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LA, Permutations, and the Hajós Calculus

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

Abstract

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.

Keywords

Hamiltonian Path Permutation Matrix Propositional Formula Graph Isomorphism Pigeonhole Principle 
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.

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References

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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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