We investigate the connections between the fields of distributed computing and measurable combinatorics by considering complexity classes of locally checkable labeling problems on regular forests. We show that the most important deterministic complexity classes from the LOCAL model of distributed computing exactly coincide with well-studied classes in measurable combinatorics. Namely, first we show that a locally checkable labeling problem admits a continuous solution if and only if it can be solved by a deterministic local algorithm with complexity $O(\log^* n)$. Second, our main result states that, surprisingly, a locally checkable labeling problem admits a Baire measurable solution if and only if it can be solved by a local algorithm with complexity $O(\log n)$. These theorems suggest the existence of deeper connections between the two frameworks. Furthermore, the latter result relies on a complete combinatorial characterization of the classes in question, and as a by-product, it shows that membership in these classes is decidable.

翻译：本文通过研究正则森林上局部可检验标记问题的复杂度类别，探讨了分布式计算与可测组合学两个领域之间的联系。我们证明了分布式计算LOCAL模型中最重要的确定性复杂度类别恰好对应于可测组合学中已深入研究的类别。具体而言，首先我们证明：一个局部可检验标记问题存在连续解当且仅当它能被复杂度为$O(\\log^* n)$的确定性局部算法求解。其次，我们的主要结果表明：令人惊讶的是，一个局部可检验标记问题存在贝尔可测解当且仅当它能被复杂度为$O(\\log n)$的局部算法求解。这些定理暗示了两个框架之间存在更深层次的联系。此外，后一结果依赖于对相关类别的完全组合刻画，作为推论，它表明这些类别的成员判定问题是可判定的。