Preparation for Second Midterm - Answers

Answers

What is the negation of each of the following sentences:

1. for-every x, exists y: son(x) = y
2. for-every x, for-every y: son(x) = y <=> father(y) = x

In your answers, any "not" may only appear after the last appearance of a universal or existential quantifier.

Answer:

1. exists x, for every y: son(x) != y
2. exists x, exists y: not (son(x) = y <=> father(y) = x)

Textbook exercise 8.8 (in both second and third edition).
Answer for exercise 8.8 in the 2nd edition:

There is an extra axiom needed:
```
(Male(x) and Spouse(x, y)) => Female(y)
```
Answer for exercise 8.8 in the 3nd edition:

No, not(Spouse(George, Laura)) does not follow from the given facts. We need the additional axiom:
```
(Spouse(x, y) and (z != x)) => not(Spouse(z, y))
```
If Spouse becomes a unary function, then no additional axiom is needed. Then, the facts would be written as:
Jim != George
Spouse(Laura) = Jim
and the statement not(Spouse(Laura) = George) can be proven as follows:
((Spouse(Laura) = Jim) and (Jim != George)) => not(Spouse(Laura) = George)
If Spouse is a function, then it can only return a single value for each argument (such as Laura).
Consider the technique of propositionalization. For each of the following two knowledge bases, decide if propositionalization can be applied successfully. If not, why not?
```
KB 1:

for-every x: king(x) and greedy(x) => evil(x)
king(John)
greedy(John)
brother(Richard, John)


KB 2:
for-every x: king(x) and greedy(x) => evil(x)
king(John)
greedy(John)
brother(Richard, John)
king(father(John))
```
Answer:

For KB1, propositionalization can be applied successfully because there are no functions. For KB2, propositionalization cannot be applied successfully because there are functions.
Textbook exercise 9.19, parts (a), (b), (c) (second edition), exercise 9.24, parts (a), (b), (c) (third edition).
Answer:
(a): (A) becomes: for every natural number x there exists a natural number y such that x is greater than or equal to y.
(B) becomes: there exists a natural number y such that every natural number x is greater than or equal to y.
(b): (A) is true: any natural number is >= 0.
(c): (B) is true: If y = 0, then any natural number x is greater than or equal to 0.