1. The document discusses knowledge representation and predicate logic.
2. It explains that knowledge representation involves representing facts through internal representations that can then be manipulated to derive new knowledge. Predicate logic allows representing objects and relationships between them using predicates, quantifiers, and logical connectives.
3. Several examples are provided to demonstrate representing simple facts about individuals as predicates and using quantifiers like "forall" and "there exists" to represent generalized statements.
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Knowledge representation In Artificial IntelligenceRamla Sheikh
facts, information, and skills acquired through experience or education; the theoretical or practical understanding of a subject.
Knowledge = information + rules
EXAMPLE
Doctors, managers.
Artificial Intelligence: Introduction, Typical Applications. State Space Search: Depth Bounded
DFS, Depth First Iterative Deepening. Heuristic Search: Heuristic Functions, Best First Search,
Hill Climbing, Variable Neighborhood Descent, Beam Search, Tabu Search. Optimal Search: A
*
algorithm, Iterative Deepening A*
, Recursive Best First Search, Pruning the CLOSED and OPEN
Lists
Search techniques in ai, Uninformed : namely Breadth First Search and Depth First Search, Informed Search strategies : A*, Best first Search and Constraint Satisfaction Problem: criptarithmatic
Artificial Intelligence: Introduction, Typical Applications. State Space Search: Depth Bounded
DFS, Depth First Iterative Deepening. Heuristic Search: Heuristic Functions, Best First Search,
Hill Climbing, Variable Neighborhood Descent, Beam Search, Tabu Search. Optimal Search: A
*
algorithm, Iterative Deepening A*
, Recursive Best First Search, Pruning the CLOSED and OPEN
Lists
Knowledge representation In Artificial IntelligenceRamla Sheikh
facts, information, and skills acquired through experience or education; the theoretical or practical understanding of a subject.
Knowledge = information + rules
EXAMPLE
Doctors, managers.
Artificial Intelligence: Introduction, Typical Applications. State Space Search: Depth Bounded
DFS, Depth First Iterative Deepening. Heuristic Search: Heuristic Functions, Best First Search,
Hill Climbing, Variable Neighborhood Descent, Beam Search, Tabu Search. Optimal Search: A
*
algorithm, Iterative Deepening A*
, Recursive Best First Search, Pruning the CLOSED and OPEN
Lists
Search techniques in ai, Uninformed : namely Breadth First Search and Depth First Search, Informed Search strategies : A*, Best first Search and Constraint Satisfaction Problem: criptarithmatic
Artificial Intelligence: Introduction, Typical Applications. State Space Search: Depth Bounded
DFS, Depth First Iterative Deepening. Heuristic Search: Heuristic Functions, Best First Search,
Hill Climbing, Variable Neighborhood Descent, Beam Search, Tabu Search. Optimal Search: A
*
algorithm, Iterative Deepening A*
, Recursive Best First Search, Pruning the CLOSED and OPEN
Lists
Propositional Logic
CMSC 56 | Discrete Mathematical Structure for Computer Science
August 17, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Analysis of algorithms is the determination of the amount of time and space resources required to execute it. Usually, the efficiency or running time of an algorithm is stated as a function relating the input length to the number of steps, known as time complexity, or volume of memory, known as space complexity.
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Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named “Event Management System” is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
2. • To solve complex problems we need:
1. Large amount of knowledge
2. Mechanism for representation and manipulation
of existing knowledge to create new solution.
Knowledge Representation
– Facts: Things we want to represent. Truth in some
relevant world.
– Representation of facts.
6. 6
Representation and Mapping
• Spot is a dog
dog(Spot)
• Every dog has a tail
x: dog(x) hastail(x)
hastail(Spot)
Spot has a tail
7. • Fact-representation mapping is not one-to-one.
• Good representation can make a reasoning program
trivial.
The Multilated Checkerboard Problem
“Consider a normal checker board from which two
squares, in opposite corners, have been removed.
The task is to cover all the remaining squares exactly
with donimoes, each of which covers two squares. No
overlapping, either of dominoes on top of each other or
of dominoes over the boundary of the multilated board
are allowed.
