Polylogarithmic Cuts in Models of V^0

TL;DR

This paper demonstrates that polylogarithmic cuts of models of V^0 are stronger than the original models, leading to implications in proof complexity such as sub-exponential simulation of Frege systems.

Contribution

It formalizes a proof that polylogarithmic cuts of V^0 models are models of VNC^1, strengthening previous results and connecting model theory with proof complexity.

Findings

Polylogarithmic cuts of V^0 models are models of VNC^1.

Frege proof systems can be sub-exponentially simulated by bounded depth Frege.

An average case separation of Resolution from AC0-Frege is established.

Abstract

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.