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). Also it has been shown (Alekseev et al 2020) that the probabilistically verifiable Ideal Proof System (IPS) (Grochow and Pitassi 2018) together with eBVP polynomially simulates a similar semialgebraic proof system. In this paper we consider Polynomial Calculus with the algebraic version of Tseitin's extension rule (Ext-PC). 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 in sharp contrast with the result of (Alekseev 2021) that shows an exponential lower bound on the size of Ext-PC derivations of the Binary Value Principle from its square. On the other hand, we demonstrate that eBVP probably does not help in proving exponential lower bounds for Boolean formulas: we show that an Ext-PC (even with the Square Root Rule) derivation of any unsatisfiable Boolean formula in CNF from eBVP must be of exponential size.

翻译：(extend) 二进制值原则 (eBVP:$\sum ⁇ i=1 ⁇ n x_i2 ⁇ i-1} = -k$, $k>0美元和$x%2_i=x_i美元) 最近受到了很多关注。 在此纸张中, 我们考虑了若干下限, 包括了 Tseitin 扩展规则( Ext- PC) 的代数版本的多元计算。 与 IPS 相反, 这是 Cook- Reckhow 校验系统。 在此情况下, eBVP 仍然可以模拟类似的 e- e- gregic 系统 (Grocchow 和 Pitassi 201818) 和 eBVP 多边规则 模拟类似的半成型数校正校正校程系统。 我们还证明, 在 IMAL Excal Excion C 的代数中, 它可以模拟 20 的直径的直立度 。