Typical forcings, NP search problems and an extension of a theorem of Riis

We define typical forcings encompassing many informal forcing arguments in bounded arithmetic and give general conditions for such to produce models of the universal variant relativized T21. apply this result study relative complexity total (type 2) NP search problems associated finitary combinatorial principles. Complexity theory compares with respect polynomial time many-one or Turing reductions. From a logical perspective are graded according theories that prove their totality. The analogue reduction is totality one problem from another. link between two perspectives tight what we call variants arithmetics. strengthen theorem Buss Johnson (2012) infers depth Frege proofs reducibility. As an application our method derive strong form Riis' finitization (1993). extend it by exhibiting simple model-theoretic property implies independence T21 plus weak pigeonhole principle. More generally, show does not (the to) principle one. Being properties based on behavior principles finite structures only partially defined.