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DOI: 10.4230/LIPIcs.CSL.2012.213

Definability of linear equation systems over groups and rings

Authors: Anuj Dawar, Erich Grädel, Bjarki Holm, Eryk Kopczynski, and Wied Pakusa

Published in: LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

Abstract

Motivated by the quest for a logic for PTIME and recent insights that the descriptive complexity of problems from linear algebra is a crucial aspect of this problem, we study the solvability of linear equation systems over finite groups and rings from the viewpoint of logical (inter-)definability. All problems that we consider are decidable in polynomial time, but not expressible in fixed-point logic with counting. They also provide natural candidates for a separation of polynomial time from rank logics, which extend fixed-point logics by operators for determining the rank of definable matrices and which are sufficient for solvability problems over fields. Based on the structure theory of finite rings, we establish logical reductions among various solvability problems. Our results indicate that all solvability problems for linear equation systems that separate fixed-point logic with counting from PTIME can be reduced to solvability over commutative rings. Further, we prove closure properties for classes of queries that reduce to solvability over rings. As an application, these closure properties provide normal forms for logics extended with solvability operators.

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Anuj Dawar, Erich Grädel, Bjarki Holm, Eryk Kopczynski, and Wied Pakusa. Definability of linear equation systems over groups and rings. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 213-227, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{dawar_et_al:LIPIcs.CSL.2012.213,
  author =	{Dawar, Anuj and Gr\"{a}del, Erich and Holm, Bjarki and Kopczynski, Eryk and Pakusa, Wied},
  title =	{{Definability of linear equation systems over groups and rings}},
  booktitle =	{Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL},
  pages =	{213--227},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-42-2},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{16},
  editor =	{C\'{e}gielski, Patrick and Durand, Arnaud},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.213},
  URN =		{urn:nbn:de:0030-drops-36749},
  doi =		{10.4230/LIPIcs.CSL.2012.213},
  annote =	{Keywords: inite model theory, logics with algebraic operators}
}

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Definability of linear equation systems over groups and rings

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