artificial intelligence knowledge representation problem

Artificial Intelligence

Knowledge Representation Problem

Knowledge bases

Knowledge base = set of sentences in a formal language

Stages of Knowledge Use Acquisition

structure of facts integration of old & new knowledge

Retrieval (recall) roles of linking and chunking means of improving recall efficiency

Representation

Set of syntactic and semantic conventions which make it possible to describe things

Syntax specific symbols allowed and rules allowed

Semantics how meaning is associated with symbol

arrangements allowed by syntax

Knowledge Representation Schemas

Logic based representation – first order predicate logic, Prolog

Procedural representation – rules, production system

Network representation – semantic networks, conceptual graphs

Structural representation – scripts, frames, objects

Conceptual Graphs each concept has got its type and an instance

general concept – a concept with a wildcard instance

specific concept – a concept with a concrete instance

there exsists a hierarchy of types subtype:

concept w is specialisation of concept v iftype(v)>type(w) or instance(w)::type(v)

dog:Emma browncolour

dog:*X browncolour

animal

dog cat

Types of Knowledge Objects

both physical & concepts Events

usually involve time maybe cause & effect relationships

Performance how to do things

META Knowledge knowledge about how to use knowledge

Proposition logic

Basic connectives and truth tables

statements (propositions): declarative sentences that areeither true or false--but not both.

Eg. Ahmed Hassan wrote Gone with the Wind. 2+3=5.

not statements:

What a beautiful morning!Get up and do your exercises.

Fundamentals of Logic

"The number x is an integer."

is not a statement because its truth value cannot be determined until a numerical value is assigned for x.

Propositional logic Logical constants: true, false Propositional symbols: P, Q, S, ... (atomic

sentences) Sentences are combined by connectives:

...and [conjunction] ...or [disjunction] ...implies [implication / conditional] ..is equivalent [biconditional] ...not [negation]

Literal: atomic sentence or negated atomic sentence

Truth Tables

p q p q p q p q p qp q

0 0 0 0 0 1 1

0 1 0 1 1 1 0

1 0 0 1 1 0 0

1 1 1 1 0 1 1

Examples of PL sentences P means “It is hot.” Q means “It is humid.” R means “It is raining.” (P Q) R

“If it is hot and humid, then it is raining” Q P

“If it is humid, then it is hot” A better way:

Hot = “It is hot”

Humid = “It is humid”

Raining = “It is raining”

Example

s: Aya goes out for a walk.t: The moon is out.u: It is snowing.

( )t u s : If the moon is out and it is not snowing, thenAya goes out for a walk.

If it is snowing and the moon is not out, then Aya will not go out for a walk. ( )u t s

Logical Equivalence

0011

0101

1100

1101

1101

p q p p q p q

s s1 2

Logical equivalence

Two sentences are logically equivalent} iff true in same models: α ≡ ß iff α╞ β and β╞ α

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Tables of Logical Equivalences

Identity lawsLike adding 0

Domination lawsLike multiplying by 0

Idempotent lawsDelete redundancies

Double negation“I don’t like you, not”

Commutativity Like “x+y = y+x”

AssociativityLike “(x+y)+z = y+(x+z)”

DistributivityLike “(x+y)z = xz+yz”

De Morgan

Tables of Logical Equivalences

Excluded middle Negating creates opposite Definition of implication in

terms of Not and Or

Fundamentals of Logic

A compound statement is called a tautology(T0) if it is true for all truth value assignments for its component statements. If a compound statement is false for all such assignments, then it is called a contradiction(F0).

p p q

p p q

( )

( )

: tautology

: contradiction

Propositional Logic - 2 more defn…A tautology is a proposition that’s always TRUE.

A contradiction is a proposition that’s always FALSE.

p p p p p p

T F

F T

T

T

F

F

21

Tautology example

Demonstrate that

[¬p (p q )]q

is a tautology in two ways:

1. Using a truth table – show that [¬p (p q )]q is always true

2. Using a proof (will get to this later).

Tautology by truth tablep q ¬p p q ¬p (p q ) [¬p (p q )]q

T T

T F

F T

F F


T T F

T F F

F T T

F F T


T T F T

T F F T

F T T T

F F T F


T T F T F

T F F T F

F T T T T

F F T F F


T T F T F T

T F F T F T

F T T T T T

F F T F F T

Derivational Proof TechniquesEG: consider the compound proposition

(p p ) ((sr)t) ) (qr )

Q: Why is this a tautology?

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Derivational Proof TechniquesA: Part of it is a tautology (p p ) and the

disjunction of True with any other compound proposition is still True:

(p p ) ((sr)t )) (qr ) T ((sr)t )) (qr ) TDerivational techniques formalize the

intuition of this example.

