In this paper, we discuss algorithms for the problem of finding read-once resolution refutations of unsatisfiable 2CNF formulas within the resolution refutation system. Broadly, a read-once resolution refutation is one in which each constraint (input or derived) is used at most once. Read-once resolution refutations have been widely studied in the literature for a number of constraint system-refutation system pairs. For instance, read-once resolution has been analyzed for boolean formulas in conjunctive normal form (CNF) and read-once cutting planes have been analyzed for polyhedral systems. By definition, read-once refutations are compact, and hence valuable in applications that place a premium on visualization. The satisfiability problem (SAT) is concerned with finding a satisfying assignment for a boolean formula in CNF. While SAT is NP-complete in general, there exist some interesting subclasses of CNF formulas, for which it is decidable in polynomial time. One such subclass is the class of 2CNF formulas, i.e., CNF formulas in which each clause has at most two literals. The existence of efficient algorithms for satisfiability checking in 2CNF formulas (2SAT), makes this class useful from the perspective of modeling selected logic programs. The work in this paper is concerned with the read-once refutability problem (under resolution) in this subclass. Although 2SAT is decidable in polynomial time, the problem of finding a read-once resolution refutation of an unsatisfiable 2CNF formula is NP-complete. We design non-trivial, parameterized and exact exponential algorithms for this problem. Additionally, we study the computational complexity of finding a shortest read-once resolution refutation of a 2CNF formula.

在本文中，我们讨论了在解析反驳系统中寻找不可满足的 2CNF 公式的一次性解析反驳问题的算法。从广义上讲，一次性解析反驳是其中每个约束（输入或派生）最多使用一次的反驳。已在文献中针对许多约束系统-反驳系统对广泛研究了一次性解析反驳。例如，已经分析了合取范式 (CNF) 中布尔公式的一次性分辨率，并分析了多面体系统的一次性切割平面。根据定义，一次性反驳是紧凑的，因此在重视可视化的应用程序中很有价值。可满足性问题 (SAT) 与为 CNF 中的布尔公式找到令人满意的分配有关。虽然 SAT 是一般来说，NP-complete存在一些有趣的 CNF 公式子类，对于这些子类，它在多项式时间内是可判定的。一个这样的子类是 2CNF 公式类，即每个子句最多有两个文字的 CNF 公式。2CNF 公式 (2SAT) 中存在有效的可满足性检查算法，从建模所选逻辑程序的角度来看，该类很有用。本文的工作涉及该子类中的只读可反驳性问题（未解决）。虽然 2SAT 在多项式时间内是可判定的，但找到一个不可满足的 2CNF 公式的 read-once 解决反驳的问题是NP 完全的. 我们为这个问题设计了非平凡的、参数化的和精确的指数算法。此外，我们研究了寻找 2CNF 公式的最短读取一次分辨率反驳的计算复杂性。