上下文无关语言沿正则轨迹的洗牌
Shuffles of Context-Free Languages along Regular Trajectories
AI总结：
本文研究了沿正则轨迹的洗牌操作，探讨了如何通过轨迹约束将两个非正则上下文无关语言（CFL）洗牌为非CFL，并为确定性CFL提供了轨迹三分类。
摘要：
在单核处理器中，并发性要求调度器将多个进程交错到单一线程中。对应的语言理论操作是两个语言的洗牌：通过任意字母方式交错每个语言中的单词得到的单词集合。已知正则语言在洗牌下封闭，而上下文无关语言（CFL）则不封闭。本文考虑沿正则“轨迹”进行的洗牌，即受自动机表达的调度约束限制。显然，某些轨迹允许CFL洗牌为CFL（例如简单拼接），而其他轨迹则不允许。本文提供了一种稳健的工具集，以证明给定轨迹总是将两个非正则CFL洗牌为非CFL。在确定性CFL（DCFL）的情况下，提供了根据轨迹如何洗牌DCFL的显著三分类。值得注意的是，这些结果基于关于推导自动机（PDA）在接受非正则CFL或DCFL时必须如何使用堆栈的独立引理。后者情况依赖于Jančar和Šíma（MFCS'2021）最近的结果；回答其中的开放问题，表明该结果不能推广到任意CFL，从而导致为两种情况专门设计的机器。
英文摘要：
In single-core processors, concurrency requires that multiple processes be interleaved into a single thread of execution by a scheduler. The language-theoretic operation that corresponds to this is the shuffle of two languages: the set of words obtained by interleaving a word from each language in an arbitrary, letter-wise fashion. It is well known that regular languages are closed under shuffles, while context-free languages (CFLs) are not. Following an established line of research, this paper considers shuffles according to regular ``trajectories,'' that is, subject to scheduling constraints expressed by an automaton. Unsurprisingly, some trajectories allow for CFLs to be shuffled into CFLs (e.g., simple concatenation of the two words), while others do not. This paper provides a robust toolset to show that a given trajectory would always shuffle two nonregular CFLs into a nonCFL. In the case of deterministic CFLs (DCFLs), a salient trichotomy of trajectories depending on how they shuffle DCFLs is provided. Notably, these results are based on lemmata of independent interest regarding how pushdown automata (PDA) must invoke the stack when accepting a nonregular CFL or DCFL. The latter case relies on a recent result of Jančar and Šíma (MFCS'2021); answering an open question therein, it is demonstrated that said result cannot be generalized to arbitrary CFLs, leading to dedicated machinery for both cases.
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。