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Tautology by proof[¬p (p q )]q

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[(¬p p)(¬p q)]q Distributive

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[ F (¬p q)]q ULE

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[ F (¬p q)]q ULE [¬p q ]q Identity

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¬ [¬p q ] q ULE

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¬ [¬p q ] q ULE

[¬(¬p) ¬q ] q DeMorgan

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¬ [¬p q ] q ULE


[p ¬q ] q Double Negation

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¬ [¬p q ] q ULE



p [¬q q ] Associative

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¬ [¬p q ] q ULE




p [q ¬q ] Commutative

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¬ [¬p q ] q ULE




p [q ¬q ] Commutative p T ULE

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¬ [¬p q ] q ULE




p [q ¬q ] Commutative p T ULE T Domination

Examples1. “I don’t study well and fail” is logically

equivalent to “If I study well, then I don’t fail”

2. Write a C program that represents the compound proposition (pq)r

Use truth table to find - P Q P R P -Q R P R - Q A B -C D E F

Limitations of propositional logic So far we studied propositional logic

Some English statements are hard to model in propositional logic:

“If your roommate is wet because of rain, your roommate must not be carrying any umbrella”

Pathetic attempt at modeling this:

RoommateWetBecauseOfRain => (NOT(RoommateCarryingUmbrella0) AND NOT(RoommateCarryingUmbrella1) AND NOT(RoommateCarryingUmbrella2) AND …)

Problems with propositional logic No notion of objects

No notion of relations among objects

RoommateCarryingUmbrella0 is instructive to us, suggesting there is an object we call Roommate,

there is an object we call Umbrella0,

there is a relationship Carrying between these two objects

Formally, none of this meaning is there Might as well have replaced RoommateCarryingUmbrella0 by

P

First-Order Logic

Syntax

Constants Constants refer to objects, functions and relationships.

Ahmed, Mona, loves, happy, Simple sentences express relationships among objects.

loves(Ahmed, Mona) They are called atoms.

Compound sentences capture relationships among relations.loves(x,y) loves(y,x)loves(x,y) loves(y,x) happy(x)

Relations can be unary as well.tall(Tomy)

Elements of first-order logic Objects: can give these names such as Umbrella0,

Person0, John, Earth, …

Relations: Carrying(., .), IsAnUmbrella(.)

Carrying(Person0, Umbrella0), IsUmbrella(Umbrella0)

Relations with one object = unary relations = properties

Functions: Roommate(.)

Roommate(Person0) Equality: Roommate(Person0) = Person1

Example with Functions

E.g. How about saying that Ahmed has a big nose?

Ahmed is an object and

nose_of (Ahmed)

is a function that constructs an object from the argument object.

Then, we can write:

big(nose_of (Ahmed))

E.g. Mona loves her dog.

loves(Mona, dog_of (Mona))

Note: We are allowed to relate sentences only.

So, we can say:

loves(Mona, dog_of (Mona)) loves(Mona, cat_of (Mona))

But not,

loves(Mona,

dog_of (Mona) cat_of (Mona))

First-Order Logic: , The language that we have described so far, consisting of atoms and the

connectives (,,,,,) is typically called predicate logic. To extend it to first-order logic, we need to add quantifiers. The purpose of quantifiers is to allow us to say things about sets of

objects. To say that Heba loves everything we write:

x. loves (Heba, x)We can think of as a big conjunction. For example, if there are only three objects Heba, dog, and cat, what the above asserts is:

loves (Heba, dog) loves (Heba, cat) loves (Heba, Heba) To say that Hassan loves something we write:

x. loves (Hassan, x)We can think of as a big disjunction. For example, if there are only three objects as above, then what we are asserting is:

loves (Hassan, dog) loves (Hassan, cat) loves (Hassan, Hassan)

First Order Predicate Logic – enriched by variables, predicates, functions quantifiers ,

friends(father(david),father(andrew)) Y friends(Y, petr) X likes(X,ice_cream) X Y Z parent(X,Y) parent(X,Z)

siblings(Y,Z)

Reasoning about many objects at once

Variables: x, y, z, … can refer to multiple objects New operators “for all” and “there exists”

Universal quantifier and existential quantifier

for all x: CompletelyWhite(x) => NOT(PartiallyBlack(x)) Completely white objects are never partially black

there exists x: PartiallyWhite(x) AND PartiallyBlack(x) There exists some object in the world that is partially white and

partially black

Practice converting English to first-order logic

“John has an umbrella” there exists y: (Has(John, y) AND IsUmbrella(y)) “Anything that has an umbrella is not wet” for all x: ((there exists y: (Has(x, y) AND

IsUmbrella(y))) => NOT(IsWet(x))) “Any person who has an umbrella is not wet” for all x: (IsPerson(x) => ((there exists y: (Has(x, y)

