We construct automata with input(s) in Fibonacci representation (also known as Zeckendorf representation) recognizing some basic arithmetic relations and study their number of states. We also consider some basic operations on Fibonacci-automatic sequences and discuss their state complexity. Furthermore, as a consequence of our results, we improve a bound in a recent paper of Bosma and Don. 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.

翻译：我们构造了具有斐波那契表示（也称为齐肯多夫表示）输入的自动机，这些自动机识别一些基本算术关系，并研究了其状态数量。我们还考虑了斐波那契自动序列的一些基本运算，并讨论了它们的状态复杂性。此外，作为我们结果的一个推论，我们改进了Bosma和Don近期论文中的一个界。我们还讨论了使用布奇算术的合理解释来实际构造一些用于识别关系的自动机时的状态复杂性和运行时间复杂度。