For every $n >0$, we show the existence of a CNF tautology over $O(n^2)$ variables of width $O(\log n)$ such that it has a Polynomial Calculus Resolution refutation over $\{0,1\}$ variables of size $O(n^3polylog(n))$ but any Polynomial Calculus refutation over $\{+1,-1\}$ variables requires size $2^{\Omega(n)}$. This shows that Polynomial Calculus ... more >>>

In this paper, we obtain new size lower bounds for proofs in the
Polynomial Calculus (PC) proof system, in two different settings.

1. When the Boolean variables are encoded using $\pm 1$ (as opposed
to $0,1$): We establish a lifting theorem using an asymmetric gadget
$G$, showing ... more >>>

For every prime p > 0, every n > 0 and ? = O(logn), we show the existence
of an unsatisfiable system of polynomial equations over O(n log n) variables of degree O(log n) such that any Polynomial Calculus refutation over F_p with M extension variables, each depending on at ... more >>>

A major open problem in proof complexity is to prove super-polynomial lower bounds for AC^0[p]-Frege proofs. This system is the analog of AC^0[p], the class of bounded depth circuits with prime modular counting gates. Despite strong lower bounds for this class dating back thirty years (Razborov, '86 and Smolensky, '87), ... more >>>