We introduce three formal theories of increasing strength for linear algebra in order to study the complexity of the concepts needed to prove the basic theorems of the subject. We give what is apparently the first feasible proofs of the Cayley–Hamilton theorem and other properties of the determinant, and study the propositional proof complexity of matrix identities such as AB=I→BA=I.

We prove assorted properties of matrices over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Z}_{2}}$$\end{document}, and outline the complexity of the concepts required to prove these properties. The goal of this line of research is to establish the proof complexity of matrix algebra. It also presents a different approach to linear algebra: one that is formal, consisting in algebraic manipulations according to the axioms of a ring, rather than the traditional semantic approach via linear transformations.

Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

Pudlák shows that bounded arithmetic proves an upper bound on the Ramsey number Rr . We will strengthen this result by improving the bound. We also investigate lower bounds, obtaining a non-constructive lower bound for the special case of 2 colors , by formalizing a use of the probabilistic method. A constructive lower bound is worked out for the case when the monochromatic set size is fixed to 3 . The constructive lower bound is used to prove two “reversals”. To (...) explain this idea we note that the Ramsey upper bound proof for k=3 uses the weak pigeonhole principle in a significant way. The Ramsey upper bound proof for the case in which the upper bound is not explicitly mentioned, uses the totality of the exponentiation function in a significant way. It turns out that the Ramsey upper bounds actually imply the respective principles used to prove them, indicating that these principles were in some sense necessary. (shrink)