1. Propositional logic
It represents real-world facts as logical propositions in well formed formulas. It presented as an atomic propositions, and complex sentences can be created using AND, OR, and other operators.
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2. Example It is raining RAINING It is sunny SUNNY It is windy WINDY If it is raining, then it is not sunny RAINING  ¬SUNNY
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3. Classical sentences Socrates is a man SOCRATESMAN Plato is a man
PLATOMAN
MAN (SOCRATES)
MAN (PLATO)
Now is reflects the structure of representation reflects the structure of knowledge itself.
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4. Predicate logic
In represents the logic there are objects, properties, functions (Relations) are involved. It serves a useful way of representing and manipulating some of kinds of knowledge.
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5. Resolution
Resolution is a technique for proving theorems in predicate calculus
Resolution proves a theorem by negating the statement to be proved and adding the negated goal to the set of axioms that are known or have been assumed to be true
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6. Example We wish to prove that “Fido will die” from the statements that
“Fido is a dog” and
“all dogs are animals” and
“all animals will die”
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7. Equivalent Reasoning by Resolution
Convert these predicates to clause form
Predicate Form
Clause Form
Dog(fido)
x: [dog(x)animal(x)]
¬ dog(x) V animal(x)
y:[animal(y) die(y)]
¬ animal(y) V die(y)
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8. Apply Resolution Negate the conclusion that fido will die ¬die(fido)
¬dog(x) V animal(x) ¬animal(y) V die(y)
[Y/X]
dog(fido) ¬dog(y) V die(y)
[fido/Y]
die(fido) ¬die(fido)
a Null clauseHence fido will die
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9. Steps in Resolution Put the premises or axioms into clause form
Add the negations of what is to be proved in clause form, to the set of axioms
Resolve these clauses together, producing new clauses that logically follow from them
Produce a contradiction by generating the empty clause
The substitutions used to produce the empty clause are those under which the opposite of the negated goal is true
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10. Marcus was a man Man (Marcus) Marcus was a Pompeian Pompeian(Marcus) All Pompeian were Romans x:Pompeian(x)  Roman(x) Ceaser was a ruler Ruler(Ceaser) All Romans were either loyal to Ceaser or Hated him x: Roman(x)  loyalto(x,Caesar) V hate (x, Caesar) Everyone is loyal to someone x: y: loyalto (x,y) People only try to assassinate rulers they are not loyol to x: y : Person(x) ᴧ ruler(y) ᴧ tryassassinate (x,y  ¬ loyalto(x,y)
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11. Marcus tried to assassinate Caesar. Tryassassinate (Marcus, Caeser)
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12. ] Question : Was Marcus loyal to Ceasar? [
Prove this : Marcus was not loyal to Ceasar
(If we got Marcus was not loyal to Ceasor otherwise Marcus was Loyal. )
Proven
1. Man (Marcus) 2. Pompeian(Marcus) 3. x:Pompeian(x)  Roman(x) 4. Ruler(Ceaser) 5. x: Roman(x)  loyalto(x,Caesar) V hate (x, Caesar) 6. x: y: loyalto (x,y) 7. x: y : Person(x) ᴧ ruler(y) ᴧ tryassassinate (x,y) ¬ loyalto(x,y) 8. Tryassassinate (Marcus, Caeser) All men are people 9. x: man (x)  person(x) [ ]
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13. Another Example
Anyone passing the Artificial Intelligence exam and winning the lottery is happy. But anyone who studies or is lucky can pass all their exams. Gopal did not study but he is lucky. Anyone who is lucky wins the lottery. Is Gopal happy?
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14. Another Example
Anyone passing the AI Exam and winning the lottery is happy X:[pass(x,AI) Λ win(x, lottery) happy(x)] Anyone who studies or is lucky can pass all their exams X : Y [studies(x) V lucky(x) pass(x,y)] gopal did not study but he is lucky ¬ study(gopal) Λ lucky(gopal) Anyone who is lucky wins the lottery X: [lucky(x) win(x,lottery)]
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15. Change to clause Form ¬pass(X,AI) V ¬win(X,lottery) V happy(X)
¬study(Y) V pass(Y,Z)
¬lucky(W) V pass(W,V)
¬study(gopal)
Lucky(gopal)
¬lucky(u) V win(u,lottery)
Add negation of the conclusion ¬happy(gopal)
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16. ¬pass(X,AI) V ¬win(X,lottery) V happy(X) win(u,lottery) V ¬lucky(u) u/v ¬pass(u,AI) V happy(u) V ¬lucky(u) gopal/u ¬happy(gopal) ¬pass(gopal,AI) V ¬lucky(gopal) lucky(gopal) ¬ pass(gopal,AI) gop/v,AI/w ¬lucky(v) V pass(V,W) ¬lucky(gopal) lucky(gopal)
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NULL

17. Example Consider the following logic program.
GRANDFATHER (x, y)  FATHER (x, z) , PARENT (z, y)
PARENT (x, y)  FATHER (x, y)
PARENT (x, y)  MOTHER (x, y)
FATHER ( abraham, robert)
FATHER ( robert, mike)
In FOL above program is represented as a set of clauses as:
S = { GRANDFATHER (x, y) V ~FATHER (x, z) V ~ PARENT (z, y), PARENT(x, y) V ~ FATHER (x, y),
PARENT(x, y) V ~ MOTHER (x, y),
FATHER ( abraham, robert), }

18. Example i. GRANDFATHER(x, y) V ~FATHER(x, z) V ~PARENT(z, y)
Let us number the clauses of S as follows:
i. GRANDFATHER(x, y) V ~FATHER(x, z) V ~PARENT(z, y)
ii. PARENT(x, y) V ~ FATHER (x, y)
iii. PARENT(x, y) V ~ MOTHER (x, y),
iv. FATHER ( abraham, robert)
v. FATHER ( robert, mike)
Question : “Is abraham a grandfather of mike ?"

19. Ground Query “Is abraham a grandfather of mike ?"
 GRANDFATHER (abraham, mike).
In FOPL, ~GRANDFATHER(abraham, mike) is negation of goal { GRANDFATHER (abraham, mike).
Include {~goal} in the set S and show using resolution refutation that S  {~ goal} is unsatisfiable in order to conclude the goal.
Let ~ goal is numbered as ( vi) in continuation of first five clauses of S listed above.
vi. ~ GRANDFATHER (abraham, mike)
Resolution tree is given on next slide:

20. Resolution ( i ) ( vi ) {x / abraham, y / mike}
( iv) ~ FATHER(abraham, z) V ~PARENT(z, mike)
{z / robert}
~ PARENT (robert, mike) ( ii)
~ FATHER (robert, mike) (v)
Answer: Yes

21. Alpha-beta pruning example
Unevaluated nodes F
Minimax without pruning
Depth-first search
Visit C, A, F,
Visit G, heuristics evaluates to 2 Visit H, heuristics evaluates to 3
Back up {2,3} to F. max(F)=3
Back up to A. β(A)=3. Temporary min(A) is 3.
3 is the ceiling for node A's score.
Visit B according to depth-first order.
Visit I. Evaluates to 5.
Max(B)>=5. α(B)=5.
It does not matter what the value of J is, min(A)=3. β-prune J.
Alpha-beta pruning improves search efficiency of minimax without sacrificing accuracy. J G H I