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Weak theories of linear algebra

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Abstract.

We investigate the theories of linear algebra, which were originally defined to study the question of whether commutativity of matrix inverses has polysize Frege proofs. We give sentences separating quantified versions of these theories, and define a fragment in which we can interpret a weak theory V1 of bounded arithmetic and carry out polynomial time reasoning about matrices - for example, we can formalize the Gaussian elimination algorithm. We show that, even if we restrict our language, proves the commutativity of inverses.

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Acknowledgments.

The authors would like to thank Steve Cook for the very helpful conversations that led to this work.

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Authors and Affiliations

  1. St Hilda’s College, Cowley Place, Oxford, OX4 1DY, UK

    Neil Thapen

  2. Department of Computing and Software, McMaster University, 1280 Main Street West, Hamilton, Ontario, L8S 4K1, Canada

    Michael Soltys

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  1. Neil Thapen
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  2. Michael Soltys
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Correspondence to Neil Thapen.

Additional information

This work was done while a postdoctoral research fellow at the Department of Computer Science, University of Toronto, Canada.

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Thapen, N., Soltys, M. Weak theories of linear algebra. Arch. Math. Logic 44, 195–208 (2005). https://doi.org/10.1007/s00153-004-0249-8

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  • DOI: https://doi.org/10.1007/s00153-004-0249-8

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