Wheeler 双射
Wheeler Bisimulations
AI总结：
本文研究了双射与数据压缩的关系，提出Wheeler双射概念，展示如何通过线性时间构建最小Wheeler NFA，改进了传统双射方法。
摘要：
多年来，双射已成为一种普遍范式，应用于并发理论、模型检查、自动机理论、逻辑、编程语言和范畴论等多个领域。本文建立了双射与数据压缩之间的联系。更具体地说，我们研究了双射与Wheeler自动机（Alanko等人，SODA 2020）之间的关系，这是一种近年来受到广泛关注的自动机类。标准的双射概念不适用，因此我们引入了Wheeler双射，即尊重所考虑Wheeler自动机凸结构的双射。我们证明Wheeler双射诱导了一个唯一的最小Wheeler NFA（类似于标准双射）。特别是，在确定性情况下，我们获得了给定语言的最小Wheeler确定自动机。我们还证明由Wheeler双射诱导的最小Wheeler NFA可以在线性时间内构建。这与标准双射不同，后者对应的最小NFA可以通过适应Paige-Tarjan分区细化算法在$ O(m \log n) $时间内构建（其中$ m $是边数，$ n $是状态数）。与之前的状态减少技术相比，我们的双射诱导构造是第一个满足（i）我们获得一个规范的Wheeler NFA，（ii）所得到的Wheeler NFA可以在线性时间内构建的。
英文摘要：
Over the years, bisimulations have emerged as a pervasive paradigm, finding applications in numerous areas, including concurrency theory, model checking, automata theory, logic, programming languages and category theory. In this paper, we establish a connection between bisimulations and data compression. More precisely, we study the relationship between bisimulations and Wheeler automata (Alanko et al., SODA 2020), a class of automata that has received considerable attention in recent years. The standard notion of bisimulation is not appropriate, so we introduce Wheeler bisimulations, that is, bisimulations that respect the convex structure of the considered Wheeler automata. We show that Wheeler bisimilarity induces a unique minimal Wheeler NFA (analogously to standard bisimulations). In particular, in the deterministic case, we retrieve the minimal Wheeler deterministic automaton of a given language. We also show that the minimal Wheeler NFA induced by Wheeler bisimulations can be built in linear time. This is in contrast with standard bisimulations, for which the corresponding minimal NFA can be built in $ O(m \log n) $ time (where $ m $ is the number of edges and $ n $ is the number of states) by adapting Paige-Tarjan partition refinement algorithm. Compared to previous state-reduction techniques, our bisimulation-induced construction is the first for which (i) we obtain a canonical Wheeler NFA and (ii) the resulting Wheeler NFA can be built in linear time.
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