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arXiv cs/0107027cs.LOcs.AI
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逻辑程序语义的固定参数复杂性

Fixed-parameter complexity of semantics for logic programs

Zbigniew Lonc, Miroslaw Truszczynski

摘要:

如果一个决策问题的输入是一对字符串,那么这个决策问题被称为参数化问题。其中一个字符串称为参数。问题:给定一个命题逻辑程序P和一个非负整数k,判定P是否有大小不超过k的稳定模型,是以k为参数的参数化决策问题的一个例子。NP完全的参数化问题,如果参数是固定的,通常在多项式时间内就可以求解。决定一个程序P是否有一个大小不超过k的稳定模型的问题,其中k是固定的,不是输入的一部分,可以在时间O(mn^k)内解决,其中m是P的大小,n是P中的原子数。关键问题是是否存在更好的算法(多项式的次数不依赖于k)。为了解决这个问题,我们使用了固定参数复杂度的框架。建立了逻辑程序模型、支持模型和稳定模型的参数化决策问题的固定参数复杂性。我们还建立了固定参数的复杂性,这些问题的变种限制注意霍恩程序和纯负的程序。大多数的问题,考虑在文件中有很高的固定参数的复杂性。因此,固定模型(支持的模型,稳定的模型)的边界不太可能导致快速算法来决定这些模型的存在。

英文摘要:

A decision problem is called parameterized if its input is a pair of strings. One of these strings is referred to as a parameter. The problem: given a propositional logic program P and a non-negative integer k, decide whether P has a stable model of size no more than k, is an example of a parameterized decision problem with k serving as a parameter. Parameterized problems that are NP-complete often become solvable in polynomial time if the parameter is fixed. The problem to decide whether a program P has a stable model of size no more than k, where k is fixed and not a part of input, can be solved in time O(mn^k), where m is the size of P and n is the number of atoms in P. Thus, this problem is in the class P. However, algorithms with the running time given by a polynomial of order k are not satisfactory even for relatively small values of k. The key question then is whether significantly better algorithms (with the degree of the polynomial not dependent on k) exist. To tackle it, we use the framework of fixed-parameter complexity. We establish the fixed-parameter complexity for several parameterized decision problems involving models, supported models and stable models of logic programs. We also establish the fixed-parameter complexity for variants of these problems resulting from restricting attention to Horn programs and to purely negative programs. Most of the problems considered in the paper have high fixed-parameter complexity. Thus, it is unlikely that fixing bounds on models (supported models, stable models) will lead to fast algorithms to decide the existence of such models.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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