Modulo-Counting First-Order Logic on Bounded Expansion Classes

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 count...

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Bibliographic Details
Main Authors Nesetril, J, de Mendez, P. Ossona, Siebertz, S
Format Journal Article
LanguageEnglish
Published 07.11.2022
Subjects
Computer Science - Logic in Computer Science
Mathematics - Combinatorics
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Summary: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.
DOI:10.48550/arxiv.2211.03704