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
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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.

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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。
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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.
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