Lets imagine we have a satisfiable formula $F(A_0, A_1,...A_k,S_0,...,S_n)$ The problem to solve is "Is there an assignment for variables $(S_0,...,S_n)$ which will make F unsatisfiable?". One way of solving is to find all solutions for F in terms of variables $S_0,...,S_n$ and if the count is < $2^n$, the missing solution will be the answer, but the complexity of this algorithm is huge, if the number of such assignments is small.

My questions are:

  • Is there a way to solve the problem with less SAT solver calls?
  • Is it a well-known problem in theory (What I should google to read about it)?
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    $\begingroup$ "which will make F unsatisfiable" -- that does not make sense. Do you simply mean "does not satisfy F"? Then you are talking about the problem TAUTOLOGY (resp. it's complement). $\endgroup$ – Raphael May 29 '15 at 15:26
  • $\begingroup$ Ignoring the fact that the question doesn't make sense, I think trying to find a solution to $\neg F(A_0,A_1,\ldots, A_k,S_0,\ldots, S_n)$ might be what you are looking for. $\endgroup$ – Dave Clarke May 29 '15 at 15:34
  • $\begingroup$ Maybe I wasn't clear. After applying assignments for $(S_0,...,S_n)$ we will have another formula $G(A_0 ,..., A_k)$ and this must be unsatisfiable. $\endgroup$ – Grigor Aghanyan May 29 '15 at 15:36

Your problem is the canonical $\Sigma_2^P$-complete problem: $$ \exists \vec{S} \forall \vec{A} \lnot F(\vec{A},\vec{S}). $$ As such, it is thought to be more difficult than SAT (which is $\Sigma_1^P$). Solving it with a few SAT-oracle calls is akin to solving SAT itself efficiently (the P vs. NP question), though it could be that $\Sigma_2^P = \Sigma_1^P$ while $P \neq NP$, so in some sense there is more hope for your problem than for SAT itself.

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  • $\begingroup$ Exactly. Thank you for the answer. So the solution with $2^n$ solver calls is "not a bad solution" for it? Some link for papers about this problem will help me lot. $\endgroup$ – Grigor Aghanyan May 29 '15 at 15:58
  • $\begingroup$ Practically speaking there could be heuristics that work well for some problems, but I'm unaware of any. The polynomial hierarchy (which contains $\Sigma_2^P$) should be covered in any textbook on computational complexity. $\endgroup$ – Yuval Filmus May 29 '15 at 16:00

This is a well-known problem: it is the 2QBF problem. Unfortunately, it's significantly harder than SAT. There are QBF solvers available. You could try finding a QBF solver (or, even better yet, a 2QBF solver) and seeing if it can solve your formula. However, QBF solvers don't scale as well as SAT solvers; QBF is significantly harder than SAT.

See https://cstheory.stackexchange.com/q/11022/5038 and http://www.qbflib.org/ for some resources that might be helpful.

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