# Placing quantified variants of 3-SAT and Not-All-Equal 3-SAT in the polynomial hierarchy

## TL;DR

This paper explores the complexity of quantified variants of 3-SAT and Not-All-Equal 3-SAT, establishing their placement in the polynomial hierarchy under various constraints, and identifying cases that are NP-complete, co-NP-complete, or polynomial-time solvable.

## Contribution

It provides new complexity classifications for quantified 3-SAT and NAE-3-SAT variants, including tight bounds and conditions for their hardness or tractability.

## Key findings

- -completeness of ^P for certain quantified 3-SAT variants
- NP-completeness for variants with universal variables appearing once
- co-NP-complete or polynomial-time solvable variants of quantified NAE-3-SAT

## Abstract

The complexity of variants of 3-SAT and Not-All-Equal 3-SAT is well studied. However, in contrast, very little is known about the complexity of the problems' quantified counterparts. In the first part of this paper, we show that $\forall \exists$ 3-SAT is $\Pi_2^P$-complete even if (1) each variable appears exactly twice unnegated and exactly twice negated, (2) each clause is a disjunction of exactly three distinct variables, and (3) the number of universal variables is equal to the number of existential variables. Furthermore, we show that the problem remains $\Pi_2^P$-complete if (1a) each universal variable appears exactly once unnegated and exactly once negated, (1b) each existential variable appears exactly twice unnegated and exactly twice negated, and (2) and (3) remain unchanged. On the other hand, the problem becomes NP-complete for certain variants in which each universal variable appears exactly once. In the second part of the paper, we establish $\Pi_2^P$-completeness for $\forall \exists$ Not-All-Equal 3-SAT even if (1') the Boolean formula is linear and monotone, (2') each universal variable appears exactly once and each existential variable appears exactly three times, and (3') each clause is a disjunction of exactly three distinct variables that contains at most one universal variable. On the positive side, we uncover variants of $\forall \exists$ Not-All-Equal 3-SAT that are co-NP-complete or solvable in polynomial time.