We define typical forcings encompassing many informal forcing arguments in bounded arithmetic and give general conditions for such forcings to produce models of the universal variant of relativized ${\mathsf{T}}_2^1$. We apply this result to study the relative complexity of total (type 2) NP search problems associated to finitary combinatorial principles.

Complexity theory compares such problems with respect to polynomial time many-one or Turing reductions. From a logical perspective such problems are graded according to the bounded arithmetic theories that prove their totality. The logical analogue of a reduction is to prove the totality of one problem from the totality of another. The link between the two perspectives is tight for what we call universal variants of relativized bounded arithmetics. We strengthen a theorem of Buss and Johnson (2012) that infers relative bounded depth Frege proofs of totality from polynomial time Turing reducibility.

As an application of our general forcing method we derive a strong form of Riis' finitization theorem (1993). We extend it by exhibiting a simple model-theoretic property that implies independence from the universal variant of relativized ${\mathsf{T}}_2^1$ plus the weak pigeonhole principle. More generally, we show that the universal variant of relativized ${\mathsf{T}}_2^1$ does not prove (the totality of the total NP search problem associated to) a strong finitary combinatorial principle from a weak one. Being weak or strong are simple model-theoretic properties based on the behavior of the principles with respect to finite structures that are only partially defined.

我们定义了有界算术中包含许多非正式强迫论证的典型强迫，并为此类强迫产生了广义相对论模型提供了一般条件 ${\mathsf{Ť}}_2^{\mathrm{1个}}$。我们将此结果用于研究与最终组合原则相关的总（类型2）NP搜索问题的相对复杂性。

复杂度理论将这些问题与多项式时间多一或图灵约简进行了比较。从逻辑的角度来看，这些问题是根据证明其整体性的有限算术理论进行分级的。归约的逻辑模拟是从一个问题的总和中证明另一个问题的总和。对于我们称为相对化有界算术的通用变体，这两种观点之间的联系是紧密的。我们加强了Buss and Johnson（2012）的一个定理，该定理从多项式时间Turing可归约性推断总体的相对有界深度弗雷格证明。

作为一般强迫方法的一种应用，我们得出了Riis的定理定理（1993）的一种强形式。我们通过展示一个简单的模型理论性质来扩展它，该性质暗示了相对于相对化的普遍变种的独立性${\mathsf{Ť}}_2^{\mathrm{1个}}$加上弱信鸽原理。更笼统地说，我们表明相对论的普遍变体${\mathsf{Ť}}_2^{\mathrm{1个}}$不能证明一个弱的强联合组合原理（总的NP搜索问题的总和）。弱或强是简单的模型理论性质，基于关于不完整定义的有限结构的原理的行为。