We define a new strict and computable hierarchy for the family of automaton semigroups, which reflects the various asymptotic behaviors of the state-activity growth. This hierarchy extends that given by Sidki for automaton groups, and also gives new insights into the latter. Its exponential part coincides with a notion of entropy for some associated automata.We prove that the Order Problem is decidable whenever the state-activity is bounded. The Order Problem remains open for the next level of this hierarchy, that is, when the state-activity is linear. Gillibert showed that it is undecidable in the whole family.We extend the aforementioned hierarchy via a semi-norm making it more coarse but somehow more robust and we prove that the Order Problem is still decidable for the first two levels of this alternative hierarchy.

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自动机半群的新层次

我们为自动机半群族定义了一个新的严格且可计算的层次结构，它反映了状态活动增长的各种渐近行为。这种层次结构扩展了 Sidki 为自动机群提供的层次，也为后者提供了新的见解。它的指数部分与一些相关自动机的熵概念相吻合。我们证明只要状态活动有界，订单问题是可判定的。对于该层次结构的下一个级别，即当状态活动是线性的时，订单问题仍然是开放的。Gillibert 表明它在整个家族中是不可判定的。我们通过半范数扩展上述层次结构，使其更粗略但在某种程度上更健壮，我们证明对于这个替代层次结构的前两个级别，顺序问题仍然是可判定的。