What can be decided locally without identifiers

Do unique node identifiers help in deciding whether a network G has a prescribed property P? We study this question in the context of distributed local decision, where the objective is to decide whether G has property P by having each node run a constant-time distributed decision algorithm. In a yes-instance all nodes should output yes, while in a no-instance at least one node should output no. Recently, Fraigniaud et al. (OPODIS 2012) gave several conditions under which identifiers are not needed, and they conjectured that identifiers are not needed in any decision problem. In the present work, we disprove the conjecture. More than that, we analyse two critical variations of the underlying model of distributed computing: (B): the size of the identifiers is bounded by a function of the size of the input network, (-B): the identifiers are unbounded, (C): the nodes run a computable algorithm, (-C): the nodes can compute any, possibly uncomputable function. While it is easy to see that under (B, C) identifiers are not needed, we show that under all other combinations there are properties that can be decided locally if and only if identifiers are present.