Archive for Mathematical Logic

, Volume 44, Issue 2, pp 195–208 | Cite as

Weak theories of linear algebra

  • Neil ThapenEmail author
  • Michael Soltys


We investigate the theories Open image in new window 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 Open image in new window 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, Open image in new window proves the commutativity of inverses.


Polynomial Time Mathematical Logic Linear Algebra Time Reasoning Matrix Inverse 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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


  1. 1.
    Berkowitz, S.J.: On computing the determinant in small parallel time using a small number of processors. Inf. Proc. Lett. 18 (3), 147–150 (1984)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bonet M.L., Buss, S.R., Pitassi, T.: Are there hard examples for frege proof systems? In: P. Clote, J. Remmel (eds.), Feasible mathematics II, Birkhauser, Boston, 1995, pp. 30–56Google Scholar
  3. 3.
    Buss, S.R.: Bounded Arithmetic. Bibliopolis, Naples, 1986Google Scholar
  4. 4.
    Cook, S.A.: A taxonomy of problems with fast parallel algorithms. Inf. Com. 64 (13), 2–22 (1985)zbMATHGoogle Scholar
  5. 5.
    Dummit, D.S., Foote, R.M.: Abstract algebra. Prentice Hall, 1991Google Scholar
  6. 6.
    Soltys, M.: The complexity of derivations of matrix identities. PhD thesis, University of Toronto, 2001. Available from the ECCC server;see
  7. 7.
    Soltys, M., Cook, S.A.: The proof complexity of linear algebra. In: Seventeenth Annual IEEE Symposium on Logic in Computer Science (LICS 2002). IEEE Computer Society, 2002Google Scholar
  8. 8.
    Zambella, D.: Notes on polynomially bounded arithmetic. J. Symbolic Logic 61 (3), 942–966 (1996)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.St Hilda’s CollegeOxfordUK
  2. 2.Department of Computing and SoftwareMcMaster UniversityHamiltonCanada

Personalised recommendations