Synchronizing words and monoid factorization, yielding a new parameterized complexity class?

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2022-02-01 DOI:10.1017/S0960129522000184
H. Fernau, Jens Bruchertseifer
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引用次数: 1

Abstract

Abstract The concept of a synchronizing word is a very important notion in the theory of finite automata. We consider the associated decision problem to decide if a given DFA possesses a synchronizing word of length at most k, where k is the standard parameter. We show that this problem DFA-SW is equivalent to the problem Monoid Factorization introduced by Cai, Chen, Downey, and Fellows. Apart from the known $\textsf{W}[2]$ -hardness results, we show that these problems belong to $\textsf{A}[2]$ , $\textsf{W}[\textsf{P}],$ and $\textsf{WNL}$ . This indicates that DFA-SW is not complete for any of these classes, and hence, we suggest a new parameterized complexity class $\textsf{W}[\textsf{Sync}]$ as a proper home for these (and more) problems. We present quite a number of problems that belong to $\textsf{W}[\textsf{Sync}]$ or are hard or complete for this new class.
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同步单词和monoid因子分解,产生一个新的参数化复杂度类?
摘要同步字的概念是有限自动机理论中一个非常重要的概念。我们考虑相关的决策问题来决定给定的DFA是否拥有长度最多为k的同步字,其中k是标准参数。我们证明了这个问题DFA-SW等价于蔡、陈、唐尼和费罗斯提出的单调因子分解问题。除了已知的$\textsf{W}[2]$硬度结果外,我们还证明了这些问题属于$\textsf{A}[2]$、$\textsf{W}[\textsf{P}]、$和$\textsf{WNL}$。这表明DFA-SW对于这些类中的任何一个都不完整,因此,我们建议使用一个新的参数化复杂性类$\textsf{W}[\textsf{Sync}]$作为这些(以及更多)问题的合适归宿。我们提出了相当多的问题,这些问题属于$\textsf{W}[\textsf{Sync}]$,或者对于这个新类来说是困难的或完全的。
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来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
期刊介绍: Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.
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