Can this task be done?” 7
9. Good Knowledge representation should exhibit:
1. Representational adequacy-
Ability to represent all kinds of knowledge that are needed in the
domain.
2. Inferential adequacy-
Ability to manipulate representational structures such that new
knowledge can be derived/inferred from the old.
3. Inferential efficiency-
Ability to incorporate additional information into an existing
knowledge base that can be used to focus the attention of inference
mechanisms in the most promising direction.
4. Acquisitional efficiency-
Ability to easily acquire new information. 9
10. 10
Approaches to KR
1. Simple relational knowledge:
• Provides very weak inferential capabilities.
• May serve as the input to powerful inference engines.
Fails to infer “which right handed player can best face a
particular bowler” .
Player Height Weight handed
Peter 6-0 180 right
Ajay 5-10 170 left
John 6-2 215 left
Vickey 6-3 205 right
11. 11
Approaches to KR
Inheritable knowledge:
• Objects are organized into classes and classes are
organized in a generalization hierarchy.
• Inheritance is a powerful form of inference, but not
adequate.
• Ex. Property inheritance inference mechanism.
Adult male Person
isa
Peter
instance
Right
handed
Tata Consultancy
Works_at
12. 12
Approaches to KR
Inferential knowledge:
• Facts represented in a logical form, which facilitates
reasoning.
• An inference engine is required.
ex. 1. “Marcus is a man”
2. “All men are mortal”
Implies:
3. “Marcus is mortal”
13. 13
Approaches to KR
Procedural knowledge:
• Representation of “how to make it” rather than “what
it is”.
• May have inferential efficiency, but no inferential
adequacy and acquisitional efficiency.
• Ex. Writing LISP programs
14. 14
Issues in KR
1. Important Attributes: Isa and instance attributes.
2. Relationships among attributes: inverses, existence in a Isa
hierarchy, single-valued attributes, techniques for reasoning
about values.
3.Choosing the Granularity: High-level facts may not be adequate
for inference. Low-level primitives may require a lot of storage.
• Ex: “john spotted sue”
[representation: spotted(agent(john),object(sue))]
Q1: “who spotted sue?” Ans1: “john”.
Q2: “Did john see sue?” Ans2: NO ANSWER!!!!
• Add detailed fact: spotted(x,y)-->saw(x,y) then Ans2: “Yes”.
15. • 4.Representing Set of Objects:
• 5. finding the right structure as needed.:
Ex: word “fly” can have multiple meanings:
1. “John flew to new york”
2. “John flew into a rage” [idiom]
3. “john flew a kite”
SELF: Please read frame problem pg. 96-97, Rich & Knight,3rd
edition.
15
16. Propositional logic
• Statements used in mathematics.
• Proposition :is a declarative sentence whose value is
either true or false.
Examples:
• “The sky is blue.” [Atomic Proposition]
• “The sky is blue and the plants are green.”
[Molecular/Complex Proposition]
• “Today is a rainy day” [Atomic Proposition]
• “Today is Sunday” [Atomic Proposition]
• “ 2*2=4” [Atomic Proposition]
16
17. Terminologies in propositional algebra:
Statement: sentence that can be true/false.
Properties of statement:
Satisfyability: a sentence is satisfyable if there is an
interpretation for which it is true.
Eg.”we wear woollen cloths”
Contradiction: if there is no interpretation for which
sentence is true.
Eg. “Japan is capital of India”
Validity: a sentence is valid if it is true for every
interpretation.
