嵌套限制和嵌套异常理论的复杂性
Complexity of Nested Circumscription and Nested Abnormality Theories
摘要：
需要一个限制的形式主义，允许简单而优雅的模块化问题表示，导致Lifschitz（AIJ，1995）介绍嵌套异常理论（NAT）作为一种工具，模块化知识表示，适用于应用限制，以尽量减少异常情况。抽象从这个特定的目标，我们提出了L_{CIRC}，这是一个扩展的通用命题限制，允许命题组合和嵌套的限制理论。如图所示，NAT自然嵌入到这种语言中，并且实际上具有相同的表达能力。然后，我们分析了L_{CIRC}和NAT的复杂性，特别是嵌套的影响。后者被发现是复杂性的来源，它爬上多项式层次的嵌套深度的增加，并达到PSPACE-完全在一般情况下。我们还确定有意义的NAT的语法片段，具有较低的复杂性。特别是，我们证明了在NAT框架中的Horn限制的泛化仍然是CONP-完全的，并且没有固定字母的Horn NAT可以有效地转化为等价的Horn CNF，这意味着主要推理任务的多项式可解性。最后，我们还研究了NAT的扩展，并简要地解决了一阶情况下的复杂性。我们的研究结果提供了使用L_{CIRC}的“成本”的见解（分别为：NATs）作为表达其他形式主义（如行动理论、叙事或空间理论）的宿主语言。
英文摘要：
The need for a circumscriptive formalism that allows for simple yet elegant modular problem representation has led Lifschitz (AIJ, 1995) to introduce nested abnormality theories (NATs) as a tool for modular knowledge representation, tailored for applying circumscription to minimize exceptional circumstances. Abstracting from this particular objective, we propose L_{CIRC}, which is an extension of generic propositional circumscription by allowing propositional combinations and nesting of circumscriptive theories. As shown, NATs are naturally embedded into this language, and are in fact of equal expressive capability. We then analyze the complexity of L_{CIRC} and NATs, and in particular the effect of nesting. The latter is found to be a source of complexity, which climbs the Polynomial Hierarchy as the nesting depth increases and reaches PSPACE-completeness in the general case. We also identify meaningful syntactic fragments of NATs which have lower complexity. In particular, we show that the generalization of Horn circumscription in the NAT framework remains CONP-complete, and that Horn NATs without fixed letters can be efficiently transformed into an equivalent Horn CNF, which implies polynomial solvability of principal reasoning tasks. Finally, we also study extensions of NATs and briefly address the complexity in the first-order case. Our results give insight into the ``cost'' of using L_{CIRC} (resp. NATs) as a host language for expressing other formalisms such as action theories, narratives, or spatial theories.
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