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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)?
Grigor Aghanyan
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2 Answers2

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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.

Yuval Filmus
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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.

Community
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D.W.
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