Constant time testability of first-order logic with modulo counting on finitary graphs
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arXiv
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| Format: | Preprint |
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2026
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| author | Adler, Isolde Stimpson, Jenny |
| author_facet | Adler, Isolde Stimpson, Jenny |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_10841 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Constant time testability of first-order logic with modulo counting on finitary graphs Adler, Isolde Stimpson, Jenny Logic in Computer Science Logic F.4.1; F.2.2 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. |
| title | Constant time testability of first-order logic with modulo counting on finitary graphs |
| topic | Logic in Computer Science Logic F.4.1; F.2.2 |
| url | https://arxiv.org/abs/2605.10841 |