Operational complexity and pumping lemmas

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS Acta Informatica Pub Date : 2022-07-20 DOI:10.1007/s00236-022-00431-3
Jürgen Dassow, Ismaël Jecker
{"title":"Operational complexity and pumping lemmas","authors":"Jürgen Dassow,&nbsp;Ismaël Jecker","doi":"10.1007/s00236-022-00431-3","DOIUrl":null,"url":null,"abstract":"<div><p>The well-known pumping lemma for regular languages states that, for any regular language <i>L</i>, there is a constant <i>p</i> (depending on <i>L</i>) such that the following holds: If <span>\\(w\\in L\\)</span> and <span>\\(\\vert w\\vert \\ge p\\)</span>, then there are words <span>\\(x\\in V^{*}\\)</span>, <span>\\(y\\in V^+\\)</span>, and <span>\\(z\\in V^{*}\\)</span> such that <span>\\(w=xyz\\)</span> and <span>\\(xy^tz\\in L\\)</span> for <span>\\(t\\ge 0\\)</span>. The minimal pumping constant <span>\\({{{\\,\\mathrm{mpc}\\,}}(L)}\\)</span> of <i>L</i> is the minimal number <i>p</i> for which the conditions of the pumping lemma are satisfied. We investigate the behaviour of <span>\\({{{\\,\\mathrm{mpc}\\,}}}\\)</span> with respect to operations, i. e., for an <i>n</i>-ary regularity preserving operation <span>\\(\\circ \\)</span>, we study the set <span>\\({g_{\\circ }^{{{\\,\\mathrm{mpc}\\,}}}(k_1,k_2,\\ldots ,k_n)}\\)</span> of all numbers <i>k</i> such that there are regular languages <span>\\(L_1,L_2,\\ldots ,L_n\\)</span> with <span>\\({{{\\,\\mathrm{mpc}\\,}}(L_i)=k_i}\\)</span> for <span>\\(1\\le i\\le n\\)</span> and <span>\\({{{\\,\\mathrm{mpc}\\,}}(\\circ (L_1,L_2,\\ldots ,L_n)=~k}\\)</span>. With respect to Kleene closure, complement, reversal, prefix and suffix-closure, circular shift, union, intersection, set-subtraction, symmetric difference,and concatenation, we determine <span>\\({g_{\\circ }^{{{\\,\\mathrm{mpc}\\,}}}(k_1,k_2,\\ldots ,k_n)}\\)</span> completely. Furthermore, we give some results with respect to the minimal pumping length where, in addition, <span>\\(\\vert xy\\vert \\le p\\)</span> has to hold.</p></div>","PeriodicalId":7189,"journal":{"name":"Acta Informatica","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2022-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00236-022-00431-3.pdf","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Informatica","FirstCategoryId":"94","ListUrlMain":"https://link.springer.com/article/10.1007/s00236-022-00431-3","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INFORMATION SYSTEMS","Score":null,"Total":0}
引用次数: 1

Abstract

The well-known pumping lemma for regular languages states that, for any regular language L, there is a constant p (depending on L) such that the following holds: If \(w\in L\) and \(\vert w\vert \ge p\), then there are words \(x\in V^{*}\), \(y\in V^+\), and \(z\in V^{*}\) such that \(w=xyz\) and \(xy^tz\in L\) for \(t\ge 0\). The minimal pumping constant \({{{\,\mathrm{mpc}\,}}(L)}\) of L is the minimal number p for which the conditions of the pumping lemma are satisfied. We investigate the behaviour of \({{{\,\mathrm{mpc}\,}}}\) with respect to operations, i. e., for an n-ary regularity preserving operation \(\circ \), we study the set \({g_{\circ }^{{{\,\mathrm{mpc}\,}}}(k_1,k_2,\ldots ,k_n)}\) of all numbers k such that there are regular languages \(L_1,L_2,\ldots ,L_n\) with \({{{\,\mathrm{mpc}\,}}(L_i)=k_i}\) for \(1\le i\le n\) and \({{{\,\mathrm{mpc}\,}}(\circ (L_1,L_2,\ldots ,L_n)=~k}\). With respect to Kleene closure, complement, reversal, prefix and suffix-closure, circular shift, union, intersection, set-subtraction, symmetric difference,and concatenation, we determine \({g_{\circ }^{{{\,\mathrm{mpc}\,}}}(k_1,k_2,\ldots ,k_n)}\) completely. Furthermore, we give some results with respect to the minimal pumping length where, in addition, \(\vert xy\vert \le p\) has to hold.

查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
操作复杂性和泵浦引理
众所周知的正则语言抽吸引理指出,对于任何正则语言L,都存在一个常数p(取决于L),使得以下条件成立:如果\(w\in L\)和\(\vert w\vert \ge p\),则存在单词\(x\in V^{*}\)、\(y\in V^+\)和\(z\in V^{*}\),使得\(t\ge 0\)对应\(w=xyz\)和\(xy^tz\in L\)。L的最小抽运常数\({{{\,\mathrm{mpc}\,}}(L)}\)是满足抽运引理条件的最小数p。我们研究了\({{{\,\mathrm{mpc}\,}}}\)在操作方面的行为,即,对于一个n元正则性保持操作\(\circ \),我们研究了所有数字k的集合\({g_{\circ }^{{{\,\mathrm{mpc}\,}}}(k_1,k_2,\ldots ,k_n)}\),使得\(1\le i\le n\)和\({{{\,\mathrm{mpc}\,}}(\circ (L_1,L_2,\ldots ,L_n)=~k}\)都有正则语言\(L_1,L_2,\ldots ,L_n\)和\({{{\,\mathrm{mpc}\,}}(L_i)=k_i}\)。对于Kleene闭包、补包、反转、前缀和后缀闭包、圆移位、并并、交集、集减法、对称差分和连接,我们完全确定了\({g_{\circ }^{{{\,\mathrm{mpc}\,}}}(k_1,k_2,\ldots ,k_n)}\)。此外,我们给出了一些关于最小泵浦长度的结果,此外,\(\vert xy\vert \le p\)必须保持不变。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
Acta Informatica
Acta Informatica 工程技术-计算机:信息系统
CiteScore
2.40
自引率
16.70%
发文量
24
审稿时长
>12 weeks
期刊介绍: Acta Informatica provides international dissemination of articles on formal methods for the design and analysis of programs, computing systems and information structures, as well as related fields of Theoretical Computer Science such as Automata Theory, Logic in Computer Science, and Algorithmics. Topics of interest include: • semantics of programming languages • models and modeling languages for concurrent, distributed, reactive and mobile systems • models and modeling languages for timed, hybrid and probabilistic systems • specification, program analysis and verification • model checking and theorem proving • modal, temporal, first- and higher-order logics, and their variants • constraint logic, SAT/SMT-solving techniques • theoretical aspects of databases, semi-structured data and finite model theory • theoretical aspects of artificial intelligence, knowledge representation, description logic • automata theory, formal languages, term and graph rewriting • game-based models, synthesis • type theory, typed calculi • algebraic, coalgebraic and categorical methods • formal aspects of performance, dependability and reliability analysis • foundations of information and network security • parallel, distributed and randomized algorithms • design and analysis of algorithms • foundations of network and communication protocols.
期刊最新文献
Editorial 2024: moving forwards in the electronic age Serial and parallel algorithms for order-preserving pattern matching based on the duel-and-sweep paradigm Linear-size suffix tries and linear-size CDAWGs simplified and improved Parameterized aspects of distinct Kemeny rank aggregation Word-representable graphs from a word’s perspective
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1