We prove that, on bounded expansion classes, every first-order formula with modulo counting is equivalent, in a linear-time computable monadic expansion, to an existential first-order formula. As a consequence, we derive, on bounded expansion classes, that first-order transductions with modulo counting have the same encoding power as existential first-order transductions. Also, modulo-counting first-order model checking and computation of the size of sets definable in modulo-counting first-order logic can be achieved in linear time on bounded expansion classes. As an application, we prove that a class has structurally bounded expansion if and only if it is a class of bounded depth vertex-minors of graphs in a bounded expansion class. We also show how our results can be used to implement fast matrix calculus on bounded expansion matrices over a finite field.

翻译：我们证明，在有界扩展类上，每个带模计数的一阶公式在线性时间可计算的幺半扩张中等价于一个存在性一阶公式。由此可得，在有界扩展类上，带模计数的一阶转录与存在性一阶转录具有相同的编码能力。此外，在有界扩展类上，模计数一阶逻辑的模型检测以及可在模计数一阶逻辑中定义的集合大小的计算均可在线性时间内完成。作为应用，我们证明一个类具有结构上有界扩展当且仅当它是有界扩展类中图的有限深度顶点子图类。我们还展示了如何将我们的结果用于在有界扩展矩阵上实现有限域上的快速矩阵计算。