We present an intimate connection among the following fields: (a) distributed local algorithms: coming from the area of computer science, (b) finitary factors of iid processes: coming from the area of analysis of randomized processes, (c) descriptive combinatorics: coming from the area of combinatorics and measure theory. In particular, we study locally checkable labellings in grid graphs from all three perspectives. Most of our results are for the perspective (b) where we prove time hierarchy theorems akin to those known in the field (a) [Chang, Pettie FOCS 2017]. This approach that borrows techniques from the fields (a) and (c) implies a number of results about possible complexities of finitary factor solutions. Among others, it answers three open questions of [Holroyd et al. Annals of Prob. 2017] or the more general question of [Brandt et al. PODC 2017] who asked for a formal connection between the fields (a) and (b). In general, we hope that our treatment will help to view all three perspectives as a part of a common theory of locality, in which we follow the insightful paper of [Bernshteyn 2020+] .

翻译：我们揭示了以下领域之间的深层联系：(a) 分布式局部算法——源自计算机科学领域，(b) 独立同分布过程的有限因子——源自随机过程分析领域，(c) 描述组合学——源自组合学与测度论领域。具体而言，我们从上述三个视角系统研究了网格图中的局部可检测标记问题。我们的主要成果集中于视角(b)，在该领域我们证明了与(a)领域已知结果[Chang, Pettie FOCS 2017]类似的时间层级定理。这种借鉴(a)和(c)领域技术的分析方法，推导出关于有限因子解可能复杂度的若干结论。其中，我们不仅回应了[Holroyd et al. Annals of Prob. 2017]提出的三个开放问题，还解答了[Brandt et al. PODC 2017]关于建立(a)与(b)领域形式关联的更一般性提问。总体而言，我们期望本研究能助力将三个视角统合为局部性理论体系——这一思想源于[Bernshteyn 2020+]的深刻洞见。