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Some sources state that an algorithm that solves the SAT problem not only needs to decide whether a given existentially-quantified formula is satisfiable or not, but, additionally, in the case where the formula is satisfiable, it does need to provide a satisfying assignment. Is this true or not?

I am bit confused, since I thought the SAT problem would just be an existential question.

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2 Answers 2

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The SAT problem is a decision problem. It means that an algorithm that solves SAT must answer true or false, and not necessarily find a satisfying assignment.

However, as many decision problems, there is a functional problem associated to SAT, that asks to find a satisfying assignment when there exists one.

There is a polynomial time reduction from the functional problem to the decision problem. Check here to see what it looks like.

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If you have variables x1 to xn, and you decide in polynomial time whether the problem can be satisfied or not, then if it can be satisfied, you set x1 = true and check if it can still be satisfied. If not you set x1 = false, which can be satisfied.

Then you set x2 = true, and if that cannot be satisfied you set x2 = false. And so on, and so on. So if SAT could be solved in polynomial time, you could find a solution in polynomial time quite easily. A similar approach could be used for TSP, for example.

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