Eg. “Delhi is the capital of India” 17
20. INFERENCE RULES IN PROPOSITIONAL LOGIC
1. Idempotent rule:
P ˄ P ==> P
P ˅ P ==> P
2. Commutative rule:
P ˄ Q ==> Q ˄ P
P ˅ Q ==> Q ˅ P
3.Associative rule:
P ˄ (Q ˄ R) ==> (P ˄ Q) ˄ R
P ˅ (Q ˅ R) ==> (P ˅ Q) ˅ R
20
22. 7. Bidirectional Implication elimination:
( P Q ) ==> ( P Q ) ˄ (Q P)
8. Contrapositive rule:
P Q => ךP ךQ
9. Double Negation rule:
(ך ךP) => P
10. Absorption Rule:
P ˅ ( P ˄ Q) => P
P ˄ ( P ˅ Q) => P
22
23. 11.Fundamental identities:
P ˄ ך p => F [contradiction]
P ˅ ךP => T [Tautology]
P ˅ T => P
P ˅ F => P
P ˅ ך T => P
P ˄ F => F
P ˄ T => P
23
24. 12. Modus Ponens:
If P is true and PQ then we can infer Q is also true.
P
PQ
__________
Hence, Q
13. Modus Tollens:
If ךP is true and PQ then we can infer ךQ .
ךP
PQ
__________
Hence, ךQ 24
25. 14. Chain rule:
If pq and qr then pr
15. Disjunctive Syllogism:
if ךp and p˅q we can infer q is true.
16. AND elimination:
Given P and Q are true then we can deduce P and Q
seperately: P ˄ Q P
P ˄ Q Q
25
26. 17. AND introduction:
Given P and Q are true then we deduce P ˄ Q
18. OR introduction:
Given P and Q are true then we can deduce P and Q
separately:
P P ˅ Q
Q P ˅ Q
26
27. • Example:
“I will get wet if it rains and I go out of the house”
Let Propositions be:
W : “I will get wet “
R : “it rains “
S : “I go out of the house”
(S ˄ R) W
27
28. 28
Using Propositional Logic
Representing simple facts
It is raining
RAINING
It is sunny
SUNNY
It is windy
WINDY
If it is raining, then it is not sunny
RAINING SUNNY
29. Normal Forms in propositional Logic
1. Conjunctive normal form (CNF):
e.g. ( P ˅ Q ˅ R ) ˄ (P ˅ Q ) ˄ (P ˅ R ) ˄ P
It is conjunction (˄) of disjunctions (˅)
Where disjunctions are:
1. ( P ˅ Q ˅ R )
2. (P ˅ Q )
3. (P ˅ R ) clauses
4. P
29
30. 2. Disjunctive normal form (DNF):
e.g. ( P ˄ Q ˄ R ) ˅ (P ˄ Q ) ˅ (P ˄ R ) ˅ P
It is disjunction (˅) of conjunctions (˄)
30
31. Procedure to convert a statement to CNF
1. Eliminate implications and biconditionals using formulas:
• ( P Q ) ==> ( P Q ) ˄ (Q P)
• P Q => ךP ˅ Q
2. Apply De-Morgan’s Law and reduce NOT symbols so as to bring negations
before the atoms. Use:
• (ךP ˅ Q) ==> ךP ˄ ךQ
• ך (P ˄ Q) ==> ך P ˅ ךQ
3. Use distributive and other laws & equivalent formulas to obtain Normal forms.
31
32. Conversion to CNF example
Q. Convert into CNF : ( ( PQ )R )
Solution:
Step 1: ( ( PQ )R ) ==> ( ( ךP ˅ Q)R)
==> ך ( ךP ˅ Q) ˅ R
Step 2: ך ( ךP ˅ Q) ˅ R ==> (P ˄ ך Q ) ˅ R
Step 3: (P ˄ ך Q ) ˅ R ==> ( P ˅ R ) ˄ (ך Q ˅ R )
CNF
32
33. Resolution in propositional logic
Proof by Refutation / contradiction.
• Used for theorem proving / rule of inference.
• Method: Say we have to prove proposition A
• Assume A to be false i.e. ךA
• Continue solving the algorithm starting from ךA
• If you get a contradiction (F) at the end it means your initial
assumption i.e. ךA is false and hence proposition A must
be true.
• Clause: disjunction of literals is called clause.
33
34. • How it works?
• E.g. “ If it is Hot then it is Humid. If it is humid then it will rain. It
is hot.” prove that “ it will rain.”
