In: Computer Science
Joe, Sally, Bill, and Ellen are the only members of the club.
Joe is married to Sally.
Bill is Ellen’s brother.
The spouse of every married person in the club is also in the club.
From the paragraph, many people could conclude that Ellen is not married.
[Hint] The properties of equality can be used.
1) Represent the above sentences into FOL(First Order Logic) with your ontology.
2) For many people for to conclude that Ellen is not married, what knowledge must be added to the knowledge base? Give the additional FOL formulas.
3) Using the augmented knowledge base, prove that Ellen is not married by refutation-resolution.
Ans(1) :
The above sentences in First Order Logic (FOL) :
1. ∀x Member(x) → x = Joe ∨ x = Sally ∨ x = Bill ∨ x = Ellen
2. M arried(Joe, Sally)
3. Brother(Ellen, Bill)
4. ∀x M arried(x) → Member(Spouse(x))
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Ans(2) :
To conclude that Ellen is not married, the below knowledge must be added to the knowledge base :
5. ∀x, y M arried(x, y) → House(x) = House(y)
6. ∀x, y M arried(x, y) → M arried(y, x)
7. ∀x, y Brother(x, y) → ¬M arried(x, y)
8. ∀x, y M arried(x, y) ⇔ Spouse(x) = y
Ans(3) :
To prove that ¬Married(Ellen) will more included. From the contrapositive of sentence 4, it is sufficient to demonstrate ¬Member(Spouse(Ellen)). Keep in mind that the Skolem function Spouse is characterized for all people, counting those who are not married. Actuality Ellen is not married, hence value of Spouse(Ellen) is totally self-assertive. So, we can prove ¬Member(Spouse(Ellen)) using sentence 1 =>
we must prove all four of the following:
Spouse(Ellen) ≠ Joe
Spouse(Ellen) ≠ Sally
Spouse(Ellen) ≠ Bill
Spouse(Ellen) ≠ Ellen.
To define Spouse(Ellen) ≠ Joe, we can define ¬M arried(Ellen, Joe), by sentence 8. This objective can be decreased to ¬M arried(Joe, Ellen) by sentence 7 then to Spouse(Joe) ≠ Ellen by sentence 8 .
By above statements since Sally ≠ Ellen, it is sufficient to prrove that Spouse(Joe) = Ellen, which follows from sentences 2 and 8.
Comparative lines of reasoning appear that Ellen’s theoretical spouse cannot be any of the individuals of the bridge club.So it is prooved that Ellen is not married.