Proving "QUESTION" is NP-Complete by reduction from n-variable 3SAT

Question

I'm struggling with a problem in my theory of computation course that asks us to prove "QUESTION" is NP-complete by reduction from n-variable 3SAT. I've done a number of other similar reductions but I keep getting stumped on this particular problem.

We define a question as a string over the alphabet $\{0,1, ?\}$ and say that a question covers all of the strings where substituting ?? with 0s and 1s yields a string such as $0??1$ covers the four strings $0001$, $0011$, $0101$, $0111$.

We have to reduce the 3SAT problem to QUESTION= {A : A is a set of questions, each of length $n$, such that there exists a string $w$ of length $n$ where no question in A covers $w$}. To show that it is NP-complete.

I recognize that some instance of QUESTION will "cover" less than $2^n$ strings but I'm pretty stumped on how to go forward as all of the things I've tried end up not working out.

Answer 1

The reduction is straightforward, but what's likely tripping you up is that while 3CNF clauses seem to describe what's needed for a satisfying assignment, what they do more directly is describe what assignments won't work, and that's the insight you need to do the reduction.

Each clause will be converted into a QUESTION string of length $n$, where $n$ is the number of variables in the 3CNF formula. Number the variables in the formula from 1 to $n$. For each clause output a QUESTION string with a 0, 1 or ? in the position of the string corresponding to that variable's number. Use 1 if the variable is negated in the clause, 0 if it is not negated and ? if it does not appear in the clause at all.

Example: the 3CNF formula has seven variables ($x_1$ ... $x_7$) and there is a clause ($x_1 \lor \lnot x_4 \lor x_7$) you would output the QUESTION string 0??1??0 .

Once all the clauses are converted to QUESTION strings, any $w$ string not covered by the QUESTION strings can be directly converted to a satisfying assignment for the 3CNF formula. As in the QUESTION strings each position in $w$ corresponds to previously numbered variable. But in $w$ each 0 represents a variable that should be negated in the satisfying assignment, and each 1 represents a variable that should not be negated.