The canonical pair of a proof system P is the pair of disjoint NP sets where one set is the set of all satisfiable CNF formulas and the other is the set of CNF formulas that have P-proofs bounded by some polynomial. We give a combinatorial characterization of the canonical pairs of depth d Frege systems. Our characterization is based on certain games, introduced in this article, that are parametrized by a number k, also called the depth. We show that the canonical pair of a depth d Frege system is polynomially equivalent to the pair $(A_{d+2},B_{d+2})$ where $A_{d+2}$ (respectively, $B_{d+1}$) are depth $d+1$ games in which Player I (Player II) has a positional winning strategy. Although this characterization is stated in terms of games, we will show that these combinatorial structures can be viewed as generalizations of monotone Boolean circuits. In particular, depth 1 games are essentially monotone Boolean circuits. Thus we get a generalization of the monotone feasible interpolation for Resolution, which is a property that enables one to reduce the task of proving lower bounds on the size of refutations to lower bounds on the size of monotone Boolean circuits. However, we do not have a method yet for proving lower bounds on the size of depth d games for $d>1$.

证明系统P的规范对是一对不相交的NP集，其中一个集合是所有可满足的CNF公式的集合，另一个是具有P证明以某个多项式为边界的CNF公式的集合。我们给出深度d Frege系统的规范对的组合特征。我们的表征是基于本文介绍的某些游戏，这些游戏由数字k（也称为深度）参数化。我们证明深度d Frege系统的规范对与该对在多项式上等效$（{\mathrm{一种}}_{d+2}，{乙}_{d+2}）$ 哪里 ${\mathrm{一种}}_{d+2}$ （分别， ${乙}_{d+\mathrm{1个}}$）是深度 $d+\mathrm{1个}$玩家I（玩家II）具有定位获胜策略的游戏。尽管这种特征是用游戏来描述的，但我们将证明这些组合结构可以看作是单调布尔电路的概括。特别是，深度1游戏本质上是单调布尔电路。因此，我们得到了用于分辨率的单调可行插值的一般化，它是一种特性，使人们可以减少将证明大小的下限证明为单调布尔电路大小的下限的任务。但是，我们还没有一种方法可以证明深度d游戏的大小的下限$d>\mathrm{1个}$。