Converting Not All Equal 2-Sat Pr0blem to an equivalent 2-SAT pr0blem

Question

I'm looking over previous exam papers, and have come across this question which has confused me.

Question:

Convert the Not-All-Equal 2-SAT problem given by the clauses {x1, x2}, {x2, x3}, {x3, x4}, {x4, x5}, {x5, x1} to an equivalent 2-SAT problem. (Hint: the 2-SAT problem contains 10 clauses).

From my understanding, is this simply finding the negation for each literal in every clause? So for example, {x1, x2} = {-x1, -x2}, and this is done for each clause? Is this correct?

Answer 1

That is correct. Specifically, replace all clauses (x ∨ y) with (x ∨ y) ∧ (~x ∨ ~y). That literally says "x or y have to be true, and x or y have to be false", or equivalently "satisfy (x ∨ y) while making sure that one of x and y is false".

To prove the equivalence, let us first assume that the NAE 2-SAT problem is satisfiable. Let A be a satisfying assignment and let {x, y} be one arbitrary clause. Since exactly one of x and y are true, this implies that (x ∨ y) ∧ (~x ∨ ~y) is true. Hence, the corresponding two clauses in the 2-SAT formula are satisfied. Since {x, y} was chosen arbitrarily, we conclude that A satisfies all the clauses in the 2-SAT formula.

Conversely, let us assume that the NAE 2-SAT is not satisfiable. That is, for any assignment, there exists some clause {x, y} for which x and y are either both true or both false. Let A be an arbitrarily chosen assignment and let {x, y} be a clause that A does not satisfy (in the NAE 2-SAT). Since x = y, this implies that (x ∨ y) ∧ (~x ∨ ~y) is false (because one of the halves of the conjunction will be false). Hence, A does not satisfy the 2-SAT formula. Since A was chosen arbitrarily, we conclude that no assignment satisfies the 2-SAT formula.