Classification of Local Problems on Paths from the Perspective of Descriptive Combinatorics

We classify which local problems with inputs on oriented paths have so-called Borel solution and show that this class of problems remains the same if we instead require a measurable solution, a factor of iid solution, or a solution with the property of Baire. Together with the work from the field of distributed computing [Balliu et al. PODC 2019], the work from the field of descriptive combinatorics [Gao et al. arXiv:1803.03872, Bernshteyn arXiv:2004.04905] and the work from the field of random processes [Holroyd et al. Annals of Prob. 2017, Greb'ik, Rozhovn arXiv:2103.08394], this finishes the classification of local problems with inputs on oriented paths using complexity classes from these three fields. A simple picture emerges: there are four classes of local problems and most classes have natural definitions in all three fields. Moreover, we now know that randomness does not help with solving local problems on oriented paths.