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Knowledge representation

- Epistemology or theory of knowledge is the branch of philosophy that studies the nature, methods, limitations, and validity of knowledge and belief.
- In other words, epistemology primarily addresses the following questions:
- "What is knowledge?",
- "How is knowledge acquired?", and
- "What do people know?"

Wikipedia

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Knowledge representation

- EpistemologyAddresses
- The structure used to describe the elements of knowledge.
- The interpretive process required to use the described knowledge.

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Representation Criteria

- Transparency
- Explicitness
- Naturalness
- Efficiency
- Adequacy
- Modularity

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Knowledge Representation Schemes

Mypoulos and Lavesque (1984) classification:

1) Logical representation schemes

First order predicate logic for knowledge representation (scheme)

Prolog for implementing (media)

2) Procedural representation schemes

Production systems

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Knowledge Representation Schemes

3) Network representation schemes

Capture knowledge as a graph in which the nodes represent the objects or concepts in the problem domain and the arcs represent the relations or associations between them.

- Semantic networks
- Conceptual graphs

4) Structured representation schemes

Each node is considered as a complex data structure consisting of named slots with attached values

- Frame-based systems

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Formal Logic

- Logic provides
- A representation of knowledge &
- Automation of the inferencing process
- Formal Logic
- Propositional Logic
- Predicate Logic

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Propositional Logic

- One of the oldest & simplest type of formal logic
- Formal indicates logic is concerned with form of logical statements as opposed to meaning

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Propositional Logic

- Propositional symbols denote propositions or statements about the world that may be either true or false.
- Examples

It is raining.

raining

It is sunny.

sunny

If it is raining then it is not sunny.

raining Þ Ø sunny

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Propositional Logic

- The truth table for the implication connective is shown below.

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Propositional Logic

For propositional expressions P, Q, and R:

- Ø(ØP) = P
- (PÚQ) = (ØPÞQ)
- De Morgan\'s Law: Ø(PÚQ) = (ØPÙØQ)
- De Morgan\'s Law: Ø(PÙQ) = (ØPÚØQ)
- distributive law: PÚ(QÙR) = (PÚQ)Ù(PÚR)
- distributive law: PÙ(QÚR) = (PÙQ)Ú(PÙR)
- commutative law: (PÙQ) = (QÙP)
- commutative law: (PÚQ) = (QÚP)
- associative law: PÙ(QÙR) = (PÙQ) ÙR
- associative law: PÚ(QÚR) = (PÚQ)ÚR
- contrapositive law: (PÞQ) = (ØQÞØP)

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Propositional Logic

Doesn’t matter if X is

apples, planes or Ships.

Only the form is important.

PREMISE: All X are Y

PREMISE: Z is a X

Conclusion: Z is a Y

- Separating form from meaning is what gives logic its power as a tool
- Propositional logic is a symbolic logic for manipulating propositions or logical variables
- Propositions are classified as either T or F.
- KBS course is brilliant - (definitely T)
- Fish is good to eat - ( uncertain truth as depends on person)

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Propositional Logic

- He is tall - (variable involved so can’t assign value. Also tall is hard concept to define. Can’t be done in pred. Calculus).
- Proposition logic: assign a truth value to statements
- Propositional logic provides a mechanism for assigning a truth value to compound proposition based on value of individual propositions and connective involved

Propositional logic connectives

Conjunction AND

Disjunction OR

Negation NOT A’

Material implication If-Then -->

Material equivalence Equals

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Propositional Logic

- Prop Logic
- can only deal with complete sentences that is, it can not examine the internal structure of a statement.
- too simple for complex domains
- no support for inferencing
- doesn’t handle fuzzy concepts
- Solution - Predicate Logic

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First order predicate logic

Predicate logic was developed in order to analyze more general cases. Propositional logic is a subset of predicate logic.

- Concerned with internal structure of sentences
- Quantifiers - all, some, no - make sentence more exact.
- Wider scope of expression.

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an alphabet

- predicates represent the relations within the domain: man(tom)
- variables - Dog, Color
- functions - father(CHILD), plus (2, 3)
- constants - rover(a_dog), blue(a_color)
- connectives - and, or, not, is equivalent to..
- quantifiers - "X, $ X :

"x [ PERSON(X) ® NEED-AIR(X)]

- delimiters

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Language

A legally constructed formula in the language is called a well-formed formula (wff)

Evaluation is done by the use of truth tables.

