We study the space complexity of the following problem: For a fixed regular language $L$, test membership of a sliding window of length $n$ to $L$ over a given stream of symbols. For deterministic streaming algorithms we prove a trichotomy theorem, namely that the (optimal) space complexity is either constant, logarithmic or linear, measured in the window length $n$. Additionally, we provide natural language-theoretic characterizations of the space classes. We then extend the results to randomized streaming algorithms and we show that in this setting, the space complexity of any regular language is either constant, doubly logarithmic, logarithmic or linear. Finally, we introduce sliding window testers, which can distinguish whether a window belongs to the language $L$ or has Hamming distance $> \epsilon n$ to $L$. We prove that every regular language has a deterministic (resp., randomized) sliding window tester that requires only logarithmic (resp., constant) space.

中文翻译：

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滑动窗口模型中的正则语言

我们研究以下问题的空间复杂度：对于固定的正则语言 $L$，测试给定符号流上长度为 $n$ 到 $L$ 的滑动窗口的成员资格。对于确定性流算法，我们证明了三分定理，即（最佳）空间复杂度是常数、对数或线性，以窗口长度 $n$ 进行测量。此外，我们还提供空间类的自然语言理论特征。然后，我们将结果扩展到随机流算法，并表明在这种情况下，任何常规语言的空间复杂度要么是常数、双对数、对数或线性。最后，我们引入滑动窗口测试器，它可以区分窗口是否属于语言 $L$ 或具有汉明距离 $> \epsilon n$ 到 $L$。我们证明每种常规语言都有一个确定性（或随机）滑动窗口测试器，仅需要对数（或常数）空间。