We study first-order logic over unordered structures whose elements carry a finite number of data values from an infinite domain. Data values can be compared wrt.\ equality. As the satisfiability problem for this logic is undecidable in general, we introduce a family of local fragments. They restrict quantification to the neighbourhood of a given reference point that is bounded by some radius. Our first main result establishes decidability of the satisfiability problem for the local radius-1 fragment in presence of one ``diagonal relation''. On the other hand, extending the radius leads to undecidability. In a second part, we provide the precise decidability and complexity landscape of the satisfiability problem for the existential fragments of local logic, which are parameterized by the number of data values carried by each element and the radius of the considered neighbourhoods. Altogether, we draw a landscape of formalisms that are suitable for the specification of systems with data and open up new avenues for future research.

翻译：我们研究无序结构上的一阶逻辑，其中元素携带来自无限域的有限个数据值。数据值可相对于相等性进行比较。由于该逻辑的可满足性问题通常不可判定，我们引入了一族局部片段。这些片段将量化限制在以给定参考点为中心、由某一半径界定的邻域内。我们的第一个主要结果表明，在存在一个“对角关系”的情况下，局部半径为1的片段的可满足性问题是可判定的。另一方面，扩展半径会导致不可判定性。在第二部分中，我们提供了局部逻辑存在性片段的可满足性问题的精确可判定性与复杂性全景，这些片段由每个元素携带的数据值数量和所考虑邻域的半径参数化。总体而言，我们绘制了一组适用于带数据系统规范的形式化方法的全景图，并为未来研究开辟了新的途径。