Which of the following is TRUE about formulae in Conjunctive Normal Form?
(A)
For any formula, there is a truth assignment for which at least half the clauses evaluate to true.
(B)
For any formula, there is a truth assignment for which all the clauses evaluate to true
(C)
There is a formula such that for each truth assignment, at most one-fourth of the clauses evaluate to true.
(D)
None of the above
Answer: (A)
Explanation:
We can easily prove that for any formula, there is a truth assignment for which at least half the clauses evaluate to true .
Proof : Consider an arbitrary truth assignment. For each of its clause ‘j’ , introduce a random variable. Xj = 1 if clause ‘j’ is satisfied Xj = 0 otherwise
Then, X = summation of (j * Xj) is the number of satisfied clauses.
Given any clause ’c’ , it is unsatisfied only if all of its ‘k’ constituent literals evaluates to false as they are joined by OR operator.
Now, because each literal within a clause has a 1/2 chance of evaluating to true independently of any of the truth value of any of the other literals, the probability that they are all false is (1 / 2)k .
Thus, the probability that ‘c’ is satisfied = 1 − (1 / 2)k
So, E(Xj) = 1 * (1 / 2)k = (1 / 2)k
Therefore, E(Xj) >= 1/2 Summation on both sides to get E(X).
Therefore, we have E(X) = summation of (j * Xj) >= m/2 where ‘m’ is the number of clauses.
E(X) represents expected number of satisfied clauses.
Thus, there must exist an assignment that satisfies at least half of the clauses.
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