In formal languages and automata theory, the magic number problem can be formulated as follows: for a given integer n, is it possible to find a number d in the range [n,2^n] such that there is no minimal deterministic finite automaton with d states that can be simulated by an optimal nondeterministic finite automaton with exactly n states? If such a number d exists, it is called magic. In this paper, we consider the magic number problem in the framework of deterministic automata with output, which are known to characterize automatic sequences. More precisely, we investigate magic numbers for periodic sequences viewed as either automatic, regular, or constant-recursive.

翻译：在形式语言与自动机理论中，魔法数问题可表述如下：对于给定整数n，是否存在范围在[n, 2^n]内的整数d，使得不存在一个具有d个状态的极小确定性有限自动机，能够被恰好具有n个状态的最优非确定性有限自动机模拟？若这样的整数d存在，则称之为魔法数。本文在带输出的确定性自动机框架下（该类自动机已知可刻画自动序列）研究魔法数问题。具体而言，我们将探讨将周期序列视为自动序列、正则序列或常递归序列时的魔法数。