In this paper we study an extension of the Polynomial Calculus proof system where we can introduce new variables and take a square root. We prove that an instance of the subset-sum principle, the bit-value principle 1 + x 1 + 2x 2 + . . . 2 n−1 x n = 0 (BVP n ), requires refutations of exponential bit size over Q in this system.Part and Tzameret [18] proved an exponential lower bound on the size of Res-Lin (Resolution over linear equations [21]) refutations of BVP n . We show that our system p-simulates Res-Lin and thus we get an alternative exponential lower bound for the size of Res-Lin refutations of BVP n .
The (extended) Binary Value Principle (eBVP: n i=1 x i 2 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 [AGHT20, Ale21, PT21], and a polynomial simulation of a strong semialgebraic proof system in IPS+eBVP has been shown [AGHT20]. In this paper we consider 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 [GH03], 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 √ derivation of any such tautology from eBVP must be of exponential size.
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