The Dimension of Nonterminating Resampling Computations

2026-07-19

AI Summary

This study investigates the structure and complexity of exceptional random tapes that induce infinite executions in almost surely terminating randomized algorithms. Employing tools from Hausdorff dimension, Kolmogorov complexity, spectral analysis of repair matrices, and conditional block min-entropy, the work characterizes the measure-theoretic, dimensional, and information-theoretic properties of non-terminating behaviors. Key contributions include establishing the first precise connections between non-termination dimension and trace growth as well as block min-entropy; demonstrating that under identical stopping-time distributions, non-termination dimensions can be extremally separated; showing that with finite-band randomness, distinct repair rules can drive the non-termination dimension arbitrarily close to either 0 or 1; and proving that for bounded-dependence k-SAT, super-threshold block min-entropy guarantees exponential termination, with tree and clique formulas achieving maximal-degree dimension and graph-specific one-step thresholds, respectively.

Abstract

A randomized algorithm may terminate almost surely even though exceptional random tapes make it run forever. This paper studies the survival tail, the Kolmogorov complexity of one such tape, and the Hausdorff dimension of all of them. For each $s>0$ at which the powered repair matrices commute, the main theorem bounds $\sum_wP[w]^s$ over surviving prefixes $w$, uniformly over deterministic nonanticipating selectors. The case $s=1$ controls termination; the full family gives weak-source and dimension bounds. The source powers contain information absent even from the ordinary repair kernel and the complete stopping-time law. Under one common finite tape source, two overlapping disagreement-repair rules on a four-vertex path have the same ordinary kernels and the same stopping-time law for every selector, yet their nontermination dimensions can be arbitrarily close to zero and one. At one common source-power level, the same dominated tape source makes one rule run forever but gives the other an exponential stopping tail. The separation is caused by action labels that produce the same state transition and are therefore invisible at power one. For bounded-dependence $k$-SAT, conditional block min-entropy above the trace-growth threshold gives exponential termination, and the effective dimension of an individual infinite run is bounded by the trace growth induced by the clauses repaired infinitely often. Tree formulas asymptotically attain the maximum-degree dimension and global source bounds, while clique formulas attain the graph-specific one-step threshold in the stated regime. An exact backward likelihood identity complements these setwise results with tail and coding bounds for each run.

Problem

• nontermination
• Hausdorff dimension
• randomized algorithms
• survival tail
• Kolmogorov complexity

Innovation

• Hausdorff dimension
• source powers
• nontermination
• repair matrices
• effective dimension