AND IsUmbrella(y))) => NOT(IsWet(x))))

More practice converting English to first-order logic

“John has at least two umbrellas” there exists x: (there exists y: (Has(John, x) AND

IsUmbrella(x) AND Has(John, y) AND IsUmbrella(y) AND NOT(x=y))

“John has at most two umbrellas” for all x, y, z: ((Has(John, x) AND IsUmbrella(x) AND

Has(John, y) AND IsUmbrella(y) AND Has(John, z) AND IsUmbrella(z)) => (x=y OR x=z OR y=z))

Even more practice converting English to first-order logic…

“Duke’s basketball team defeats any other basketball team”

for all x: ((IsBasketballTeam(x) AND NOT(x=BasketballTeamOf(Duke))) => Defeats(BasketballTeamOf(Duke), x))

“Every team defeats some other team” for all x: (IsTeam(x) => (there exists y: (IsTeam(y)

AND NOT(x=y) AND Defeats(x,y))))

Reverse translation• Translate the following into English.

x hesitates(x) lost(x)• He who hesitates is lost.

x business(x) like(x,Showbusiness)• There is no business like show business.

x glitters(x) gold(x)• Not everything that glitters is gold.

x t person(x) time(t) canfool(x,t)• You can fool some of the people all the time.

Translating English to FOLEvery gardener likes the sun.

x gardener(x) likes(x,Sun) You can fool some of the people all of the time.

x t person(x) time(t) can-fool(x,t)You can fool all of the people some of the time.

x t (person(x) time(t) can-fool(x,t))x (person(x) t (time(t) can-fool(x,t)))

All purple mushrooms are poisonous.x (mushroom(x) purple(x)) poisonous(x)

No purple mushroom is poisonous.x purple(x) mushroom(x) poisonous(x) x (mushroom(x) purple(x)) poisonous(x)

There are exactly two purple mushrooms.x y mushroom(x) purple(x) mushroom(y) purple(y) ^ (x=y) z

(mushroom(z) purple(z)) ((x=z) (y=z)) Clinton is not tall.

tall(Clinton) X is above Y iff X is on directly on top of Y or there is a pile of one or more other

objects directly on top of one another starting with X and ending with Y.x y above(x,y) ↔ (on(x,y) z (on(x,z) above(z,y)))

Equivalent

Equivalent

Resolution for first-order logic for all x: (NOT(Knows(John, x)) OR IsMean(x) OR

Loves(John, x)) John loves everything he knows, with the possible exception of mean

things

for all y: (Loves(Jane, y) OR Knows(y, Jane)) Jane loves everything that does not know her

What can we unify? What can we conclude?

Use the substitution: {x/Jane, y/John}

Get: IsMean(Jane) OR Loves(John, Jane) OR Loves(Jane, John)

Complete (i.e., if not satisfiable, will find a proof of this), if we can remove literals that are duplicates after unification Also need to put everything in canonical form first

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

“There is a person who loves everyone in the world” “Everyone in the world is loved by at least one person”

Quantifier duality: each can be expressed using the other x Likes(x,IceCream) x Likes(x,IceCream) x Likes(x,Broccoli) x

Likes(x,Broccoli)

y x Loves(x,y) x y Loves(x,y)

Using FOL

Brothers are siblings x,y Brother(x,y) Sibling(x,y)

One's mother is one's female parent m,c Mother(c) = m (Female(m) Parent(m,c))

“Sibling” is symmetric x,y Sibling(x,y) Sibling(y,x)

A first cousin is a child of a parent’s sibling x,y FirstCousin(x,y) p,ps Parent(p,x) Sibling(ps,p)

Parent(ps,y)

An example1. Sameh is a lawyer.

2. Lawyers are rich.

3. Rich people have big houses.

4. Big houses are a lot of work. We would like to conclude that Sameh’s house is a

lot of work. Natural languages are ambiguous so we can have

different axiomatizations.

Axiomatization 11. lawyer(Sameh)2. x lawyer(x) rich(x)3. x rich(x) y house(x,y) 4. x,y rich(x) house(x,y) big(y)5. x,y ( house(x,y) big(y) work(y) ) 3 and 4, say that rich people do have at least one house and all

their houses are big. Conclusion we want to show:

house(Sameh, S_house) work(Sameh, S_house) Or, do we want to conclude that John has at least one house

that needs a lot of work? I.e. y house(Sameh,y) work(y)

Amir and the cat Everyone who loves all animals is loved by

someone. Anyone who kills an animal is loved by no

one. Mohamed loves all animals. Either Mohamed or Amir killed the cat, who

is named SoSo. Did Amir kill the cat?

artificial intelligence knowledge representation problem

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