忙碌海狸值BB(15)的难度
Hardness of busy beaver value BB(15)
AI总结：
本文通过构造一个15状态、2符号图灵机，证明知道BB(15)至少与解决Erdős提出的关于2的幂的三进制表示中数字2存在性的Collatz相关猜想一样困难，使BB(15)成为目前与自然未解数学问题相关的最小忙碌海狸值。
摘要：
忙碌海狸值BB(n)是指任何n状态、2符号的确定性停机图灵机从空白纸带开始运行所能执行的最大步数。忙碌海狸函数$n \mapsto \text{BB}(n)$是不可计算的，并且从下方看，迄今为止只知道其中4个值，即BB(1) ... BB(4)。这引出一个问题：从上方看，编码了一个重大数学挑战的最小BB值是多少？Yedidia和Aaronson [28]已经证明，知道BB(4,888)至少与解决哥德巴赫猜想一样困难，随后有一个尚未发表的改进，将结果改进到BB(27) [4,1]。我们证明，知道BB(15)至少与解决以下由Erdős提出的、自1979年以来一直未解决的Collatz相关猜想一样困难[9]：对所有n > 8，$2^n$的三进制表示中至少有一个数字2。我们通过构造一个显式的15状态、2符号图灵机来证明这一点，该图灵机停机当且仅当该猜想为假。这个2符号图灵机模拟了我们首先构造的一个概念上更简单的5状态、4符号机器。这使得BB(15)成为迄今为止与数学中一个自然未解决问题相关的最小忙碌海狸值，揭示了追求了解忙碌海狸值背后众多挑战之一。
英文摘要：
The busy beaver value BB(n) is the maximum number of steps made by any n-state, 2-symbol deterministic halting Turing machine starting on blank tape. The busy beaver function $n \mapsto \text{BB}(n)$ is uncomputable and, from below, only 4 of its values, BB(1) ... BB(4), are known to date. This leads one to ask: from above, what is the smallest BB value that encodes a major mathematical challenge? Knowing BB(4,888) has been shown by Yedidia and Aaronson [28] to be at least as hard as solving Goldbach's conjecture, with a subsequent improvement, as yet unpublished, to BB(27) [4,1]. We prove that knowing BB(15) is at least as hard as solving the following Collatz-related conjecture by Erdős, open since 1979 [9]: for all n > 8 there is at least one digit 2 in the base 3 representation of $2^n$. We do so by constructing an explicit 15-state, 2-symbol Turing machine that halts if and only if the conjecture is false. This 2-symbol Turing machine simulates a conceptually simpler 5-state, 4-symbol machine which we construct first. This makes, to date, BB(15) the smallest busy beaver value that is related to a natural open problem in mathematics, bringing to light one of the many challenges underlying the quest of knowing busy beaver values.
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。
