State complexity of inversion operations

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• 作者：Da-Jung Cho^a ; ^{dajung@cs.yonsei.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Yo-Sub Han^a ; ^{emmous@cs.yonsei.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Sang-Ki Ko^a ; ^{narame7@cs.yonsei.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Kai Salomaa^b ; ^{ksalomaa@cs.queensu.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：State complexity ; Inversion operations ; Finite automata ; Regular languages
• 刊名：Theoretical Computer Science
• 出版年：2016
• 出版时间：11 January 2016
• 年：2016
• 卷：610
• 期：part_PA
• 页码：2-12
• 全文大小：447 K

文摘

The reversal operation is well-studied in the literature and the deterministic (respectively, nondeterministic) state complexity of reversal is known to be 2ⁿ$2^n$ (respectively, n). We consider the inversion operation where some substring of the given string is reversed. Formally, the inversion (respectively, prefix-inversion) of a language L   consists of all strings ux^Rv$u{x}^{R}v$ such that uxv∈L$uxv\in L$ (respectively, all strings u^Rx$u^{R}x$ where ux∈L$ux\in L$). We show that the nondeterministic state complexity of prefix-inversion is Θ(n²)$\mathrm{\Theta }(n^2)$ and that of inversion is Θ(n³)$\mathrm{\Theta }(n^3)$. We show that the deterministic state complexity of prefix-inversion is at most 2^{n⋅log⁡n+n}$2^{n\cdot \log n+n}$ and has lower bound 2^Ω(nlog⁡n)$2^{\mathrm{\Omega }(n\log n)}$. The same lower bound holds for the state complexity of inversion, but for inversion we do not have a matching upper bound. We also study the state complexity of other variants of the inversion operation.