Common definitions of the "standard" LOCAL model tend to be sloppy and even self-contradictory on one point: do the nodes update their state using an arbitrary function or a computable function? So far, this distinction has been safe to neglect, since problems where it matters seem contrived and quite different from e.g. typical local graph problems studied in this context. We show that this question matters even for locally checkable labeling problems (LCLs), perhaps the most widely studied family of problems in the context of the LOCAL model. Furthermore, we show that assumptions about computability are directly connected to another aspect already recognized as highly relevant: whether we have any knowledge of $n$, the size of the graph. Concretely, we show that there is an LCL problem $Π$ with the following properties: 1. $Π$ can be solved in $O(\log n)$ rounds if the LOCAL model is uncomputable. 2. $Π$ can be solved in $O(\log n)$ rounds in the computable model if we know any upper bound on $n$. 3. $Π$ requires $Ω(\sqrt{n})$ rounds in the computable model if we do not know anything about $n$. We also show that the connection between computability and knowledge of $n$ holds in general: for any LCL problem $Π$, if you have any bound on $n$, then $Π$ has the same round complexity in the computable and uncomputable models.

翻译：“标准” LOCAL 模型的常见定义往往不够严谨，甚至在一个关键点上自相矛盾：节点更新其状态时使用的是任意函数还是可计算函数？迄今为止，这种区分一直被安全地忽略，因为受其影响的问题似乎都是人为构造的，并且与此背景下研究的典型局部图问题（例如）截然不同。我们证明，即使对于局部可检查标记问题（LCLs）——可能是 LOCAL 模型背景下研究最广泛的问题族——这个问题也至关重要。此外，我们证明关于可计算性的假设直接关联到另一个已被公认高度相关的方面：我们是否对图的大小 $n$ 有任何了解。具体而言，我们证明存在一个 LCL 问题 $Π$，其具有以下性质：1. 如果 LOCAL 模型是不可计算的，则 $Π$ 可在 $O(\log n)$ 轮内求解。2. 如果我们知道 $n$ 的任意上界，则 $Π$ 在可计算模型中可在 $O(\log n)$ 轮内求解。3. 如果我们对 $n$ 一无所知，则 $Π$ 在可计算模型中需要 $Ω(\sqrt{n})$ 轮。我们还证明，可计算性与对 $n$ 的了解之间的关联在一般情况下成立：对于任何 LCL 问题 $Π$，如果你对 $n$ 有任何界限，那么 $Π$ 在可计算和不可计算模型中具有相同的轮复杂度。