GATE | GATE-CS-2007 | Question 48

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.

Please comment below if you find anything wrong in the above post.

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