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.


Hamiltonian Path Permutation Matrix Propositional Formula Graph Isomorphism Pigeonhole Principle 
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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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