Classification of Local Problems on Paths from the Perspective of Descriptive Combinatorics
Abstract
We classify which local problems with inputs on oriented paths have socalled 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, Grebík, Rozhoň 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 \emph{not} help with solving local problems on oriented paths.
 Publication:

arXiv eprints
 Pub Date:
 March 2021
 DOI:
 10.48550/arXiv.2103.14112
 arXiv:
 arXiv:2103.14112
 Bibcode:
 2021arXiv210314112G
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Data Structures and Algorithms;
 Mathematics  Logic;
 Mathematics  Probability