单带$n\log n$阈值下的接受路径计数
Accepting-Path Counting at the One-Tape $n\log n$ Threshold
- Faculty of Electrical Engineering, Czech Technical University in Prague（布拉格捷克理工大学电气工程学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明单带图灵机的$n\log n$时间阈值同时是接受路径计数的阈值，低于时生成函数有理，达到时出现$\\#\mathsf P_1$完全性和不可计算增长率。
AI中文摘要：
我们观察到，单带图灵机的经典$n\log n$时间阈值也是其接受路径计数的阈值。低于该阈值时，每个在强$o(n\log n)$时间内运行的非确定性单带机器，其接受路径计数具有有理普通生成函数。在强$O(n\log n)$时间内，情况完全改变：存在一个固定的单带机器，其接受路径函数在简约多项式时间计数归约下对于$\\#\mathsf P_1$（$\\#\mathsf P$的 tally 类比）是完全的。同一时间界限内的第二个构造给出了具有不可计算指数增长率的正接受路径计数。合理性结果结合了单带时间间隙与 Tadaki、Yamakami 和 Lin 的线性时间计数定理。完全性证明将 Beame 等人的线性时间通用计数机器适应到单带设置。
英文摘要：
We observe that the classical $n\log n$ time threshold for one-tape Turing machines is also a threshold for their accepting-path counts. Below it, every nondeterministic one-tape machine running in strong $o(n\log n)$ time has a rational ordinary generating function of accepting-path counts. At strong $O(n\log n)$ time, the situation changes completely: there is a fixed one-tape machine whose accepting-path function is complete for $\#\mathsf P_1$, the tally analogue of $\#\mathsf P$, under parsimonious polynomial-time tally reductions. A second construction within the same time bound gives positive accepting-path counts with a noncomputable exponential growth rate. The rationality result combines the one-tape time gap with the linear-time counting theorem of Tadaki, Yamakami and Lin. The completeness proof adapts the linear-time universal counting machine of Beame et al. to the one-tape setting.
