1. Knowledge Representation
Lecture 2
Knowledge Representation
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2. 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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3. Knowledge representation
Epistemology Addresses
The structure used to describe the elements of knowledge.
The interpretive process required to use the described knowledge.
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4. Representation Criteria
Transparency
Explicitness
Naturalness
Efficiency
Adequacy
Modularity
Transparency- The extent to which we can easily identify stored knowledge. How much the system is understandable both to the system developer and the system user. Expert systems often separate domain specific knowledge from general problem solving knowledge to improve transparency.
. Explicitness- The extent to which knowledge is directly represented.
. Naturalness- The extent to which knowledge is represented in its native form.
. Efficiency- The ease of access to specific knowledge during execution.
. Adequacy-The extent to which a given structure can be used to represent all knowledge required for the system.
. Modularity- The extent to which knowledge fragments can be stored independently from one another.
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5. 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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6. 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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7. Formal Logic Logic provides Formal Logic
A representation of knowledge &
Automation of the inferencing process
Formal Logic
Propositional Logic
Predicate Logic
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8. 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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9. 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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10. Propositional Logic
The truth table for the implication connective is shown below.
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11. 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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12. Propositional Logic PREMISE: All X are Y Doesn’t matter if X is
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)
Doesn’t matter if X is
apples, planes or Ships.
Only the form is important.
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13. Propositional Logic Propositional logic connectives Conjunction AND
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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14. Propositional Logic Prop Logic Solution - Predicate 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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15. 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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16. First order predicate logic
an alphabet
a formal language
axioms
inference rules
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17. 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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18. Language Evaluation is done by the use of truth tables.
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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19. 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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20. 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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21. 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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22. 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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23. 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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24. 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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25. Inference Process
There are three reasoning methods that can be applied to a set of premises.
Deduction
Abduction
Induction
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26. "x, "y, "z larger(x, y) Ù larger(y, z) ® larger(x, z)
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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27. 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 and B is true
Abduction allows us to say A is possibly true.
Reasoning under uncertainty.
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28. 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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29. 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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30. 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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31. 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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32. 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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33. 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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34. 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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35. 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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36. 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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37. 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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38. 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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39. 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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40. Inference Process
Prove: hate(Marcus, Caesar)
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41. Inference Process Prove: loyalto(Marcus, Caesar)
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42. 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) V hate(y2, x5)
10. hate(x, y) ® persecute(y, x) ~hate(x6, y3) V persecute(y3, x6)
To detect that there is no contradiction we must discover that the only resolvents that can be generated have been generated before.
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43. 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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44. 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(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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45. 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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46. 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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47. 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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48. 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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