发表机构
National University of Singapore（新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究非终止重采样计算的维度，通过主定理界定相关量，探讨源幂信息，分析不同修复规则及\(k\)-SAT 等情况，得出有限磁带源下规则的非终止维度差异、\(k\)-SAT 终止条件及公式维度界等结论。
AI 中文摘要
随机算法即便存在异常随机磁带使其永远运行，也可能几乎必然终止。本文研究了生存尾部、此类磁带之一的柯尔莫哥洛夫复杂度以及所有磁带的豪斯多夫维度。对于动力修复矩阵可交换的每个\(s>0\)，主定理界定了在生存前缀\(w\)上的\(\sum_wP[w]^s\)，在确定性非预期选择器上一致。\(s = 1\)的情况控制终止；完整族给出弱源和维度界。源幂包含普通修复核和完整停止时间定律中都没有的信息。在一个常见的有限磁带源下，四顶点路径上的两个重叠分歧修复规则对于每个选择器具有相同的普通核和相同的停止时间定律，但它们的非终止维度可以任意接近零和一。在一个共同的源幂水平上，相同的主导磁带源使一个规则永远运行，但给另一个指数停止尾部。这种分离是由产生相同状态转换的动作标签引起的，因此在幂为一时不可见。对于有界依赖\(k\)-SAT，高于迹增长阈值的条件块最小熵给出指数终止，单个无限运行的有效维度由无限次修复的子句引起的迹增长界定。树公式渐近达到最大度维度和全局源界，而团公式在所述 regime 中达到特定于图的一步阈值。一个精确的反向似然恒等式用每次运行的尾部和编码界补充了这些逐集结果。
英文摘要
How much does an algorithm's running-time distribution under independent randomness reveal about its behavior when independence is no longer guaranteed? We study sources satisfying $ν[w]\le DP[w]^s$ for every finite prefix $w$, where $P$ is an independent reference law, $0<s\le1$, and $D\ge1$. The constraint controls complete-prefix probabilities while allowing individual choices to be predictable, even fully determined by the past. For retry tasks, all deterministic history-dependent selectors have the same independent-source running-time law. Yet two orders have worst-case failure probabilities $1$ and $\exp[-Θ(n)]$ at the same linear deadline under the same source constraint. We identify a static priority rule that is optimal at every deadline and every $D$. For the standard local walk on a $k$-CNF with at least $r$ true literals per clause under some assignment, $k/2<r<k$, we determine the sharp source threshold $s_*$. At and above it, the expected flip count is $O_{k,r}(\min\{L^3,L/(s-s_*)\})$, where $L=h+\log D+1$, $h$ is the initial Hamming distance to that assignment, and $L/0=\infty$. The bound allows arbitrary clause overlap and history-dependent clause selection. Matching instances admit one source forcing this delay with probability one for every selector. At criticality and fixed $D$, the delay is cubic despite a linear independent-source expectation. Variable-depth prefix covers, together with classical tree max-flow/min-cut, yield an exact criterion for restoring exponential tails by restarting on the same tape. We synthesize updates and restarts for explicit finite-state processes. Under a sufficient prefix guarantee, we also obtain noisy predecessor search with error at most $η$ and expected query count polynomial in the correct leaf's depth and $\log(D/η)$, without knowing the depth or tree height.
