It is known that the following problem is complete in $\Sigma_2^p$:

$\Sigma_{2}SAT$ : Given a quantified boolean formula $\theta = \exists x_1,...,x_l\forall y_1,...,y_m\psi$, where $\psi$ is a boolean propositional formula over the variables $x_1,...,x_l,y_1,...,y_m$ , is $\theta$ valid?

Is it still complete when it is assumed that $\psi$ is in CNF?

It is mentioned in "Computational Complexity: A Modern Approach" by Sanjeev Arora and Boaz Barak that it can be assumed, but no proof is given: https://www.iith.ac.in/~subruk/4510/phchap_arora.pdf (Example 5.9)


1 Answer 1


No. Since universal quantifiers commute with conjunctions, it is easy to see that $\Sigma_2$-SAT with $\psi$ CNF is in NP. If it's really written like this in the book, it's an error.

However, the problem is $\Sigma_2$-complete for $\psi$ a 3-DNF.

  • 1
    $\begingroup$ In the book (OP has a link to its draft) the error is in Example 5.6. $\endgroup$ Commented Oct 25, 2021 at 17:51
  • $\begingroup$ Thank you. Do you know a reference for completeness when assuming $\phi$ is in 3-DNF? $\endgroup$ Commented Oct 26, 2021 at 9:04
  • 5
    $\begingroup$ @Naama Shamash Hal: Celia Wrathall ("Complete sets and the polynomial hierarchy", Theoret. Comp. Sci., 3:23-33, 1976) proves completness for DNF-formulas in Corollary 6 on page 32; the step from DNF to 3-DNF is straightforward. $\endgroup$
    – Gamow
    Commented Oct 26, 2021 at 11:19

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