  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+] .