• Solution:
• Let us denote these statements with propositions H,O and R:
– H: “ It is humid”.
– O: “ It is Hot”. And R: “It will rain”.
• Formulas corresponding to the sentences are:
• 1. “if it is hot then it is humid” [ OH] ==> ךO ˅ H
• 2. “If it is humid then it will rain”. [ HR] ==> ךH ˅ R
• 3. “ It is Hot” [ O ] ==> O
•
• To prove: R.
34
35. • Let us assume “it will NOT rain” [ ךR ]
• [ ךR ] [ ךH ˅ R]
ךH [ ךO ˅ H]
ך O O
E [EMPTY CLAUSE / CONTRADICTION ]
35
36. • Since an empty clause ( E ) has been deduced we say that
our assumption is wrong and hence we have proved:
“It will rain”
Using Prepositional Logic:
• Theorem proving is decidable BUT
• It Cannot represent objects and quantification.
• Hence we go for PREDICATE LOGIC
36
37. PREDICATE LOGIC
• Can represent objects and quantification
• Theorem proving is semi-decidable
37
38. Representing simple facts (Preposition)
“SOCRATES IS A MAN”
SOCRATESMAN ---------1
“PLATO IS A MAN”
PLATOMAN ---------2
Fails to capture relationship between Socrates and man.
We do not get any information about the objects involved
Ex:
if asked a question : “who is a man?” we cannot get
answer.
Using Predicate Logic however we can represent above
facts as: Man(Socretes) and Man(Plato)
38
41. • Quantifiers:
• 2 types:-
• Universal quantifier ()
• x: means “for all” x
• It is used to represent phrase “ for all”.
• It says that something is true for all possible values of
a variable.
• Ex. “ John loves everyone”
•
41
42. • Quantifiers:
• 2 types:-
• Universal quantifier ()
• x: means “for all” x
• It is used to represent phrase “ for all”.
• It says that something is true for all possible values of
a variable.
• Ex. “ John loves everyone”
x: loves(John , x)
42
43. • Existential quantifier ( ):
• Used to represent the fact “ there exists some”
• Ex:
• “some people like reading and hence they gain good
knowledge”
x: { [person(x) like (x , reading)] gain(x, knowledge) }
• “lord Haggins has a crown on his head”
• x: crown(x) onhead (x , Haggins)
43
44. Nested Quantifiers
• We can use both and seperately
• Ex: “ everybody loves somebody ”
x: y: loves ( x , y)
• Connection between and
• “ everyone dislikes garlic”
x: like ( x , garlic )
This can be also said as:
“there does not exists someone who likes garlic”
x: like (x, garlic)
44
45. 3. All Romans were either loyal to Caesar or hated him.
x: Roman(x) loyalto (x, Caesar) hate(x, Caesar)
4. Every one is loyal to someone.
x: y: loyalto(x, y) y: x: loyalto(x, y)
5. People only try to assassinate rulers they are not loyal to.
x: y: person(x) ruler(y) tryassassinate(x, y)
loyalto(x, y)
45
46. 46
6. “All Pompeians were Romans”
x: Pompeian(x) Roman(x)
