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Definability of linear equation systems over groups and rings Dawar, Anuj Grädel, Erich Holm, Bjarki Kopczynski, Eryk Pakusa, Wied inite model theory logics with algebraic operators 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Anuj Dawar and Erich Grädel and Bjarki Holm and Eryk Kopczynski and Wied Pakusa 2012 Is Part Of LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CSL.2012.213 urn:nbn:de:0030-drops-36749 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.213 eng https://creativecommons.org/licenses/by-nc-nd/3.0/legalcode