The document details proof by resolution in first-order logic, providing examples involving characters like Rimi and Raja, as well as the relationships between happiness and graduating individuals. It outlines a structured approach to converting statements into first-order logic and then into conjunctive normal form (CNF), followed by negating the conclusion and drawing a resolution graph to prove contradictions. The explanations are methodical, demonstrating the application of logic in artificial intelligence and reasoning.

1. Topic To Be Covered:
First Order Logic(Part-04)
Proof By Resolution In FOL Examples
Jagdamba Education Society's
SND College of Engineering & Research Centre
Department of Computer Engineering
SUBJECT: Artificial Intelligence & Robotics
Lecture No-13(UNIT-03)
Logic & Reasoning
Prof.Dhakane Vikas N

2. Proof By Resolution In FOL Examples
Resolution in FOL: Example
Example:
a. Rimi is Hungry
b. If Rimi is hungry she barks
c. If Rimi is barking then Raja is angry
Prove by resolution that:
Raja is angry
Step-1: Conversion of Facts into FOL
In the first step we will convert all the given statements into its first order
logic.

3. Proof By Resolution In FOL Examples
Resolution in FOL: Example
Example:
a. Rimi is Hungry
b. If Rimi is hungry she barks
c. If Rimi is barking then Raja is angry
Prove by resolution that: Raja is angry
Step-1: Conversion of Facts into FOL
In the first step we will convert all the given statements into its first order
logic.
a. Hungry(Rimi)
b. hungry(Rimi) ->barks(Rimi)
c. barks(Rimi) ->angry(Raja)

4. Resolution in FOL
a. Hungry(Rimi)
b. hungry(Rimi) ->barks(Rimi)
c. barks(Rimi) ->angry(Raja)
Step-2: Conversion of FOL into CNF
In First order logic resolution, it is required to convert the FOL into CNF as CNF form
makes easier for resolution proofs.
a. Hungry(Rimi)
b. ~hungry(Rimi) V barks(Rimi)
c. ~barks(Rimi) V angry(Raja)

5. Resolution in FOL
Step-3: Negate the statement to be proved
 In this statement, we will apply negation to the conclusion statements,
which will be written as
~Angry(Raja

6. Resolution in FOL
Step-4: Draw Resolution graph:
 Now in this step, we will solve the problem by resolution tree using
substitution. For the above problem, it will be given as follows:
 Hence the negation of the conclusion has been proved as a complete
contradiction with the given set of statements.
a. Hungry(Rimi)
b. ~hungry(Rimi) V barks(Rimi)
c. ~barks(Rimi) V angry(Raja)
~Angry(Raja

7. Resolution in FOL
Step-4: Draw
Resolution graph:
a. Hungry(Rimi)
b. ~hungry(Rimi) V
barks(Rimi)
c. ~barks(Rimi) V
angry(Raja)
~Angry(Raja

8. Proof By Resolution In FOL Examples
Resolution in FOL: Example
Example:
a. All people who are graduating are happy
b. All happy people smile
c. Someone is graduating
Prove by resolution that:
Someone is smiling
Step-1: Conversion of Facts into FOL
In the first step we will convert all the given statements into its first order
logic.

9. Proof By Resolution In FOL Examples
Resolution in FOL: Example
Example:
a. All people who are graduating are happy
b. All happy people smile
c. Someone is graduating
Prove by resolution that: Someone is smiling
Step-1: Conversion of Facts into FOL
In the first step we will convert all the given statements into its first order
logic. ∀x∃x
a. ∀x :graduating(x)->happy(x)
b. ∀x: happy(x) ->smile(x)
c. ∃x :graduating(x)
∃x:smiling(x)

10. Resolution in FOL
a. ∀x :graduating(x)->happy(x)
b. ∀x: happy(x) ->smile(x)
c. ∃x :graduating(x)
∃x:smiling(x)
Step-2: Conversion of FOL into CNF
In First order logic resolution, it is required to convert the FOL into CNF as CNF form
makes easier for resolution proofs.
a. ~graduating(x) V happy(x)
b. ~happy(x1) V smile(x1)
c. graduating(x2)
smiling(x3)

11. Resolution in FOL
Step-3: Negate the statement to be proved
 In this statement, we will apply negation to the conclusion statements,
which will be written as
~smiling(x3)

12. Resolution in FOL
Step-4: Draw Resolution graph:
 Now in this step, we will solve the problem by resolution tree using
substitution. For the above problem, it will be given as follows:
 Hence the negation of the conclusion has been proved as a complete
contradiction with the given set of statements.
a. ~graduating(x) V happy(x)
b. ~happy(x1) V smile(x1)
c. graduating(x2)
~smiling(x3)

13. Resolution in FOL
Step-4: Draw
Resolution graph:
a. ~graduating(x)
V happy(x)
b. ~happy(x1) V
smile(x1)
c. graduating(x2)
~smiling(x3)