Polylogarithmic Cuts in Models of V

We study initial cuts of models of weak two-sorted Bounded Arithmetics with respect to the strength of their theories and show that these theories are stronger than the original one. More explicitly we will see that polylogarithmic cuts of models of V are models of VNC by formalizing a proof of Nepomnjascij’s Theorem in such cuts. This is a strengthening of a result by Paris and Wilkie [17] and [18]. We can then exploit our result in Proof Complexity to observe that Frege proof systems can be sub exponentially simulated by bounded depth Frege proof systems. This result has recently been obtained by Filmus, Pitassi and Santhanam [10] in a direct proof. As an interesting observation we also obtain an average case separation of Resolution from AC-Frege by applying a recent result with Tzameret [15]. Mathematical Subject Classification 2000: 03B30, 03B70, 03C62, 03D15.