Unique SAT complexity clarification

Question

Unique SAT is defined as:

Given any SAT problem, does the SAT problem have exactly 1 solution?

As I understand it is co-NPHard. I am unclear how it is in co-NP

1. Assuming the problem has more than 1 solution, any 2 solutions can be a certificate of it being nonUnique. That part is coNP. Now, given a problem can be unsatisfiable, how do we get a certificate of no solution?

2. Its also said that an NP Oracle can solve this problem in polynomial time, but an NP Oracle just tells if the problem is satisfiable. So how is that possible?

Apologies but I have no clue regarding this version of SAT.

Answer 1

1. To prove coNP-hardness, you don't need a certificate (that would be needed only if you wanted to show coNP-completeness). We just need some coNP-hard problem that reduces to UniqueSAT. UniqueSAT is indeed likely not in coNP.

2. You can ask an oracle multiple queries. You could begin by asking if the formula has a satisfying assignment (and, using self-reduction, find a solution if one exists). Perhaps you can then use further queries to determine if another solution exists?

Answer 2

1. You say it is coNP-hard. However, it tells nothing about whether it is contained in $\mathsf{coNP}$. It only says that it is at least as hard as any problem in $\mathsf{coNP}$.

This problem is known to be US-hard. $\mathsf{US}$ is not known to equal $\mathsf{coNP}$. It is only known to be a subset of $\mathsf{P^{NP}}$.

2. You can find a solution $p$, add a clause that excludes that solution and see if it's satisfiable.

Let's say the formula $\Phi(x,~y,~z,~w)$ has a positive assignment $x,~\overline y,~\overline z,~w$. Now you just solve $\Phi \land (\overline x\lor y\lor z\lor\overline w)$. If it's satisfiable, your formula is not unique-SAT.