F. Drewes, M. Holzer, Sebastian Jakobi, Brink van der Merwe
{"title":"确定性有限自动机上切操作的紧界","authors":"F. Drewes, M. Holzer, Sebastian Jakobi, Brink van der Merwe","doi":"10.3233/FI-2017-1577","DOIUrl":null,"url":null,"abstract":"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.","PeriodicalId":56310,"journal":{"name":"Fundamenta Informaticae","volume":"6 1","pages":"45-60"},"PeriodicalIF":0.4000,"publicationDate":"2015-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"Tight Bounds for Cut-Operations on Deterministic Finite Automata\",\"authors\":\"F. Drewes, M. Holzer, Sebastian Jakobi, Brink van der Merwe\",\"doi\":\"10.3233/FI-2017-1577\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"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.\",\"PeriodicalId\":56310,\"journal\":{\"name\":\"Fundamenta Informaticae\",\"volume\":\"6 1\",\"pages\":\"45-60\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2015-09-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Fundamenta Informaticae\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.3233/FI-2017-1577\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, SOFTWARE ENGINEERING\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fundamenta Informaticae","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.3233/FI-2017-1577","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
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.
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.