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Artificial Intelligence
Knowledge Representation Problem
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Knowledge bases
Knowledge base = set of sentences in a formal language
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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
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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
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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
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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
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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
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Proposition logic
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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.
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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.
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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
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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
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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”
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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
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Logical Equivalence
0011
0101
1100
1101
1101
p q p p q p q
s s1 2
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Logical equivalence
Two sentences are logically equivalent} iff true in same models: α ≡ ß iff α╞ β and β╞ α
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L3 17
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
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Tables of Logical Equivalences
Excluded middle Negating creates opposite Definition of implication in
terms of Not and Or
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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
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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
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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).
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Tautology by truth tablep q ¬p p q ¬p (p q ) [¬p (p q )]q
T T
T F
F T
F F
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Tautology by truth tablep q ¬p p q ¬p (p q ) [¬p (p q )]q
T T F
T F F
F T T
F F T
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Tautology by truth tablep q ¬p p q ¬p (p q ) [¬p (p q )]q
T T F T
T F F T
F T T T
F F T F
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Tautology by truth tablep q ¬p p q ¬p (p q ) [¬p (p q )]q
T T F T F
T F F T F
F T T T T
F F T F F
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Tautology by truth tablep q ¬p p q ¬p (p q ) [¬p (p q )]q
T T F T F T
T F F T F T
F T T T T T
F F T F F T
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Derivational Proof TechniquesEG: consider the compound proposition
(p p ) ((sr)t) ) (qr )
Q: Why is this a tautology?
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L3 28
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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L3 29
Tautology by proof[¬p (p q )]q
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L3 30
Tautology by proof[¬p (p q )]q
[(¬p p)(¬p q)]q Distributive
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L3 31
Tautology by proof[¬p (p q )]q
[(¬p p)(¬p q)]q Distributive
[ F (¬p q)]q ULE
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L3 32
Tautology by proof[¬p (p q )]q
[(¬p p)(¬p q)]q Distributive
[ F (¬p q)]q ULE [¬p q ]q Identity
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L3 33
Tautology by proof[¬p (p q )]q
[(¬p p)(¬p q)]q Distributive
[ F (¬p q)]q ULE [¬p q ]q Identity
¬ [¬p q ] q ULE
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L3 34
Tautology by proof[¬p (p q )]q
[(¬p p)(¬p q)]q Distributive
[ F (¬p q)]q ULE [¬p q ]q Identity
¬ [¬p q ] q ULE
[¬(¬p) ¬q ] q DeMorgan
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L3 35
Tautology by proof[¬p (p q )]q
[(¬p p)(¬p q)]q Distributive
[ F (¬p q)]q ULE [¬p q ]q Identity
¬ [¬p q ] q ULE
[¬(¬p) ¬q ] q DeMorgan
[p ¬q ] q Double Negation
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L3 36
Tautology by proof[¬p (p q )]q
[(¬p p)(¬p q)]q Distributive
[ F (¬p q)]q ULE [¬p q ]q Identity
¬ [¬p q ] q ULE
[¬(¬p) ¬q ] q DeMorgan
[p ¬q ] q Double Negation
p [¬q q ] Associative
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L3 37
Tautology by proof[¬p (p q )]q
[(¬p p)(¬p q)]q Distributive
[ F (¬p q)]q ULE [¬p q ]q Identity
¬ [¬p q ] q ULE
[¬(¬p) ¬q ] q DeMorgan
[p ¬q ] q Double Negation
p [¬q q ] Associative
p [q ¬q ] Commutative
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L3 38
Tautology by proof[¬p (p q )]q
[(¬p p)(¬p q)]q Distributive
[ F (¬p q)]q ULE [¬p q ]q Identity
¬ [¬p q ] q ULE
[¬(¬p) ¬q ] q DeMorgan
[p ¬q ] q Double Negation
p [¬q q ] Associative
p [q ¬q ] Commutative p T ULE
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L3 39
Tautology by proof[¬p (p q )]q
[(¬p p)(¬p q)]q Distributive
[ F (¬p q)]q ULE [¬p q ]q Identity
¬ [¬p q ] q ULE
[¬(¬p) ¬q ] q DeMorgan
[p ¬q ] q Double Negation
p [¬q q ] Associative
p [q ¬q ] Commutative p T ULE T Domination
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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
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Use truth table to find - P Q P R P -Q R P R - Q A B -C D E F
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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 …)
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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
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First-Order Logic
Syntax
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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)
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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
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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))
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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)
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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)
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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
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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))))
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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))
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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))))
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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.
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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
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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
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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)
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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)
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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.
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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)
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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?