LA, Permutations, and the Hajós Calculus

  • Michael Soltys
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3142)


LA is a simple and natural field independent system for reasoning about matrices. We show that LA extended to contain a matrix form of the pigeonhole principle is strong enough to prove a host of matrix identities (so called “hard matrix identities” which are candidates for separating Frege and extended Frege). LAP is LA with matrix powering; we show that LAP extended with quantification over permutations is strong enough to prove theorems such as the Cayley-Hamilton Theorem. Furthermore, we show that LA extended with quantification over permutations expresses NP graph-theoretic properties, and proves the soundness of the Hajós calculus. A corollary is that a fragment of Quantified Permutation Frege (a novel propositional proof system that we introduce in this paper) is p-equivalent of extended Frege. Several open problems are stated.


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  1. 1.
    Stephen, A.: Cook and Michael Soltys. The proof complexity of linear algebra. In: Seventeenth Annual IEEE Symposium on Logic in Computer Science, LICS 2002 (2002)Google Scholar
  2. 2.
    Krajíček, J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory. Cambridge (1995)Google Scholar
  3. 3.
    Pitassi, T., Urquhart, A.: The complexity of the Hajós calculus. SIAM J. Disc. Math. 8(3), 464–483 (1995)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Soltys, M.: The Complexity of Derivations of Matrix Identities. PhD thesis, University of Toronto (2001)Google Scholar
  5. 5.
    Soltys, M., Cook, S.: The complexity of derivations of matrix identities. To appear in the Annals of Pure and Applied Logic (2004)Google Scholar
  6. 6.
    Soltys, M., Urquhart, A.: Matrix identities and the pigeonhole principle. Archive for Mathematical Logic 43(3), 351–357 (2004)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Michael Soltys
    • 1
  1. 1.Department of Computing and SoftwareMcMaster UniversityHamiltonCANADA

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