We exhibit a monotone function computable by a monotone circuit of quasipolynomial size such that any monotone circuit of polynomial depth requires exponential size. This is the first size-depth tradeoff result for monotone circuits in the so-called supercritical regime. Our proof is based on an analogous result in proof complexity: We introduce a new family of unsatisfiable 3-CNF formulas (called bracket formulas) that admit resolution refutations of quasipolynomial size while any refutation of polynomial depth requires exponential size.

翻译：我们展示了一个可由拟多项式规模的单调电路计算的单调函数，使得任何多项式深度的单调电路都需要指数规模。这是单调电路在所谓超临界区域中的首个规模-深度权衡结果。我们的证明基于证明复杂性中的一个类似结果：我们引入了一类新的不可满足3-CNF公式（称为括号公式），这类公式允许拟多项式规模的消解反驳，而任何多项式深度的反驳都需要指数规模。