关于计算正则语言共字典序宽度的复杂度
On the Complexity of Computing the Co-lexicographic Width of a Regular Language
AI总结：
本文研究正则语言确定型共字典序宽度的计算复杂度，提出$O(m^p)$时间的判定算法，并基于强指数时间假设给出匹配条件下界，结合已有NFA下的PSPACE完全结果，基本解决该问题复杂度。
摘要：
共字典序偏序（co-lex partial orders）由Cotumaccio等人于SODA 2021及JACM 2023中提出，是一种用于有限状态自动机索引的强大工具，可应用于正则表达式匹配。它是Wheeler序（Gagie等人，《理论计算机科学》2017）的推广，自然反映了自动机中从源点到节点路径所标记字符串的共字典序。简而言之，有限状态自动机的共字典序宽度$p$衡量的是其状态根据所接受字符串的共字典序的可排序程度。共字典序宽度为$p$的自动机每条边可压缩至$O(\text{log }p)$比特，且支持的正则表达式匹配算法每匹配一个字符的运行时间与$p^2$成正比。\n正则语言$\boldsymbol{\text{mathcal L}}$的确定型共字典序宽度，是所有识别$\boldsymbol{\text{mathcal L}}$的确定有限自动机（DFA）中，此类共字典序的最小宽度。由于共字典序宽度较小的语言在自动机压缩和模式匹配中存在高效解决方案，因此计算语言的共字典序宽度在这些应用中具有重要意义。引入共字典序的论文已确定，给定识别语言$\boldsymbol{\text{mathcal L}}$的任意DFA $\boldsymbol{\text{mathcal A}}$（其转移数即规模为$m = |\boldsymbol{\text{mathcal A}}|$），该语言的确定型共字典序宽度$p$可在与$m^{O(p)}$成正比的时间内计算得出。\n本文采用新技术证明，对于给定的最小DFA所识别的语言，可在$O(m^p)$时间内判定其确定型共字典序宽度是否严格小于某个整数$p\boldsymbol{\text{ge}} 2$。我们还基于强指数时间假设（Strong Exponential Time Hypothesis）给出了匹配的条件下界，作为对该上界的补充。已知当输入为非确定有限自动机（NFA）时，该问题是PSPACE完全的（D'Agostino等人，《理论计算机科学》2023）；因此，结合该结果，本文基本解决了该问题的复杂度问题。
英文摘要：
Co-lex partial orders were recently introduced in (Cotumaccio et al., SODA 2021 and JACM 2023) as a powerful tool to index finite state automata, with applications to regular expression matching. They generalize Wheeler orders (Gagie et al., Theoretical Computer Science 2017) and naturally reflect the co-lexicographic order of the strings labeling source-to-node paths in the automaton. Briefly, the co-lex width $p$ of a finite-state automaton measures how sortable its states are with respect to the co-lex order among the strings they accept. Automata of co-lex width $p$ can be compressed to $O(\log p)$ bits per edge and admit regular expression matching algorithms running in time proportional to $p^2$ per matched character. The deterministic co-lex width of a regular language $\mathcal L$ is the smallest width of such a co-lex order, among all DFAs recognizing $\mathcal L$. Since languages of small co-lex width admit efficient solutions to automata compression and pattern matching, computing the co-lex width of a language is relevant in these applications. The paper introducing co-lex orders determined that the deterministic co-lex width $p$ of a language $\mathcal L$ can be computed in time proportional to $m^{O(p)}$, given as input any DFA $\mathcal A$ for $\mathcal L$, of size (number of transitions) $m =|\mathcal A|$. In this paper, using new techniques, we show that it is possible to decide in $O(m^p)$ time if the deterministic co-lex width of the language recognized by a given minimum DFA is strictly smaller than some integer $p\ge 2$. We complement this upper bound with a matching conditional lower bound based on the Strong Exponential Time Hypothesis. The problem is known to be PSPACE-complete when the input is an NFA (D'Agostino et al., Theoretical Computer Science 2023); thus, together with that result, our paper essentially settles the complexity of the problem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
