We construct automata with input(s) in base $k$ recognizing some basic relations and study their number of states. We also consider some basic operations on $k$-automatic sequences and discuss their state complexity. We find a relationship between subword complexity of the interior sequence $(h'(i))_{i \geq 0}$ and state complexity of the linear subsequence $(h(ni+c))_{i \geq 0}$. We resolve a recent question of Zantema and Bosma about linear subsequences of $k$-automatic sequences with input in most-significant-digit-first format. We also discuss the state complexity and runtime complexity of using a reasonable interpretation of Büchi arithmetic to actually construct some of the studied automata recognizing relations or carrying out operations on automatic sequences.

翻译：我们构造了以$k$进制为输入的自动机来识别某些基本关系，并研究了它们的状态数。我们还考虑了$k$-自动序列的一些基本操作，并讨论了它们的状态复杂性。我们发现了内部序列$(h'(i))_{i \geq 0}$的子词复杂性与线性子序列$(h(ni+c))_{i \geq 0}$的状态复杂性之间的关系。我们解决了Zantema和Bosma最近提出的关于以最高有效位优先格式输入的$k$-自动序列的线性子序列的问题。我们还讨论了使用Büchi算术的合理解释来实际构造所研究的识别关系或对自动序列执行操作的自动机的状态复杂性和运行时复杂性。