8. Marcus tried to assassinate Caesar.
tryassassinate(Marcus, Caesar)
47. Some more examples
• “all indoor games are easy”
x: indoor_game( x) easy(x)
• “Rajiv likes only cricket”
Like(Rajiv, Cricket)
• “Any person who is respected by every person is a king”
x:y: { person(x) person(y) respects (y ,x) king( x)}
47
48. • “god helps those who helps themselves”
x: helps( god, helps(x , x))
• “everyone who loves all animals is loved by someone”
x: [ y: animal (y) loves( x , y) ]
everyone who loves all animals
z: loves( z , x ) there exist someone z and z loves x
Thus the predicate sentence is:
x: [ [y: animal (y) loves( x , y) ] [ z: loves( z , x ) ] ]
48
• “god helps those who helps themselves”
x: helps( god, helps(x , x))
• “everyone who loves all animals is loved by someone”
x: [ y: animal (y) loves( x , y) ]
everyone who loves all animals
z: loves( z , x ) there exist someone z and z loves x
Thus the predicate sentence is:
x: [ [y: animal (y) loves( x , y) ] [ z: loves( z , x ) ] ]
• “god helps those who helps themselves”
x: helps( god, helps(x , x))
• “everyone who loves all animals is loved by someone”
x: [ y: animal (y) loves( x , y) ]
everyone who loves all animals
z: loves( z , x ) there exist someone z and z loves x
Thus the predicate sentence is:
x: [ [y: animal (y) loves( x , y) ] [ z: loves( z , x ) ] ]
49. Computable functions and predicates
• “ Marcus was born in 40 A.D”
Born( Marcus, 40)
• “ All Pompeians died when volcano erupted in 79 A.D”
Erupted(volcano, 79) x: [ Pompeian (x ) Died (x , 79)]
• “ no mortal lives longer than 150 years”
• How to solve ?
• let t1 is time instance 1 and t2 is time instance 2
• We use computable function gt( … , ….) which computes
greater than.
x: t1: t2: mortal (x) born ( x, t1) gt( t2 –t1, 150 )
dead ( x, t2)
49
50. Resolution algorithm in predicate
logic
• Proof by refutation.
• INPUT: Predicate sentences in clausal form (CNF)
• (See conversion algo on next slide)
• Algorithm steps :-
Convert all the propositions of KB to clause form (S).
2. Negate and convert it to clause form. Add it to S.
3. Repeat until either a contradiction is found or no progress can be made.
a. Select two clauses ( P) and ( P).
b. Add the resolvent ( ) to S.
50
51. 51
Conversion to Clause Form
1. Eliminate .
P Q P Q
2. Reduce the scope of each to a single term.
(P Q) P Q
(P Q) P Q
x: P x: P
x: p x: P
P P
3. Standardize variables so that each quantifier binds a unique variable.
(x: P(x)) (x: Q(x))
(x: P(x)) (y: Q(y))
52. 52
4. Move all quantifiers to the left without changing their relative
order.
(x: P(x)) (y: Q(y))
x: y: (P(x) (Q(y))
5. Eliminate (Skolemization).
x: P(x) P(c) Skolem constant
x: y P(x, y) x: P(x, f(x)) Skolem function
6. Drop .
x: P(x) P(x)
53. 7. Convert the formula into a conjunction of disjuncts.
(P Q) R (P R) (Q R)
8. Create a separate clause corresponding to each conjunct.
9. Standardize apart the variables in the set of obtained
clauses.
53
54. • Example of conversion:
x: [ Roman (x) ( Pompeian( x) hate ( x, Caesar))]
After step 1: i.e. elimination of and the above stmt
becomes:
x: [ Roman (x) (Pompeian( x) hate ( x, Caesar))]
After step 2: i.e. reducing scope of the above stmt becomes:
x: [ Roman (x) (Pompeian( x) hate ( x, Caesar)) ]
x: [ Roman (x) (Pompeian( x) hate ( x, Caesar)) ]
54
55. • Example to demonstrate step 3:- i.e. standardization of
variables.
x: [ [y: animal (y) loves( x , y) ] [ y: loves( y , x ) ] ]
After step 3 above stmt becomes,
x: [ [y: animal (y) loves( x , y) ] [ z: loves( z, x ) ] ]
55
56. • Example to demostrate step 4: Move all quantifiers to the left
without changing their relative order.
