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GATE | GATE-CS-2007 | Question 48

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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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Last Updated : 28 Jun, 2021
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