Automatic sequences can be defined by DFAs with output (DFAO) in two natural ways. We propose to consider the minimal size of a corresponding DFAO as the complexity measure of the automatic sequence, for both variants. This paper compares these complexity measures and investigates their properties, such as the relationships with kernel and morphic sequences. There exist automatic sequences for which the one complexity is exponentially greater than the other one, in both directions. For both complexity measures we investigate the effect of taking basic operations on sequences, like removing or adding an initial element, combining sequences, or taking arithmetic subsequences, and observe that these operations may increase the complexity at most polynomially. For periodic sequences we give sharp bounds for both complexity measures.

自动序列可以由具有输出的 DFA (DFAO) 以两种自然方式定义。我们建议将相应 DFAO 的最小尺寸作为自动序列的复杂度度量，对于这两种变体。本文比较了这些复杂性度量并研究了它们的属性，例如与核和形态序列的关系。存在自动序列，其中一个复杂度在两个方向上都比另一个复杂度大。对于这两种复杂度度量，我们研究了对序列进行基本操作的影响，例如删除或添加初始元素、组合序列或采用算术子序列，并观察到这些操作最多可能会以多项式方式增加复杂度。对于周期性序列，我们为两种复杂性度量给出了明确的界限。