[论文解读] Answer Sets for Logic Programs with Arbitrary Abstract Constraint Atoms

Enrico Pontelli, Tran Cao Son+1 more|arXiv (Cornell University)|Oct 10, 2011

Logic, Reasoning, and Knowledge参考文献 28被引用 57

一句话总结

本文为带有任意抽象约束原子（c-atom）的逻辑程序引入了四种等价的答案集语义，将稳定模型语义推广至具有复杂约束的非单调程序。提出了两种基于不动点的方法——通过约化的答案集与通过补集的答案集，以及两种基于良好支持模型的定义——强良好支持模型与弱良好支持模型，证明了它们的等价性，并在保持良好支持模型对应关系等关键性质的同时，将经典答案集语义扩展至c-atom。

ABSTRACT

In this paper, we present two alternative approaches to defining answer sets for logic programs with arbitrary types of abstract constraint atoms (c-atoms). These approaches generalize the fixpoint-based and the level mapping based answer set semantics of normal logic programs to the case of logic programs with arbitrary types of c-atoms. The results are four different answer set definitions which are equivalent when applied to normal logic programs. The standard fixpoint-based semantics of logic programs is generalized in two directions, called answer set by reduct and answer set by complement. These definitions, which differ from each other in the treatment of negation-as-failure (naf) atoms, make use of an immediate consequence operator to perform answer set checking, whose definition relies on the notion of conditional satisfaction of c-atoms w.r.t. a pair of interpretations. The other two definitions, called strongly and weakly well-supported models, are generalizations of the notion of well-supported models of normal logic programs to the case of programs with c-atoms. As for the case of fixpoint-based semantics, the difference between these two definitions is rooted in the treatment of naf atoms. We prove that answer sets by reduct (resp. by complement) are equivalent to weakly (resp. strongly) well-supported models of a program, thus generalizing the theorem on the correspondence between stable models and well-supported models of a normal logic program to the class of programs with c-atoms. We show that the newly defined semantics coincide with previously introduced semantics for logic programs with monotone c-atoms, and they extend the original answer set semantics of normal logic programs. We also study some properties of answer sets of programs with c-atoms, and relate our definitions to several semantics for logic programs with aggregates presented in the literature.

研究动机与目标

• 为扩展了任意抽象约束原子（c-atom）的逻辑程序建立统一且通用的答案集语义框架，涵盖聚集、权重重构和基数约束。
• 解决先前c-atom语义的局限性，特别是Marek和Remmel原始方法产生的非直观答案集。
• 将基于不动点和基于良好支持模型的答案集定义，从普通逻辑程序推广至带有任意c-atom的程序。
• 确保新语义与单调c-atom的现有语义一致，并将经典答案集语义推广至普通逻辑程序。
• 为构建块结果（如分裂定理）建立理论基础，适用于逻辑编程的各种扩展。

提出的方法

• 引入一种基于解释对 (I, M) 的c-atom条件满足的即时后果算子，支持答案集的迭代计算。
• 定义两种基于不动点的语义：通过约化的答案集与通过补集的答案集，二者在处理否定即失败（naf）原子时有所不同。
• 提出两种基于良好支持模型的语义：强良好支持模型与弱良好支持模型，将良好支持模型的概念推广至c-atom。
• 使用相对于解释 I 和 M 的c-atom条件满足概念，其中 I 表示计算的当前阶段，M 表示候选答案集。
• 证明通过约化的答案集对应于弱良好支持模型，通过补集的答案集对应于强良好支持模型。
• 建立四种语义在普通逻辑程序上的等价性，并与单调和凸c-atom的先前语义保持一致。

实验结果

研究问题

• RQ1 能否为带有任意抽象约束原子的逻辑程序定义统一的答案集语义，以推广经典答案集语义？
• RQ2 在任意c-atom的背景下，如何处理否定即失败，以保持最小性与良好支持性等理想性质？
• RQ3 所提出的c-atom语义是否保留了诸如稳定模型与良好支持模型之间对应关系等关键结果？
• RQ4 新语义是否与先前为单调和凸c-atom建立的语义一致？
• RQ5 新语义能否与现有聚集和权重重构的语义相关联并加以涵盖？

主要发现

• 所提出的四种答案集语义——通过约化的答案集、通过补集的答案集、强良好支持模型与弱良好支持模型——在应用于普通逻辑程序时是等价的。
• 通过约化的答案集等价于弱良好支持模型，通过补集的答案集等价于强良好支持模型，推广了经典稳定模型与良好支持模型之间的对应关系。
• 所提出的语义扩展了普通逻辑程序的原始答案集语义，并与单调c-atom的先前语义一致。
• 该语义与FLP及Ferraris对单调聚集的语义一致，通过与FLP基于约化的答案集定义的等价性得到证明。
• 基于c-atom条件满足的即时后果算子，实现了对任意c-atom的正确且统一的答案集检查。
• 该框架支持对聚集和约束的统一处理，为未来在所有逻辑编程扩展中应用构建块结果（如分裂定理）提供了可能。