We provide upper and lower bounds for the length of controlled bad sequences over the majoring and the minoring orderings of finite sets of $\mathbb{N}^d$. The results are obtained by bounding the length of such sequences by functions from the Cichon hierarchy. This allows us to translate these results to bounds over the fast-growing complexity classes. The obtained bounds are proven to be tight for the majoring ordering, which solves a problem left open by Abriola, Figueira and Senno (Theor. Comp. Sci, Vol. 603). Finally, we use the results on controlled bad sequences to prove upper bounds for the emptiness problem of some classes of automata.

中文翻译：

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$\mathbb{N}^d$ 有限集上受控坏序列的复杂性

我们为 $\mathbb{N}^d$ 的有限集的主要和次要排序上的受控坏序列的长度提供了上限和下限。结果是通过使用 Cichon 层次结构中的函数限制此类序列的长度而获得的。这使我们能够将这些结果转换为快速增长的复杂性类的边界。所获得的界限被证明对于主序是严格的，这解决了 Abriola、Figueira 和 Senno 留下的问题（Theor. Comp. Sci, Vol. 603）。最后，我们使用受控坏序列的结果来证明某些类别自动机的空性问题的上限。