论文标题
击中公式难以解决吗？
Are Hitting Formulas Hard for Resolution?
论文作者
论文摘要
伊瓦玛（Iwama）引入的命中公式是一类不寻常的命题CNF公式。它们的可满足性不仅可以在多项式时间内确定，而且甚至可以以封闭形式计算其模型。这与其他多项式定义的类别形成鲜明对比，这些类别通常具有基于回溯和分辨率的算法，并且模型计数仍然很难，例如2-SAT和HORN-SAT。但是，这些基于分辨率的算法通常很容易地暗示着在分辨率复杂性上的上限，这对于击中公式而缺少。击中公式难以解决吗？ 在本文中，我们采取了第一步，回答了这个问题。我们表明，击中公式的分辨率复杂性由Kullmann和Zhao首先研究的所谓不可约合的击球公式主导，而该公式不能由较小的击球公式组成。但是，根据定义，很难构建大型不可理解的击中公式。甚至还不知道是否存在无限的许多。基于我们的理论结果，我们在Nauty软件包的顶部实施了有效的算法，以列举所有不可约14个条款的不可约束的击中公式。我们还通过将已知的SAT编码用于我们的目的，确定了最多13个子句的生成的击中公式的确切分辨率复杂性。我们的实验结果表明，击中公式确实很难解决。
Hitting formulas, introduced by Iwama, are an unusual class of propositional CNF formulas. Not only is their satisfiability decidable in polynomial time, but even their models can be counted in closed form. This stands in stark contrast with other polynomial-time decidable classes, which usually have algorithms based on backtracking and resolution and for which model counting remains hard, like 2-SAT and Horn-SAT. However, those resolution-based algorithms usually easily imply an upper bound on resolution complexity, which is missing for hitting formulas. Are hitting formulas hard for resolution? In this paper we take the first steps towards answering this question. We show that the resolution complexity of hitting formulas is dominated by so-called irreducible hitting formulas, first studied by Kullmann and Zhao, that cannot be composed of smaller hitting formulas. However, by definition, large irreducible unsatisfiable hitting formulas are difficult to construct; it is not even known whether infinitely many exist. Building upon our theoretical results, we implement an efficient algorithm on top of the Nauty software package to enumerate all irreducible unsatisfiable hitting formulas with up to 14 clauses. We also determine the exact resolution complexity of the generated hitting formulas with up to 13 clauses by extending a known SAT encoding for our purposes. Our experimental results suggest that hitting formulas are indeed hard for resolution.
