This paper studies algorithmic meta theorems for property testing with \emph{constant running time} in the bounded degree model. In (Adler, Harwath 2018) it was shown that on graph classes $\mathcal C^{w}_d$ consisting of all graphs with both degree at most $d$ and treewidth at most $w$, every problem expressible in monadic second-order logic with counting (CMSO) is testable with \emph{polylogarithmic} running time (where $d,w\in \mathbb N$ are fixed). It was left open whether this can be improved to \emph{constant} running time. In this paper we give a positive answer for testing CMSO on classes $\mathcal C^{c}_d$, where $d$ bounds the degree and $c$ bounds the component size. Our main result shows constant time testability of first-order logic with modulo counting (FOMOD) on $\mathcal C^{c}_d$. For our proof we tailor Hanf normal form of FOMOD to our setting, and we exhibit a number-theoretic `patchability' condition that allows to infer global information on the input graph from a local sample of constant size. We believe that our `patchability' might be of independent interest. The step from FOMOD to CMSO then follows from a result by (Eickmeyer, Elberfeld, Harwath, 2017) on the expressive power of order invariant monadic second-order logic on classes of bounded treedepth.

翻译：本文研究有界度模型中具有**常数运行时间**的性质测试的算法元定理。在Adler与Harwath（2018）的工作中，他们证明了：对于由度数不超过d且树宽不超过w的所有图构成的图类$\mathcal C^{w}_d$（其中$d,w\in \mathbb N$为固定常数），每个可用带计数的一元二阶逻辑（CMSO）表达的问题均可在**多对数**运行时间内被测试。但能否将运行时间改进为**常数**的问题仍悬而未决。本文对组件大小有界且度数有界的图类$\mathcal C^{c}_d$给出了肯定答案：CMSO可在该类上被常数时间测试。我们的主要结果证明了：一阶模计数逻辑（FOMOD）在$\mathcal C^{c}_d$上具有常数时间可测试性。在证明中，我们将FOMOD的Hanf范式定制到当前场景，并提出一种数论上的“可修补性”条件，该条件允许从常数大小的局部样本推断输入图的全局信息。我们相信，“可修补性”本身可能具有独立的研究价值。从FOMOD到CMSO的跃迁则基于Eickmeyer、Elberfeld与Harwath（2017）关于有界树深类上序不变一元二阶逻辑表达能力的结论。