Tradeoffs for small-depth Frege proofs

Abstract: We study the complexity of small-depth Frege proofs and give the first tradeoffs between the size of each line and the number of lines. Existing lower bounds apply to the overall proof size -- the sum of sizes of all lines -- and do not distinguish between these notions of complexity. For depth-$d$ Frege proofs of the Tseitin principle on the $n \times n$ grid where each line is a size-$s$ formula, we prove that $\exp(n/2^{\Omega(d\sqrt{\log s})})$ many lines are necessary. This yields new lower bounds on line complexity that are not implied by H{\aa}stad's recent $\exp(n^{\Omega(1/d)})$ lower bound on the overall proof size. For $s = \mathrm{poly}(n)$, for example, our lower bound remains $\exp(n^{1-o(1)})$ for all $d = o(\sqrt{\log n})$, whereas H{\aa}stad's lower bound is $\exp(n^{o(1)})$ once $d = \omega_n(1)$. Our main conceptual contribution is the simple observation that techniques for establishing correlation bounds in circuit complexity can be leveraged to establish such tradeoffs in proof complexity.