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.
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References
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Soltys, M. (2004). LA, Permutations, and the Hajós Calculus. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds) Automata, Languages and Programming. ICALP 2004. Lecture Notes in Computer Science, vol 3142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27836-8_97
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DOI: https://doi.org/10.1007/978-3-540-27836-8_97
Publisher Name: Springer, Berlin, Heidelberg
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