[论文解读] Elementary Sets for Logic Programs

Martin Gebser, Joohyung Lee+1 more|arXiv (Cornell University)|Jul 15, 2023

Logic, Reasoning, and Knowledge参考文献 11被引用 20

一句话总结

本文将初等循环的概念简化并扩展为 elementary sets，表明它们刻画 nondisjunctive programs 的稳定性并扩展到 disjunctive programs，给出图论表征和复杂性结果。

ABSTRACT

By introducing the concepts of a loop and a loop formula, Lin and Zhao showed that the answer sets of a nondisjunctive logic program are exactly the models of its Clark's completion that satisfy the loop formulas of all loops. Recently, Gebser and Schaub showed that the Lin-Zhao theorem remains correct even if we restrict loop formulas to a special class of loops called ``elementary loops.'' In this paper, we simplify and generalize the notion of an elementary loop, and clarify its role. We propose the notion of an elementary set, which is almost equivalent to the notion of an elementary loop for nondisjunctive programs, but is simpler, and, unlike elementary loops, can be extended to disjunctive programs without producing unintuitive results. We show that the maximal unfounded elementary sets for the ``relevant'' part of a program are exactly the minimal sets among the nonempty unfounded sets. We also present a graph-theoretic characterization of elementary sets for nondisjunctive programs, which is simpler than the one proposed in (Gebser & Schaub 2005). Unlike the case of nondisjunctive programs, we show that the problem of deciding an elementary set is coNP-complete for disjunctive programs.

研究动机与目标

• 澄清 elementary loops 的作用并引入一个更简单、通用的概念，称为 elementary sets。
• 将 Lin–Zhao 框架扩展到 disjunctive programs，避免直观上不合理的结果。
• 为 nondisjunctive programs 提供一个图论表征，称为 elementary sets。
• 分析 deciding elementariness 在 nondisjunctive 与 disjunctive programs 的计算复杂性。
• 通过聚焦 elementarily unfounded sets，凸显对 SAT-based answer set solvers 的潜在算法收益。

提出的方法

• 定义并在 nondisjunctive 与 disjunctive programs 的层面，将 loops、loop formulas 与 stability 通过 LF 联系起来。
• 引入 elementary sets 作为 nondisjunctive programs 的更简单对应项，并扩展至 disjunctive programs。
• 证明 maximal unfounded elementary sets 与 minimal nonempty unfounded sets 等价。
• 提供一个称为 elementary subgraph 的图论构造，并证明其强连通性表征 nondisjunctive programs 的 elementary sets。
• 显示对 disjunctive programs 的 elementariness 判定为 coNP 完全，对 head-cycle-free disjunctive programs 则可处理。
• 演示在某些稳定性重新表述中只需要 elementary sets 的 loop formulas，从而增强先前定理。

实验结果

研究问题

• RQ1 nondisjunctive programs 中 elementary sets 与 elementary loops 的精确关系是什么？
• RQ2 如何在不产生意外结果的情况下将 elementary sets 扩展到 disjunctive programs？
• RQ3 disjunctive 与 nondisjunctive programs 的 elementariness 判定的计算复杂性如何？
• RQ4 是否存在一种基于图的表征（elementary subgraph）能高效识别 nondisjunctive programs 中的 elementary sets？
• RQ5 elementary unfounded sets 如何提升基于 SAT 的 answer set 求解？

主要发现

• Elementary sets 为 nondisjunctive programs 提供了比 elementary loops 更简单但几乎等价的概念。
• 最大 unfounded elementary sets 与最小非空 unfounded sets 同构。
• 通过 elementary subgraph，可以在 nondisjunctive programs 中对 elementary sets 进行多项式时间的基于图的检查。
• 对于 disjunctive programs，elementary sets 的判定是 coNP-完全的，但对 head-cycle-free disjunctive programs 可行。
• 使用 elementary unfounded sets 可以减少 SAT-based 求解器中需要的 loop formulas 的数量。
• elementary sets 自然推广到 disjunctive programs，保留了期望的性质，并避免了与 GS-elementary loops 相比的非直观结果。