Weak theories of linear algebra
- 32 Downloads
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.
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.
- 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.Buss, S.R.: Bounded Arithmetic. Bibliopolis, Naples, 1986Google Scholar
- 5.Dummit, D.S., Foote, R.M.: Abstract algebra. Prentice Hall, 1991Google Scholar
- 6.Soltys, M.: The complexity of derivations of matrix identities. PhD thesis, University of Toronto, 2001. Available from the ECCC server;see http://www.eccc.uni-trier.de/eccc-local/ECCC-Theses/soltys.html
- 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