We initiate the study of the meta-mathematics of algebraic circuit lower bounds, aiming both to gain insight into the methods sufficient and necessary to prove algebraic circuit lower bounds, and to contribute to the study of bounded arithmetic as a logical foundation for complexity lower bounds. We demonstrate that while ... more >>>

Given two symbolic matrices $X$ and $Y$ of dimensions $m\times n$ and $n\times m$, respectively, the *rank principle* states that when $m = n+1$ and $A$ is a scalar matrix of rank $n+1$, the equation $XY = A$ is unsatisfiable. When $m$ is arbitrarily larger than $n$ and $A$ has ... more >>>

The proof system resolution over parities (Res($\oplus$)) operates with disjunctions of linear equations (linear clauses) over $\mathbb{F}_2$; it extends the resolution proof system by incorporating linear algebra over $\mathbb{F}_2$. Over the years, several exponential lower bounds on the size of tree-like Res($\oplus$) refutations have been established. However, proving a superpolynomial ... more >>>

We show that for every integer $k \geq 2$, the Res($k$) propositional proof system does not have the weak feasible disjunction property. Next, we generalize a recent result of Atserias and Müller [FOCS, 2019] to Res($k$). We show that if NP is not included in P (resp. QP, SUBEXP) then ... more >>>

For any unsatisfiable CNF formula we give an exponential lower bound on the size of resolution refutations of a propositional statement that the formula has a resolution refutation. We describe three applications. (1) An open question in [Atserias-Müller,2019] asks whether a certain natural propositional encoding of the above statement is ... more >>>