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 $(h(i))_{i \geq 0}$ 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：我们构建了基数为$k$的输入自动机，用于识别某些基本关系并研究其状态数。我们同时考虑$k$-自动序列$(h(i))_{i \geq 0}$上的基本运算并讨论其状态复杂度。我们揭示了内部序列$(h'(i))_{i \geq 0}$的子词复杂度与线性子序列$(h(ni+c))_{i \geq 0}$的状态复杂度之间的关系。我们解决了近期Zantema与Bosma关于采用最高位优先输入格式的$k$-自动序列线性子序列的疑问。我们还讨论了使用Büchi算术的合理解释来实际构造某些用于识别关系或对自动序列进行运算的自动机时的状态复杂度与运行时间复杂度。