Razborov [J. ACM, 2016] exhibited the following surprisingly strong trade-off phenomenon in propositional proof complexity: for a parameter $k = k(n)$, there exists $k$-CNF formulas over $n$ variables, having resolution refutations of $O(k)$ width, but every tree-like refutation of width $n^{1-\epsilon}/k$ needs size $\text{exp}\big(n^{\Omega(k)}\big)$. We extend this result to tree-like ... more >>>

The notion of closure of a set of linear forms, first introduced by Efremenko, Garlik, and Itsykson [EGI-STOC-24], has proven instrumental in proving lower bounds on the sizes of regular and bounded-depth Res($\oplus)$ refutations [EGI-STOC-24, AI-STOC-25]. In this work, we present amortized closure, an enhancement that retains the properties of ... more >>>