{"title":"Synchronizing words and monoid factorization, yielding a new parameterized complexity class?","authors":"H. Fernau, Jens Bruchertseifer","doi":"10.1017/S0960129522000184","DOIUrl":null,"url":null,"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 \n$\\textsf{W}[2]$\n -hardness results, we show that these problems belong to \n$\\textsf{A}[2]$\n , \n$\\textsf{W}[\\textsf{P}],$\n and \n$\\textsf{WNL}$\n . This indicates that DFA-SW is not complete for any of these classes, and hence, we suggest a new parameterized complexity class \n$\\textsf{W}[\\textsf{Sync}]$\n as a proper home for these (and more) problems. We present quite a number of problems that belong to \n$\\textsf{W}[\\textsf{Sync}]$\n or are hard or complete for this new class.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"189 - 215"},"PeriodicalIF":0.4000,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Structures in Computer Science","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1017/S0960129522000184","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
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.
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.