Itsykson and Sokolov identified resolution over parities, denoted by Res (), as a natural and simple fragment of AC⁰2-Frege for which no super-polynomial lower bounds on size of proofs are known. Building on a recent line of work, Efremenko and Itsykson proved lower bounds of the form exp (N^Ω (1) ), on the size of Res () proofs whose depth is upper bounded by O (N N), where N is the number of variables of the unsatisfiable CNF formula. The hard formula they used was Tseitin on an appropriately expanding graph, lifted by a 2-stifling gadget. They posed the natural problem of proving super-polynomial lower bounds on the size of proofs that are Ω (N^1+ε) deep, for any constant ε> 0. We provide a significant improvement by proving a lower bound on size of the form exp (Ω (N^ε) ), as long as the depth of the Res () proofs are O (N^2-ε), for every ε> 0. Our hard formula is again Tseitin on an expander graph, albeit lifted with a different type of gadget. Our gadget needs to have small correlation with all parities. An important ingredient in our work is to show that arbitrary distributions lifted with such gadgets fool safe affine spaces, an idea which originates in the earlier work of Bhattacharya, Chattopadhyay and Dvorak.
Bhattacharya et al. (Wed,) studied this question.