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Language

Knowledge engineering When Using Formal Logic

- Develop an understanding of the knowledge.
- Formulate the knowledge as English statements.
- Break the statements into their component parts.
- Choose symbols to represent the elements and relations in each component.
- Build wffs by using the above symbols that represent the statement.
- Predicate relation is derived by its name and its arity.

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Language

Example:

"Rover is a black dog“

"You will gain weight unless you exercise.“

“If the student’s average grade is greater than 90%, then the student will get an A in the course.”

black(rover) dog(rover)

"X [¬exercise(X) ® gain_weight(X) ]

"X, " Name [(student(Name) avrg_grade(Name, X) ge(X, 90)® final_grade(Name, “A”) ]

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Inference Process

Inference in formal logic is the process of generating new wffs from existing wffs through the application of rules of inference.

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Inference Process

Conclusion:

- Many English sentences are ambiguous.
- There is often a choice of ways of representing the knowledge.
- Even in very simple situations a set of sentences is unlikely to contain all the information necessary to reason about the topic at hand.

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Inference Process

Logical Inferences

Modus ponens and modus tolens provide the foundation for making references.

Modus ponens: ((p ® q) Ù p) ® q

If someone is snorkeling then he is wet

"x snorkeling(X) ® wet(X)

If we are given that alex is snorkeling snorkeling (alex)

we can infer wet(alex)

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Inference Process

Logical Inferences

Modus tolens: ((p ® q) Ù q) ® p

If someone is snorkeling then he is wet

"x snorkeling(X) ® wet(X)

If we are given that alex is not wet

wet(alex)

we can infer snorkeling(alex)

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Inference Process

There are three reasoning methods that can be applied to a set of premises.

- Deduction
- Abduction
- Induction

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Inference Process

Deduction is reasoning from known (premises) to unknown (logical conclusions).

"x, "y, "z larger(x, y) Ù larger(y, z) ® larger(x, z)

If our list of axioms contain the axioms

larger(house, car)

larger(car, cat)

Through deductive reasoning the wff

larger(house, cat)

Can be derived and added to our list of axioms

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Inference Process

In Abduction we begin with a conclusion and procede to derive conditions that would make the conclusion valid. In other words we try to find an explanation for the conclusion

It does not guarantee that a true conclusion results.

Therefore it s called unsound rule of inference.

If given A® B andB is true

Abduction allows us to say A is possibly true.

Reasoning under uncertainty.

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Inference Process

Induction

Inductive reasoning forms the basis of scientific discovery.

If given p(a) is true

p(b) is true ……..

Then we conclude "x, p(x) is true

If we observe alex over a period of time and note that whenever he is wet, it turns out that he has gone snorkeling.

We might induce that

"x, wet(x) ® snorkeling(x)

Like abduction, induction is also an unsound inference method.

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Inference Process

Monotonic vs non-monotonic reasoning

Deductive reasoning is a monotonic reasoning that produce as arguments that preserve truth.

Axioms are not allowed to change, since once a fact is known to be true, it is always true and can never be modified or retracted.

Most real life problems are non-monotonic

quarter(fourth)

leading(bucks)

"Team [leading(Team) Ù quarter(fourth)] ® strategy(Team, conservative)

We can deduce strategy(bucks, conservative)

What if the state changes to

leading(dolphins)

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Inference Process

RESOLUTION attempts to show that the negation of the statement produces a contradiction with the known statements.

winter V summer

~winter V cold

- deduce

summer V cold

In the above example if it is winter the first statement is true if not the second statement is true. From these two we can deduce the third statement to be true

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Inference Process

Conjuctive normal form(Davis, 1960)

The steps to convert to conjuctive normal form:

1. Eliminate ® by using the fact that a ® b is equivalent to ~a V b

2. Reduce the scope of ~

~(A V B) = ~A Ù ~B ~(A Ù B) = ~A V ~B DeMorgan\'s law

~"x P(x) = $x ~P(x)

~$x P(x) = "x ~P(x)

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Inference Process

3. Standardize variables so that each quantifier binds a unique variable.

"x P(x) V "x Q(x) would be converted to

"x P(x) V "y Q(y)

4. Move all quantifiers to the left of the formula without changing their relative order.

"x "y P(x) V Q(y)

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Inference Process

5. Eliminate existential quantifiers. A formula that contains an existentially quantified variable asserts that there is a value that can be substituted for the variable that makes the formula true.

