Developing Takeuti-Yasumoto forcing

In late 90’s G.Takeuti and Y.Yasumoto gave forcing constructions for bounded arithmetic. We will reformulate their constructions using two-sort bounded arithmetic and prove the followings. 1. Generic extensions are related with P=NP problem. 2. J.Krajicek’s forcing constructions can be given as Takeuti-Yasumoto forcing. 3. We can either satisfy or falsify the dual weak pigeonhole principles in generic extensions.