Computationally Hard Problems for Logic Programs under Answer Set Semantics

Pub Date : 2024-07-10 DOI:10.1145/3676964
Yuping Shen, Xishun Zhao
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Abstract

Showing that a problem is hard for a model of computation is one of the most challenging tasks in theoretical computer science, logic and mathematics. For example, it remains beyond reach to find an explicit problem that cannot be computed by polynomial size propositional formulas (PF). As a model of computation, logic programs (LP) under answer set semantics are as expressive as PF, and also \(\mathtt{NP}\) -complete for satisfiability checking. In this paper, we show that the PAR problem is hard for LP, i.e., deciding whether a binary string contains an odd number of \(1\) ’s requires exponential size logic programs. The proof idea is first to transform logic programs into equivalent boolean circuits, and then apply a probabilistic method known as random restriction to obtain an exponential lower bound. Based on the main result, we generalize a sufficient condition for identifying hard problems for LP, and give a separation map for a logic program family from a computational point of view, whose members are all equally expressive and share the same reasoning complexity.
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答案集语义下逻辑程序的计算难题
在理论计算机科学、逻辑学和数学领域,证明一个问题对于一个计算模型来说是难题是最具挑战性的任务之一。例如,要找到一个无法用多项式大小的命题公式(PF)计算的明确问题,仍然是遥不可及的。作为一种计算模型,答案集语义下的逻辑程序(LP)与命题公式一样富有表现力,而且在可满足性检查方面也是\(\mathtt{NP}\) -complete 的。在本文中,我们证明了 PAR 问题对于 LP 来说是很难的,也就是说,决定一个二进制字符串是否包含奇数个 \(1\) '需要指数大小的逻辑程序。证明的思路是首先将逻辑程序转化为等价的布尔电路,然后应用一种称为随机限制的概率方法来获得指数级下限。在主要结果的基础上,我们概括了确定 LP 难问题的充分条件,并从计算的角度给出了逻辑程序族的分离图,该逻辑程序族的所有成员都具有相同的表达能力和推理复杂度。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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