In this paper, we analyze Boolean formulas in conjunctive normal form (CNF) from the perspective of read-once resolution (ROR) refutation schemes. A read-once (resolution) refutation is one in which each clause is used at most once. Derived clauses can be used as many times as they are deduced. However, clauses in the original formula can only be used as part of one derivation. It is well known that ROR is not complete; that is, there exist unsatisfiable formulas for which no ROR exists. Likewise, the problem of checking if a 3CNF formula has a read-once refutation is NP-complete. This paper is concerned with a variant of satisfiability called not-all-equal satisfiability (NAE-satisfiability). A CNF formula is NAE-satisfiable if it has a satisfying assignment in which at least one literal in each clause is set to false. It is well known that the problem of checking NAE-satisfiability is NP-complete. Clearly, the class of CNF formulas which are NAE-satisfiable is a proper subset of satisfiable CNF formulas. It follows that traditional resolution cannot always find a proof of NAE-unsatisfiability. Thus, traditional resolution is not a sound procedure for checking NAE-satisfiability. In this paper, we introduce a variant of resolution called NAE-resolution which is a sound and complete procedure for checking NAE-satisfiability in CNF formulas. The focus of this paper is on a variant of NAE-resolution called read-once NAE-resolution in which each clause (input or derived) can be part of at most one NAE-resolution step. Our principal result is that read-once NAE-resolution is a sound and complete procedure for 2CNF formulas. Furthermore, we provide an algorithm to determine the smallest such NAE-resolution in polynomial time. This is in stark contrast to the corresponding problem concerning 2CNF formulas and ROR refutations. We also show that the problem of checking whether a 3CNF formula has a read-once NAE-resolution is NP-complete.

中文翻译：

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NAE 分辨率：一种新的分辨率反驳技术，用于证明不完全相等的不可满足性

在本文中，我们从一次读取解析（ROR）反驳方案的角度分析了合取范式（CNF）中的布尔公式。一次读（解析）反驳是每个子句最多使用一次的反驳。派生从句可以被多次使用。但是，原始公式中的子句只能用作一个推导的一部分。众所周知，ROR 并不完整；也就是说，存在不存在 ROR 的不可满足的公式。同样，检查 3CNF 公式是否具有只读反驳的问题是NP完全. 本文关注可满足性的一种变体，称为非均等可满足性（NAE-satisfiability）。CNF 公式是 NAE 可满足的，如果它有一个令人满意的赋值，其中每个子句中至少有一个文字被设置为错误的. 众所周知，检查 NAE 可满足性的问题是NP完全. 显然，可满足 NAE 的 CNF 公式类是可满足 CNF 公式的真子集。由此可见，传统的解决方法并不总能找到 NAE 不可满足性的证明。因此，传统的分辨率不是检查 NAE 可满足性的合理程序。在本文中，我们介绍了一种称为 NAE 分辨率的分辨率变体，它是一种用于检查 CNF 公式中 NAE 可满足性的完善且完整的程序。本文的重点是 NAE 解析的一种变体，称为一次性 NAE 解析，其中每个子句（输入或派生）最多可以是一个 NAE 解析步骤的一部分。我们的主要结果是，一次性 NAE 解析是 2CNF 公式的合理且完整的过程。此外，我们提供了一种算法来确定多项式时间内的最小此类 NAE 分辨率。这与有关 2CNF 公式和 ROR 反驳的相应问题形成鲜明对比。我们还表明，检查 3CNF 公式是否具有只读 NAE 分辨率的问题是NP完全.