确定性有限自动机上切操作的紧界

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, SOFTWARE ENGINEERING Fundamenta Informaticae Pub Date : 2015-09-09 DOI:10.3233/FI-2017-1577
F. Drewes, M. Holzer, Sebastian Jakobi, Brink van der Merwe
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引用次数: 6

摘要

我们研究了确定性有限自动机(dfa)切割和迭代切割操作的状态复杂性,回答了[M]中提出的一个开放问题。Berglund等人:正则表达式中的剪切。[j].计算机工程学报,2011。这些操作可以看作是普通串联和Kleene星型最左边最大字符串匹配的替代方法。我们证明了切割操作在分别接受n状态和m状态dfa接受的两种单独语言的dfa上具有匹配的\((n-1)\cdot m+n\)状态的上界和下界。在一元情况下,我们得到\(\max (2n-1,m+n-2)\)状态作为紧界。为了接受n状态DFA所接受的语言的迭代切割,我们找到了DFA上\(1+(n+1)\cdot \mathsf {F}(\,1,n+2,-n+2;n+1\mid -1\,)\)状态的匹配界,其中\(\mathsf {F}\)指广义超几何函数。这个边界的数量级是\(\varTheta ((n-1)!)\)。最后,一元DFA接受n状态DFA的迭代切割时,边界降为\(2n-1\),因此与一元DFA切割操作的边界相似。
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Tight Bounds for Cut-Operations on Deterministic Finite Automata
We investigate the state complexity of the cut and iterated cut operation for deterministic finite automata (DFAs), answering an open question stated in [M. Berglund, et al.: Cuts in regular expressions. In Proc. DLT, LNCS 7907, 2011]. These operations can be seen as an alternative to ordinary concatenation and Kleene star modelling leftmost maximal string matching. We show that the cut operation has a matching upper and lower bound of \((n-1)\cdot m+n\) states on DFAs accepting the cut of two individual languages that are accepted by n- and m-state DFAs, respectively. In the unary case we obtain \(\max (2n-1,m+n-2)\) states as a tight bound. For accepting the iterated cut of a language accepted by an n-state DFA we find a matching bound of \(1+(n+1)\cdot \mathsf {F}(\,1,n+2,-n+2;n+1\mid -1\,)\) states on DFAs, where \(\mathsf {F}\) refers to the generalized hypergeometric function. This bound is in the order of magnitude \(\varTheta ((n-1)!)\). Finally, the bound drops to \(2n-1\) for unary DFAs accepting the iterated cut of an n-state DFA and thus is similar to the bound for the cut operation on unary DFAs.
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来源期刊
Fundamenta Informaticae
Fundamenta Informaticae 工程技术-计算机:软件工程
CiteScore
2.00
自引率
0.00%
发文量
61
审稿时长
9.8 months
期刊介绍: Fundamenta Informaticae is an international journal publishing original research results in all areas of theoretical computer science. Papers are encouraged contributing: solutions by mathematical methods of problems emerging in computer science solutions of mathematical problems inspired by computer science. Topics of interest include (but are not restricted to): theory of computing, complexity theory, algorithms and data structures, computational aspects of combinatorics and graph theory, programming language theory, theoretical aspects of programming languages, computer-aided verification, computer science logic, database theory, logic programming, automated deduction, formal languages and automata theory, concurrency and distributed computing, cryptography and security, theoretical issues in artificial intelligence, machine learning, pattern recognition, algorithmic game theory, bioinformatics and computational biology, quantum computing, probabilistic methods, algebraic and categorical methods.
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