Complexity of Semialgebraic Proofs with Restricted Degree of Falsity

A weakened version of the Cutting Plane (CP) proof system with a restriction on the degree of falsity of intermediate inequalities was introduced by Goerdt. He proved an exponential lower bound for CP proofs with degree of falsity bounded by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\frac{n}{\log^{2n}+1}}$\end{document}, where n is the number of variables. Hirsch and Nikolenko strengthened this result by establishing a direct connection between CP and Res(k) proofs. This result implies an exponential lower bound on the proof length of the Tseitin-Urquhart tautologies, when the degree of falsity is bounded by cn for some constant c. In this paper we generalize this result for extensions of Lovász-Schrijver calculi (LS), namely for LSk+CPk proof systems introduced by Grigoriev et al. We show that any LSk+CPk proof with bounded degree of falsity can be transformed into a Res(k) proof. We also prove lower and upper bounds for the new system.