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On the Width of Regular Classes of Finite Structures

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Automated Deduction – CADE 27 (CADE 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11716))

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Abstract

In this work we introduce the notion of decisional width of a finite relational structure and the notion of regular-decisional width of a regular class of finite structures. Our main result states that the first-order theory of any regular-decisional class of finite structures is decidable. Building on the proof of this decidability result, we show that the problem of counting satisfying assignments for a first-order logic formula in a structure of constant width is fixed parameter tractable when parameterized by the width parameter and can be solved in quadratic time with respect to the length of the input representation of the structure.

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Notes

  1. 1.

    MSO\(_1\) denotes the MSO logic of graphs where edge-set quantifications are not allowed, while MSO\(_2\) is the extension of MSO\(_1\) where we can also quantify over sets of edges.

  2. 2.

    Note that Theorem 5.2 implies that the first-order theory of unlabeled grids is decidable, since unlabeled grids have constant decisional-width. Nevertheless, it is known that the first-order theory of labeled grids is undecidable. In this latter case, we may have labeled grids that require ODDs of arbitrarily high width to be represented.

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Acknowledgments

Alexsander Andrade de Melo acknowledges support from the Brazilian agencies CNPq/GD 140399/2017-8 and CAPES/PDSE 88881.187636/2018-01; and Mateus de Oliveira Oliveira acknowledges support from the Bergen Research Foundation and from the Research Council of Norway.

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Authors and Affiliations

  1. Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Alexsander Andrade de Melo

  2. University of Bergen, Bergen, Norway

    Mateus de Oliveira Oliveira

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  1. Alexsander Andrade de Melo
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  2. Mateus de Oliveira Oliveira
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Correspondence to Mateus de Oliveira Oliveira .

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  1. University of Lorraine, Villers-lès-Nancy, France

    Pascal Fontaine

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Andrade de Melo, A., de Oliveira Oliveira, M. (2019). On the Width of Regular Classes of Finite Structures. In: Fontaine, P. (eds) Automated Deduction – CADE 27. CADE 2019. Lecture Notes in Computer Science(), vol 11716. Springer, Cham. https://doi.org/10.1007/978-3-030-29436-6_2

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  • DOI: https://doi.org/10.1007/978-3-030-29436-6_2

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