•
• x: [ [y: animal (y) loves( x , y) ] [ z: loves( z, x ) ] ]
• After applying step 4 above stmt becomes:
• x: y: z: [ animal (y) loves( x , y) loves( z, x ) ]
• After first 4 processing steps of conversion are carried out
on original statement S, the statement is said to be in
PRENEX NORMAL FORM
56
57. • Example to demostrate step 5: skolemization ( i.e. elimination
of quantifier )
y: President (y)
Can be transformed into
President (S1)
where S1 is a function that somehow produces a value that
satisfies President (S1) – S1 called as Skolem constant
• Ex. 2:
y: x: leads ( y , x )
Here value of y that satisfies ‘leads’ depends on particular value
of x hence above stmt can be written as:
x: leads ( f(x) , x )
Where f(x) is skolem function. 57
58. • Example to demonstrate step 6: dropping prefix
x: y: z: [ Roman (x) know ( x, y) hate( y, z)]
• After prefix dropped becomes,
[ Roman (x) know ( x, y) hate( y, z)]
58
59. • Example to demostrate step 7: Convert the formula into a
conjunction of disjuncts.(CNF)
• Roman (x) ( ( hate (x , caesar) loyalto ( x , caesar) )
• Roman (x) ( ( hate (x , caesar) loyalto ( x , caesar) )
P Q R
• P (Q R ) ( P Q ) (P R )
CLAUSE 1 ( Roman (x) ( hate (x , caesar) )
CLAUSE 2 ( Roman (x) loyalto ( x , caesar) )
59
60. Unification
• It’s a matching procedure that compares two literals and discovers
whether there exists a set of substitutions that can make them
identical.
• E.g.
Hate( marcus , X) Hate (marcus , caesar)
caesar/ X
e.g. 2.
Hate(X,Y) Hate( john, Z) could be unified as:
John/X and y/z
60
62. Resolution algorithm
• It is used as inference mechanism.
• Pre-processing steps:
1. Convert the given English sentence into predicate sentence.
2. Not all of these sentences will be in clausal form (CNF).
If any sentence is not in clausal form then convert it into clausal form.
3. Give these sentences (clauses) as an input to resolution
algorithm.
Resolution algorithm steps:
A. Negate the proposition which is to be proved.
i.e. If we have to prove :-
like(tommy , cookies) then assume like(tommy,cookies)
Add the resultant sentence to the set of sentences from step 3
62
63. B. Repeat until contradiction is found or no progress can be
made:
i. Select two clauses , call them parent clauses and resolve them
together.
The resultant clause is called resolvant.
e.g. P(x) Q(x) R(x) P(X)
Q(x) R(x)
ii. If resolvant contains empty clause then contradiction has been found.
G(x) G(x)
E [ EMPTY CLAUSE] 63
64. iii. If step ii. Results in empty clause , it means our
assumption is wrong
and the original clause (to be proved) has to be true.
64
65. 65
Example
1. Marcus was a man.
2. Marcus was a Pompeian.
3. All Pompeians were Romans.
4. Caesar was a ruler.
5. All Pompeians were either loyal to Caesar or hated him.
6. Every one is loyal to someone.
7. People only try to assassinate rulers they are not loyal to.
8. Marcus tried to assassinate Caesar.
66. 1. “Marcus was a man”
------------- 1
2. “Marcus was a Pompeian”
------------- 2
3. “All Pompeian's were Romans”
=> x1: pompeian(x1) roman(x1).
=> x1: pompeian(x1) roman(x1)
----------------- 3
66
man(marcus)
pompeian (marcus)
pompeian (x1) roman(x1)
67. 4. “Caesar was a ruler”
---------------- 4
5. “all romans were either loyalto caesar or hated him”
=> x2: roman(x2) [ loyalto(x2 , caesar) hate(x2 , caesar) ]
=> x2: roman(x2) loyalto(x2 , caesar) hate(x2 , caesar)
=> roman(x2) loyalto(x2 , caesar) hate(x2 , caesar)
------ 5
67
ruler (caesar)
roman ( x2) loyalto (x2 , caesar) hate (x2 , caesar)
68. • “Every one is loyal to someone”
=> x3: y1: loyalto(x3, y1).
Let f(x3) be a skolem function then,
=> x3: loyalto(x3, f(x3)).
=> loyalto(x3, f(x3))
---------------- 6
68
loyalto (x3, f(x3))
69. 7. “People only try to assassinate rulers they are not loyal to.”
