Itsykson and Sokolov identified resolution over parities, denoted by $\text{Res}(\oplus)$, as a natural and simple fragment of $\text{AC}^0[2]$-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 $\text{exp}(N^{\Omega(1)})$, on the size ... more >>>

In an attempt to show that the acceptance probability of a quantum query algorithm making $q$ queries can be well-approximated almost everywhere by a classical decision tree of depth $\leq \text{poly}(q)$, Aaronson and Ambainis proposed the following conjecture: let $f: \{ \pm 1\}^n \rightarrow [0,1]$ be a degree $d$ polynomial ... more >>>

Proving super-polynomial lower bounds on the size of proofs of unsatisfiability of Boolean formulas using resolution over parities, is an outstanding problem that has received a lot of attention after its introduction by Raz and Tzamaret (2008). Very recently, Efremenko, Garlik and Itsykson (2023) proved the first exponential lower bounds ... more >>>