Exponential separation between $\mathrm{Res}(k)$ and $\mathrm{Res}(k+1)$ for $k⩽\varepsilon \log n$

Abstract

$\mathrm{Res}(k)$ is a propositional proof system that extends resolution by working with k-DNFs instead of clauses. We show that there exist constants $\beta \text{,}\gamma >0$ so that if k is a function from positive integers to positive integers so that for all n, $k(n)⩽\beta \log n$, then for each n, there exists a set of clauses ${\mathcal{C}}_n$ of size $n^{\mathrm{O}(1)}$ that has $\mathrm{Res}(k(n)+1)$ refutations of size $n^{\mathrm{O}(1)}$, yet every $\mathrm{Res}(k(n))$ refutation of ${\mathcal{C}}_n$ has size at least $2^{n^{\gamma }}$.