The power of the Binary Value Principle

Abstract

The (extended) Binary Value Principle ($\mathrm{eBVP}$, the equation ${\sum }_{i=1}^n{x}_i{2}^{i-1}=-k$ for $k>0$ and Boolean variables $x_i$) has received a lot of attention recently, several lower bounds have been proved for it [1], [2], [11]. Also it has been shown [1] that the probabilistically verifiable Ideal Proof System ($\mathrm{IPS}$) [8] together with $\mathrm{eBVP}$ polynomially simulates a similar semialgebraic proof system. In this paper we consider Polynomial Calculus with an algebraic version of Tseitin's extension rule ($\mathrm{Ext}\text{-}\mathrm{PC}$) that introduces a new variable for any polynomial. Contrary to $\mathrm{IPS}$, this is a Cook–Reckhow proof system. We show that in this context $\mathrm{eBVP}$ still allows to simulate similar semialgebraic systems. We also prove that it allows to simulate the Square Root Rule [6], which is in sharp contrast with the result of [2] that shows an exponential lower bound on the size of $\mathrm{Ext}\text{-}\mathrm{PC}$ derivations of the Binary Value Principle from its square. On the other hand, we demonstrate that $\mathrm{eBVP}$ probably does not help in proving exponential lower bounds for Boolean formulas: we show that an $\mathrm{Ext}\text{-}\mathrm{PC}$ (even with the Square Root Rule) derivation of any unsatisfiable Boolean formula in CNF from $\mathrm{eBVP}$ must be of exponential size.