We extend the theory of locally checkable labeling problems (LCLs) from the classical LOCAL model to a number of other models that have been studied recently, including the quantum-LOCAL model, finitely-dependent processes, non-signaling model, dynamic-LOCAL model, and online-LOCAL model [e.g. STOC 2024, ICALP 2023]. First, we demonstrate the advantage that finitely-dependent processes have over the classical LOCAL model. We show that all LCL problems solvable with locality $O(\log^\star n)$ in the LOCAL model admit a finitely-dependent distribution (with constant locality). In particular, this gives a finitely-dependent coloring for regular trees, answering an open question by Holroyd [2023]. This also introduces a new formal barrier for understanding the distributed quantum advantage: it is not possible to exclude quantum advantage for any LCL in the $\Theta(\log^\star n)$ complexity class by using non-signaling arguments. Second, we put limits on the capabilities of all of these models. To this end, we introduce a model called randomized online-LOCAL, which is strong enough to simulate e.g. SLOCAL and dynamic-LOCAL, and we show that it is also strong enough to simulate any non-signaling distribution and hence any quantum-LOCAL algorithm. We prove the following result for rooted trees: if we can solve an LCL problem with locality $o(\log \log n)$ in the randomized online-LOCAL model, we can solve it with locality $O(\log^\star n)$ in the classical deterministic LOCAL model. Put together, these results show that in rooted trees the set of LCLs that can be solved with locality $O(\log^\star n)$ is the same across all these models: classical deterministic and randomized LOCAL, quantum-LOCAL, non-signaling model, dynamic-LOCAL, and deterministic and randomized online-LOCAL.

翻译：我们将局部可检查标记问题（LCLs）的理论从经典LOCAL模型扩展到近期研究的多种其他模型，包括量子-LOCAL模型、有限依赖过程、非信令模型、动态-LOCAL模型和在线-LOCAL模型【例如STOC 2024, ICALP 2023】。首先，我们展示了有限依赖过程相对于经典LOCAL模型的优势。我们证明，在LOCAL模型中所有可用局部性$O(\log^\star n)$解决的LCL问题，都允许一个有限依赖分布（具有常数局部性）。特别地，这给出了正则树的有限依赖着色，回答了Holroyd [2023]的一个开放问题。这也引入了一个新的正式障碍来理解分布式量子优势：无法通过非信令论证排除任何$\Theta(\log^\star n)$复杂度类中的LCL的量子优势。其次，我们限制了所有这些模型的能力。为此，我们引入了一个称为随机化在线-LOCAL的模型，该模型足够强大以模拟例如SLOCAL和动态-LOCAL，并且我们证明它也足够强大以模拟任何非信令分布，因此也能模拟任何量子-LOCAL算法。我们证明了关于有根树的以下结果：如果能够在随机化在线-LOCAL模型中以局部性$o(\log \log n)$解决一个LCL问题，那么就能在经典确定性LOCAL模型中以局部性$O(\log^\star n)$解决它。综合这些结果，我们表明在有根树中，所有能够以局部性$O(\log^\star n)$解决的LCL集合在这些模型中是相同的：经典确定性与随机化LOCAL、量子-LOCAL、非信令模型、动态-LOCAL，以及确定性与随机化在线-LOCAL。