{"title":"Operational complexity and pumping lemmas","authors":"Jürgen Dassow, 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.
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.