Tradeoffs for small-depth Frege proofs

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 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
$\mathbf{H}\mathop{\mathbf{a}}\!\!\!\!^{\circ}\mathbf{stad}$
's recent
$\exp(n^{\Omega(1/d)})$
lower bound on the overall proof size. For
$s$
= poly
$(n)$
, for example, our lower bound remains
$\exp(n^{1-o(1)})$
for all
$d=o(\sqrt{\log n})$
, whereas
$\mathbf{H}\mathop{\mathbf{a}}\!\!\!\!^{\circ}\mathbf{stad}$
's lower bound is
$\exp(n^{o(1)})$
once
$d\ = \omega_{n}(1)$
. Our main conceptual contribution is the simple obser-vation that techniques for establishing correlation bounds in circuit complexity can be leveraged to establish such tradeoffs in proof complexity.