用确定性有限自动机实现密码计数
Enumerating Cryptarithms Using Deterministic Finite Automata
摘要：
密码是一种数学难题，其中给出了一个用字母而不是数字写的算术方程，玩家必须发现字母上的数字分配，使方程成立。在本文中，我们提出了一种方法来构建一个DFA，接受密码承认（唯一）的解决方案，为每个基地。我们实现了该方法并构造了基$k \le 7$的DFA。这些DFA可以用作密码算法的完整目录，其应用包括枚举和计数具有允许基$k$解的$n$数字的密码算法实例的确切数目$G_k(n)$。并给出了$G_2(n)$和$G_3(n)$的显式表达式。
英文摘要：
A cryptarithm is a mathematical puzzle where given an arithmetic equation written with letters rather than numerals, a player must discover an assignment of numerals on letters that makes the equation hold true. In this paper, we propose a method to construct a DFA that accepts cryptarithms that admit (unique) solutions for each base. We implemented the method and constructed a DFA for bases $k \le 7$. Those DFAs can be used as complete catalogues of cryptarithms,whose applications include enumeration of and counting the exact numbers $G_k(n)$ of cryptarithm instances with $n$ digits that admit base-$k$ solutions. Moreover, explicit formulas for $G_2(n)$ and $G_3(n)$ are given.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
