Modulo-Counting First-Order Logic on Bounded Expansion Classes

J. Nesetril; P. Ossona de Mendez; S. Siebertz

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arXiv · 2211.03704

Modulo-Counting First-Order Logic on Bounded Expansion Classes

J. Nesetril·P. Ossona de Mendez·S. Siebertz

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Abstract

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.

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J. Nesetril, P. Ossona de Mendez, S. Siebertz. 2022-11-07. Modulo-Counting First-Order Logic on Bounded Expansion Classes. https://arxiv.org/abs/2211.03704

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