Constant time testability of first-order logic with modulo counting on finitary graphs

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.