=> x4: y2: [man(x4) ruler(y2) tryassassinate(x4, y2) ]
loyalto(x4, y2)
=> x4: y2: [man(x4) ruler(y2) tryassassinate(x4, y2) ]
loyalto(x4, y2)
x4: y2: man(x4) ruler(y2) tryassassinate(x4, y2)
loyalto(x4, y2)
let f(x4) be skolem function then,
=> x4: man(x4) ruler(f(x4))
tryassassinate(x4, f(x4)) loyalto(x4, f(x4)) 69
70. man(x4) ruler(f(x4)) tryassassinate(x4, f(x4))
loyalto(x4, f(x4))
» ---------- 7
8. “Marcus tried to assassinate Caesar”
tryassassinate(marcus , caesar)
------------ 8
To prove : marcus hate caesar i.e. hate(marcus, caesar)
70
man( x4) ruler(f(x4)) tryassassinate(x4, f(x4))
loyalto(x4, f(x4))
tryassassinate( marcus , caesar )
71. • Assume hate(marcus, caesar)
71
(5)
ך hate (marcus , caesar) roman ( x2) loyalto (x2 , caesar)
hate (x2 , caesar)
x2 / marcus
roman ( marcus) loyalto (marcus, caesar)
pompeian (x1) roman(x1)
(3)
x1 / marcus
(2)
pompeian (marcus)
pompeian (marcus) loyalto (marcus, caesar)
loyalto (marcus, caesar)
72. loyalto (marcus, caesar)
(7)
x4/ marcus
f(x4)/ caesar
man( marcus) ruler( caesar ) tryassassinate( marcus , caesar )
(8)
tryassassinate( marcus , caesar )
man( marcus) ruler( caesar )
(1)
man( marcus)
ruler( caesar ) 72
73. ruler( caesar )
(4)
ruler( caesar )
E
• Since we get an empty clause i.e. contradiction our assumption
that hate(marcus, caesar) is false
hence
hate(marcus, caesar) must be true.
73
74. • Consider the following paragraph:
“ anything anyone eats is called food. Milka likes all kind of
food. Bread is a food. Mango is a food. Alka eats pizza. Alka
eats everything milka eats.”
Translate the following sentences into (WFF) in predicate logic
and then into set of clauses. Using resolution principle answer
the following:
1. Does Milka like pizza?
2. what food Alka eats? [ Question answering]
74
77. like(milka , pizza) (2)
food(y1) like( milka , y1)
pizza/ y1
food(pizza)
(1)
eats(x , y) food(y) pizza/ y
eats(x , pizza)
(5)
eats( alka, pizza)
alka/ x
E
Since like(milka , pizza) is contradiction like(milka , pizza) is true
77
78. Question to be answered : 2. “ what food Alka eats ?”
eats( alka, ??)
there exist something which Alka eats we have to find the value of x
x: eats ( alka, x)
Assume : alka does not eat anything
[x2: eats ( alka, x2)]
=> x2: eats (alka , x2)
=> eats (alka , x2) (7)
(7) (5)
eats (alka , x2) eats( alka, pizza)
pizza/ x2
E
78
79. • Therefore alka does not eat anything is false and
• Alka eats something is true.
• And x2 stores pizza
• Therefore we conclude :
eats ( alka, ??) answer is “pizza”
79
80. Instance and Isa relationship
• “ Marcus is a man”
man(marcus)
OR
instance( marcus , man) where marcus is an object/
instance of class ‘man’
“ all pompeians were romans”
x: pompeian(x) roman(x).
OR
x: instance(x, pompeian) instance(x, roman).
80
81. • Isa Predicate :
“ all pompeians were romans”
x: pompeian(x) roman(x).
OR
x: instance(x, pompeian) instance(x, roman).------(1)
• Now using isa predicate (1) becomes,
Isa( pompeian , roman)
which means pompeian is a subclass of roman class
but it also requires extra axiom :
x: y: z: isa( y, z) instance (x , y) instance ( x , z)
81
82. 82
Using Predicate Logic
• Many English sentences are ambiguous.
• There is often a choice of how to represent
knowledge.
• Obvious information may be necessary for reasoning
• We may not know in advance which statements to
deduce (P or P).