Polylogarithmic Cuts in Models of V^0

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 $\mathbf{V}^0$ are models of $\mathbf{VNC}^1$ by formalizing a proof of Nepomnjascij's Theorem in such cuts. This is a strengthening of a result by Paris and Wilkie. 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 in a direct proof. As an interesting observation we also obtain an average case separation of Resolution from AC0-Frege by applying a recent result with Tzameret.Comment: 16 page

Classes of representable disjoint NP-pairs

For a propositional proof system P we introduce the complexity class of all disjoint -pairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make canonical -pairs associated with these proof systems hard or complete for . Moreover, we demonstrate that non-equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for and the reductions between the canonical pairs exist