff neck
Means probability that a patient has meningitis
P (m) = 1/20
Then, from Bayes’ rule
P(m | s) = P(s | m) P(m)
P(s)

Application of Bayes’ Rule
P(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

References
Artificial Intelligence: A Modern Approach (2nd
Edition)
by Russell and Norvig
Chapter 7, 8, 9, 13
http://www.iep.utm.edu/p/prop-log.htm#H5
http://www.cs.yale.edu/homes/cc392/node5.html
http://www.cs.nyu.edu/courses/spring03/G22.2560-001/ho

Acknowledgement
Dr. Adel Elsayed
Research Leader, M3C Lab, University of Bolton, UK
Weiqiang Wei
PhD Student, University of Bolton, UK

Knowledge representation

Recommandé

Recommandé

Contenu connexe

Tendances

Tendances (20)

En vedette

En vedette (17)

Similaire à Knowledge representation

Similaire à Knowledge representation (15)

Plus de Rushdi Shams

Plus de Rushdi Shams (17)

Dernier

Dernier (20)

Knowledge representation