Question

Axioms:

1. Every child loves every candy.
2. Anyone who loves some candy is not a nutrition fanatic.
3. Anyone who eats any pumpkin is a nutrition fanatic.
4. Anyone who buys any pumpkin either carves it or eats it.
5. John buys a pumpkin.
6. Lifesavers is a candy.
7. (Conclusion) If John is a child, then John carves some pumpkin.

1. Translate into FOL sentences
2. Convert the FOL sentences from a into Conjunctive Normal Form (CNF)
3. Negate the conclusion
4. Use resolution to prove/disprove the conclusion

Answer 1

Note:Resolution is usedin the case , if there are number of statements are given, and we need to prove the conclusion of these statements.

Solution of our Problem:

a.Translate into FOL sentences:

Every child loves every candy.

Predicate Logic:

∀ x ∀ y (child(x) ∧ candy(y) → loves(x,y))

Anyone who loves some candy is not a nutrition fanatic.

Predicate Logic:

∀ x ( (∃ y (candy(y) ∧ loves(x,y))) → ¬ fanatic(x))

Anyone who eats any pumpkin is a nutrition fanatic.

Predicate Logic:

∀ x ( (∃ y (pumpkin(y) ∧ leat(x,y))) → ¬ fanatic(x))

Anyone who buys any pumpkin either carves it or eats it.

Predicate Logic:

∀ x ∀ y(pumpkin(y) ∧ buy(x,y))) → ¬ carve(x,y) v eat(x,y))

John buys a pumpkin.

Predicate Logic:

∃ x (pumpkin(x) ∧ buy(john,x))

Lifesavers is a candy.

Predicate Logic:

candy(lifesavers)

b.Convert the FOL sentences from a into Conjunctive Normal Form (CNF):

c.Negate the conclusion:

we need to Eliminate biconditionals and implications:

1.¬child(x) v ¬candy(y) v loves(x,y)

2.¬candy(y) v ¬loves(x,y) v ¬ fanatic(x)

3.¬pumpkin(y) v eat(x,y) v fanatic(x)

4.¬pumpkin(y) v ¬buy(x,y) v carve(x,y) v eat(x,y)

5.1. pumpkin(x) , 5.2. buy(john,x))

6.candy(lifesavers)

d.Use resolution to prove/disprove the conclusion:

(Conclusion) If John is a child, then John carves some pumpkin.

child(Jhon) v ¬pumpkin(y) v ¬carve(Jhon,y)

child(Jhon) v ¬buy(Jhon,y) v eat(Jhon,y) :resolve from no .4 of step b

child(Jhon) v eat(x,y) :resolve from no 5.2 of step b

child(Jhon) v ¬ pumpkin(y) v fanatic(Jhon) :resolve from no 3 of step b

child(Jhon) v fanatic(Jhon) :resolve from no 5 of step b

child(Jhon) v ¬ candy(Lifesavers) v ¬ loves(Jhon,lifesavers) : resolve from no 2 of step b

child(Jhon) v ¬ loves(Jhon,lifesavers) : resolve from no 6 of step b

¬ candy(lifesavers)  : resolve from no 1 of step b

Null clause  : resolve from no 6 of step b

Hence this conclude that : If John is a child, then John carves some pumpkin. (Proved)

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