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, Rozho\v{n} 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.
翻译:我们分类了在定向路径上投入的哪些本地问题有所谓的Borel解决方案,并表明,如果我们相反需要可衡量的解决方案、iid解决方案的一个要素或Baire属性的解决方案,这类问题仍然相同。 连同分布式计算领域的工作[Balliu等人,PoDC 2019]、描述性组合分析领域的工作[Gao等人,ArXiv:1803.03872,Bernshteyn arXiv:2004.04905],以及随机过程领域的工作[Holroyd等人,Prob. 2017,Greb\'ik, Rozho\v{n}arXiv:2103.08394],这与分布式计算领域的工作[Balliu等人,PoDC 2019]、描述性组合学领域的工作[Gao等:1803.03872, Bernshteyn arxiv:2004.04905]以及随机过程领域的工作[Holroyd et al. Annals, Prob. 2017, Greb, Greb\\', Greb\\'ik, Rozhoqn, Rozhov{n}arxiv: arxiv: arxn] arxn:23.394],这帮助解决了方向上的当地问题。