Anuj Dawar ; Eryk Kopczynski ; Bjarki Holm ; Erich Grädel ; Wied Pakusa - Definability of linear equation systems over groups and rings

lmcs:725 - Logical Methods in Computer Science, November 14, 2013, Volume 9, Issue 4 - https://doi.org/10.2168/LMCS-9(4:12)2013
Definability of linear equation systems over groups and rings

Authors: Anuj Dawar ORCID-iD; Eryk Kopczynski ORCID-iD; Bjarki Holm ; Erich Grädel ; Wied Pakusa

    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. Moreover, we prove closure properties for classes of queries that reduce to solvability over rings, which provides normal forms for logics extended with solvability operators. We conclude by studying the extent to which fixed-point logic with counting can express problems in linear algebra over finite commutative rings, generalising known results on the logical definability of linear-algebraic problems over finite fields.


    Volume: Volume 9, Issue 4
    Published on: November 14, 2013
    Accepted on: June 25, 2015
    Submitted on: January 29, 2013
    Keywords: Computer Science - Logic in Computer Science,Computer Science - Computational Complexity
    Fundings :
      Source : OpenAIRE Research Graph
    • Descriptive Complexity with Algebraic Operators; Funder: UK Research and Innovation; Code: EP/H026835/1

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    Source : ScholeXplorer IsCitedBy DOI 10.4230/lipics.mfcs.2021.37
    Source : ScholeXplorer IsCitedBy HANDLE 20.500.11820/cbbc77b0-6ef2-41df-8549-cd67bdf1a34d
    • 10.4230/lipics.mfcs.2021.37
    • 20.500.11820/cbbc77b0-6ef2-41df-8549-cd67bdf1a34d
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