将3- SAT和不完全相等的3- SAT的量化变体置于多项式层级中
Placing quantified variants of 3-SAT and Not-All-Equal 3-SAT in the polynomial hierarchy
摘要：
3-SAT和不全相等3-SAT的变体的复杂性得到了很好的研究。然而，相比之下，很少有人知道的问题的量化对应的复杂性。在本文的第一部分中，我们证明了$\forall \exists$ 3-SAT是$Π_2^P$ -完备的，即使（1）每个变量恰好出现两次非否定和两次否定，（2）每个子句恰好是三个不同变量的析取，（3）泛变量的数目等于存在变量的数目。此外，我们还证明了，如果（1a）每个泛变量恰好出现一次未求反和恰好出现一次求反，（1b）每个存在变量恰好出现两次未求反和恰好出现两次求反，以及（2）和（3）保持不变，则该问题仍然是$Π_2^P$ -完全的。另一方面，这个问题对于某些变量是NP完全的，其中每个泛变量只出现一次。在本文的第二部分中，我们建立了$\forall \exists$不全相等3-SAT的$Π_2^P$ -完备性，即使（1 '）布尔公式是线性单调的，（2'）每个泛变量只出现一次，每个存在变量只出现三次，（3 '）每个子句是三个不同变量的析取，其中最多包含一个泛变量。在积极的一面，我们发现的变体$\forall \exists$不全相等3-SAT的co-NP-完全或可解的多项式时间。
英文摘要：
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 $Π_2^P$-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 $Π_2^P$-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 $Π_2^P$-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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
