[论文解读] Unfounded Sets and Well-Founded Semantics of Answer Set Programs with Aggregates

Mario Alviano, Francesco Calimeri+3 more|arXiv (Cornell University)|Jan 18, 2014

Logic, Reasoning, and Knowledge参考文献 54被引用 22

一句话总结

本文提出了一类针对单调与反单调聚合（LPAma程序）的新型无根基集与稳态语义，推广了经典逻辑编程语义。证明了稳态模型在多项式时间内唯一且可计算，原型系统验证了其相比无聚合编码的显著性能提升；同时表明，一般聚合会导致coNP-完全性，从而合理化了对单调/反单调情形的限制。

ABSTRACT

Logic programs with aggregates (LPA) are one of the major linguistic extensions to Logic Programming (LP). In this work, we propose a generalization of the notions of unfounded set and well-founded semantics for programs with monotone and antimonotone aggregates (LPAma programs). In particular, we present a new notion of unfounded set for LPAma programs, which is a sound generalization of the original definition for standard (aggregate-free) LP. On this basis, we define a well-founded operator for LPAma programs, the fixpoint of which is called well-founded model (or well-founded semantics) for LPAma programs. The most important properties of unfounded sets and the well-founded semantics for standard LP are retained by this generalization, notably existence and uniqueness of the well-founded model, together with a strong relationship to the answer set semantics for LPAma programs. We show that one of the D-well-founded semantics, defined by Pelov, Denecker, and Bruynooghe for a broader class of aggregates using approximating operators, coincides with the well-founded model as defined in this work on LPAma programs. We also discuss some complexity issues, most importantly we give a formal proof of tractable computation of the well-founded model for LPA programs. Moreover, we prove that for general LPA programs, which may contain aggregates that are neither monotone nor antimonotone, deciding satisfaction of aggregate expressions with respect to partial interpretations is coNP-complete. As a consequence, a well-founded semantics for general LPA programs that allows for tractable computation is unlikely to exist, which justifies the restriction on LPAma programs. Finally, we present a prototype system extending DLV, which supports the well-founded semantics for LPAma programs, at the time of writing the only implemented system that does so. Experiments with this prototype show significant computational advantages of aggregate constructs over equivalent aggregate-free encodings.

研究动机与目标

• 将无根基集与稳态语义推广至含聚合的逻辑程序，特别是单调与反单调聚合的情形。
• 确保LPAma程序的稳态模型唯一定义且可在多项式时间内计算。
• 建立新稳态语义与LPAma程序答案集语义之间的强关联。
• 证明对一般聚合的可 tractable 稳态语义在计算上不可行，因其满足聚合的判定为coNP-完全。
• 在原型系统中实现并评估新语义，展示其相比无聚合编码的计算优势。

提出的方法

• 提出LPAma程序的新无根基集定义，推广了无聚合逻辑程序中的标准概念。
• 基于新无根基集概念定义稳态算子，其不动点即为稳态模型。
• 证明LPAma程序的稳态模型存在且唯一，保持了标准逻辑编程中的关键性质。
• 建立所提稳态语义与Pelov等人提出的D-稳态语义在更广类聚合上的正式对应关系。
• 提供LPAma程序稳态模型可多项式时间计算的正式证明。
• 开发并评估一个扩展DLV的原型系统，以支持新语义，使用真实世界与合成基准进行测试。

实验结果

研究问题

• RQ1 能否将无根基集与稳态语义有意义地推广至含单调与反单调聚合的逻辑程序？
• RQ2 LPAma程序的稳态模型是否唯一定义且可在多项式时间内计算？
• RQ3 所提稳态语义与现有语义（如D-稳态语义）之间有何关系？
• RQ4 能否在多项式时间内计算一般聚合的稳态语义？
• RQ5 使用聚合构造相比等价的无聚合编码在实际中是否具有性能优势？

主要发现

• LPAma程序的稳态模型唯一定义且可在多项式时间内计算，确保了语义的可 tractable 性。
• 所提稳态语义在LPAma程序上与Pelov等人提出的D-稳态语义一致，验证了其正确性。
• 在部分解释中判定一般LPA程序的聚合满足性为coNP-完全，表明对一般聚合实现可 tractable 稳态语义的可能性极低。
• 扩展DLV的原型系统成功实现了新语义，并在实验中显示出相比无聚合编码的显著性能提升。
• 使用聚合可带来更简洁高效的编码，在实践中表现出可测量的计算优势。