We consider two-variable first-order logic $\text{FO}^2$ and its quantifier alternation hierarchies over both finite and infinite words. Our main results are forbidden patterns for deterministic automata (finite words) and for Carton-Michel automata (infinite words). In order to give concise patterns, we allow the use of subwords on paths in finite graphs. This concept is formalized as subword patterns. Deciding the presence or absence of such a pattern in a given automaton is in $\mathbf{NL}$. In particular, this leads to $\mathbf{NL}$ algorithms for deciding the levels of the $\text{FO}^2$ quantifier alternation hierarchies. This applies to both full and half levels, each over finite and infinite words. Moreover, we show that these problems are $\mathbf{NL}$-hard and, hence, $\mathbf{NL}$-complete.

中文翻译：

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确定有限和无限词自动机的FO2交替

我们考虑二元一阶逻辑$ \ text {FO} ^ 2 $及其在有限词和无限词上的量词交替层次。我们的主要结果是确定性自动机（有限单词）和Carton-Michel自动机（无限单词）的禁止模式。为了给出简洁的模式，我们允许在有限图中的路径上使用子词。这个概念被正式化为子词模式。确定给定自动机中这种模式的存在与否以$ \ mathbf {NL} $为单位。特别地，这导致用于确定$ \ text {FO} ^ 2 $量词替换层次结构级别的$ \ mathbf {NL} $算法。这适用于全部级别和一半级别，每个级别都超过有限和无限个字。此外，我们证明这些问题是$ \ mathbf {NL} $-困难的，因此是$ \ mathbf {NL} $-完整的。