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Given a set of (propositional) formulae $\Phi$, two formulae $\phi$ and $\xi$, determine whether there exists $\Psi\subseteq \Phi$ such that $\Psi\models \phi$ and $\Psi\not\models \xi$.

Question: what is the (theoretical) complexity of this problem? Is it in DP?

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    $\begingroup$ can you describe the notation in words? what is DP? $\endgroup$
    – vzn
    Commented Jul 21, 2015 at 22:09
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    $\begingroup$ Crossposted on cstheory.SE $\endgroup$
    – chazisop
    Commented Jul 22, 2015 at 9:15
  • $\begingroup$ I found the definition of DP: cse.buffalo.edu/~regan/papers/pdf/ALRch29.pdf and search for "is the class of languages A". DP is the class of languages A such that A = A1 ∩ A2 for some languages A1 in NP and A2 in co-NP. $\endgroup$ Commented Jul 22, 2015 at 9:41
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    $\begingroup$ @D.W., they are sets so is there a reason not to read it simply as "subset"? Also you can find the definition of DP on complexity zoo. $\endgroup$
    – Kaveh
    Commented Jul 24, 2015 at 6:27
  • $\begingroup$ @Kaveh, you're right, I mis-read. Thank you! $\endgroup$
    – D.W.
    Commented Jul 24, 2015 at 16:56

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Given a propositional formula $\varphi(\vec{x}, \vec{y})$, consider $\psi = \exists \vec{y} \ \forall \vec{x} \ \varphi(\vec{x}, \vec{y})$.

Let $\Delta = \{ y_0 \leftrightarrow \top, y_0 \leftrightarrow \bot, \ldots, y_m \leftrightarrow \top, y_m \leftrightarrow \bot \}$. Then $\psi$ is true iff there exists $\Gamma \subseteq \Delta$ such that $\Gamma \nvDash \bot$ and $\Gamma \vDash \varphi(\vec{x}, \vec{y})$.

Therefore the problem is $\Sigma^\mathsf{P}_2$-hard. $\mathsf{DP} \subseteq \Delta^\mathsf{P}_2$, so unless $\mathsf{PH}$ collapses the problem is not in $\mathsf{DP}$.

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