Local problems on grids from the perspective of distributed algorithms, finitary factors, and descriptive combinatorics

Abstract

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 problems on grids from all three perspectives. Most of our results are for perspective (b) where we prove time hierarchy theorems similar to those known in the field (a) Chang and Pettie (2017) [16]. This approach, which borrows techniques from fields (a) and (c), implies a number of results about the possible complexities of finitary factor solutions. Among others, it answers three open questions of Holroyd et al. (2017) [46] or the more general question of Brandt et al. Brandt (2017) [9] 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 (2023) [6].