  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 $T^1_2$. 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 $T^1_2$ plus the weak pigeonhole principle. More generally, we
show that the universal variant of relativized $T^1_2$ 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 behaviour of the principles with
respect to finite structures that are only partially defined.
