Computationally Hard Problems for Logic Programs under Answer Set Semantics

IF 0.7 4区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

ACM Transactions on Computational Logic Pub Date : 2024-07-10 DOI:10.1145/3676964

Yuping Shen, Xishun Zhao

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引用次数: 0

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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来源期刊

ACM Transactions on Computational Logic 工程技术-计算机：理论方法

CiteScore

2.30

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： TOCL welcomes submissions related to all aspects of logic as it pertains to topics in computer science. This area has a great tradition in computer science. Several researchers who earned the ACM Turing award have also contributed to this field, namely Edgar Codd (relational database systems), Stephen Cook (complexity of logical theories), Edsger W. Dijkstra, Robert W. Floyd, Tony Hoare, Amir Pnueli, Dana Scott, Edmond M. Clarke, Allen E. Emerson, and Joseph Sifakis (program logics, program derivation and verification, programming languages semantics), Robin Milner (interactive theorem proving, concurrency calculi, and functional programming), and John McCarthy (functional programming and logics in AI). Logic continues to play an important role in computer science and has permeated several of its areas, including artificial intelligence, computational complexity, database systems, and programming languages. The Editorial Board of this journal seeks and hopes to attract high-quality submissions in all the above-mentioned areas of computational logic so that TOCL becomes the standard reference in the field. Both theoretical and applied papers are sought. Submissions showing novel use of logic in computer science are especially welcome.