On matrix rank function over bounded arithmetics

Abstract

In [Mulmuley, 1987], Mulmuley gave an algorithm reducing the computation of the matrix rank function to that of determinants, of which the proof for the verification is elementary. In this article, we formalize this argument in the bounded arithmetic $LAP$; that is, we show that \[\det(AB)=\det(A)\det(B)\] for matrices $A,B$ with $mathbb{F}(X)$-coefficients implies \[rank(M)=dim(im M),\] where $\mathbb{F}$ is the universe of the field-sort of the theory, $M$ is a matrix with $\mathbb{F}$-coefficients, and $rank(M)$ is the rank function computed by Mulmuley's algorithm. Furthermore, interpreting $LAP$ by $VNC^{2}$ with $\mathbb{F}=\mathbb{Q}$ and using the result of [Tzameret \& Cook, 2021], we see that $VNC^{2}$ can formalize $rank(M)$ and prove $rank(M)=dim(im M)$. Lastly, we give several examples of combinatorial statements provable in $VNC^{2}$, using the formalized linear algebra.