The question of "what can be computed locally?" lies at the heart of distributed computing in networks. As established in Naor and Stockmeyer's seminal paper (STOC 1993, Edsger W. Dijkstra Prize in Distributed Computing 2025), this question is undecidable, even for graph problems whose solutions can be checked locally. In this paper, we adopt a novel perspective on the question, by asking for which classes Π of problems, and for which classes G of graphs, all problems in Π can be solved efficiently in a distributed manner in all graphs of G. This paper focuses on two natural candidates for such an approach, namely the class of problems expressible in first-order logic (FO), because they possess an intrinsic form of locality thanks to Gaifman's theorem, and the class of graphs with bounded expansion, because they form a large class of graphs encompassing, e.g., planar, bounded-genus, bounded-treewidth, and bounded-degree graphs, as well as graphs excluding a fixed minor or topological minor, sparse Erdös--Rényi graphs (a.a.s.), and several network models such as stochastic block models for suitable parameter ranges.

This paper explores the relationship between broadcast abstractions and the k-set agreement (k-SA) problem in crash-prone asynchronous message-passing distributed systems. It specifically investigates whether any broadcast abstraction is computationally equivalent to k-SA in message-passing systems. A key contribution of the paper is the introduction of a clear definition of admissible broadcast abstractions, achieved by introducing two new symmetry properties: compositionality and content-neutrality. The paper's primary contribution is the demonstration that no broadcast abstraction, which is both content-neutral and compositional, is computationally equivalent to k-set agreement when 1 < k < n.