The power of the Binary Value Principle

The (extended) Binary Value Principle (eBVP: $\sum_{i=1}^n x_i2^{i-1} = -k$ for $k>0$ and $x^2_i=x_i$) has received a lot of attention recently: several lower bounds have been proved for it (Alekseev et al 2020, Alekseev 2021, Part and Tzameret 2021), and a polynomial simulation of a strong semialgebraic proof system in IPS+eBVP has been shown (Alekseev et al 2020). In this paper we consider Ext-PC: Polynomial Calculus with the algebraic version of Tseitin's extension rule. Contrary to IPS, this is a Cook--Reckhow proof system. We show that in this context eBVP still allows to simulate similar semialgebraic systems. We also prove that it allows to simulate the Square Root Rule (Grigoriev and Hirsch 2003), which is absolutely unclear in the context of ordinary Polynomial Calculus. On the other hand, we demonstrate that eBVP probably does not help in proving exponential lower bounds for Boolean tautologies: we show that an Ext-PC (even with the Square Root Rule) derivation of any such tautology from eBVP must be of exponential size.