$y President(y) can be transformed into President(S1)

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Inference Process

5. If existential quantifiers occur within the scope of a universal quantifier then the value that satisfies the predicate may depend on the values of the universally quantified variables.

"x $y fatherof(y, x) can be transformed into "x fatherof(S2(x), x)

These generated functions are called Skolem functions.

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Inference Process

6. Drop the prefix.

7. Convert the matrix into conjunction of disjuncts.

8. Call each conjunct a separate clause.

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Inference Process

The Unification Algorithm:

The object of the unification procedure is to discover at least one substitution that causes two literals to match.

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Inference Process

Example:

1. Marcus was a man.

man(marcus)

2. Marcus was a Pompeian.

pompeian(marcus)

3. All Pompeians were Romans.

"x pompeian(X) ® roman(X)

4. Caesar is a ruler.

ruler(caesar)

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Inference Process

5. All Romans were either loyal to Caesar or hated him.

"x roman(X) ® loyalto(X, casear) v hate(X, casear)

6. Everyone is loyal to someone.

"X $ Y loyalto(X, Y)

7. People only try to assassinate rulers they are not loyal to.

"X "Y person(X) Ù ruler(Y) Ù tryassassinate(X,Y) ® ~loyalto(X, Y)

8. Marcus tried to assassinate Caesar.

tryassassinate(marcus, caesar)

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Inference Process

Proof by resolution:

Given the axioms in clause form:

1. man(Marcus)

2. Pompeian(Marcus)

3. ~Pompeian(x1) V Roman(x1)

4. ruler(Caesar)

5. ~Roman(x2) V loyalto(x2, Caesar) V hate(x2, Caesar)

6. loyalto(x3, f1(x3))

7. ~person(x4) V ~ruler(y1) V ~tryassassinate(x4, y1) V ~loyalto(x4, y1)

8. tryassassinate(Marcus, Caesar)

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Inference Process

Prove: loyalto(Marcus, Caesar) - Continue

Suppose our knowledge base contained the two additional statements

9. persecute(x, y) ® hate(y, x) ~persecute(x5, y2) Vhate(y2, x5)

10. hate(x, y) ® persecute(y, x) ~hate(x6, y3) V persecute(y3, x6)

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Prolog

Prolog is the implementation language for predicate logic.

Prolog is declarative

Three main features are:

Facts

Rules

Backtracking

If something is not explicitly stated, Prolog assumes that it is false.

Prolog is non-monotonic where it deviates from pure predicate logic.

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Example

child_of(alex, nicole).

child_of(alina, nicole).

child_of(nicholas, leah).

child_of(philip, leah).

child_of(melanie, cathy).

male(alex).

male(philip).

male(alex).

male(nicholas).

female(alina).

female(leah).

female(nicole).

female(angela).

sisters(nicole, leah).

sisters(X, Z):- child_of(X, Y), child_of(Z, Y),

female(X), female(Z)

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Disadvantages of Predicate Logic as a Basis for Knowledge-Based System

- Managing uncertainty
- availability of only two levels of truth – true or false.
- certainty factors have been implemented in Prolog to mitigate this.
- Monotonic versus non-monotonic reasoning.

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Exercises:

Consider the following sentences:

- John likes all kinds of food.
- Apples are food.
- Chicken is food.
- Anything anyone eats and isn\'t killed by is food.
- Bill eats peanuts and is still alive.
- Sue eats everything Bill eats.

a. Translate these sentences into formulas in predicate logic.

b. Convert the formulas of part a into clause form.

c. Prove that John likes peanuts.

d. Use resolution to answer the question, "What food does Sue eat?"

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Exercises:

Consider the following logic statements:

"x [panther(X) ® feline(X)]

"x [house_cat(X) ® feline(X)]

"x [house_cat(X) ® docile(X)]

"x [feline(X) ® carnivore(X)]

"x [carnivore(X) ® food(X, meat)]

"x [horse(X) ® herbivore(X)]

"x [herbivore(X) ® food(X, plants)]

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Exercises:

a)Using modus ponens derive all possible relations from the following facts.

panther(sam)

house_cat(rubble)

lion(leo)

b)Using abduction and the following statement what could kitty be?

food(kitty, meat)

c)Convert all of the logic statements into clauses

d) Attempt to prove the following statements using resolution:

house_cat(rebel) food(rebel, meat)

horse(wilber) food(wilber, meat)

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