Lifting to Bounded-Depth and Regular Resolutions over Parities via Games

Proving superpolynomial lower bounds on proof size in the proof system resolution over parities (Res(λ•)) remains a significant open challenge. A recent breakthrough by Efremenko, Garlik, and Itsykson (STOC 2024) established an exponential lower bound for regular Res(λ•). In this work, we introduce a lifting technique for regular Res(λ•), applicable to a wide range of formulas. Specifically, we develop a method that transforms any formula with large resolution depth into a formula requiring exponential-size regular Res(λ•) refutations. This transformation is achieved through a combination of mixing and constant-size lifting. Using this approach, we provide an alternative and improved separation between resolution and regular Res(λ•), originally proved by Bhattacharya, Chattopadhyay, and Dvorak (CCC 2024). We construct an n-variable formula with a polynomial-size resolution refutation of depth O(√n), yet requires regular Res(λ•) refutations of size 2^ω(√n). Furthermore, we apply our technique to establish an exponential lower bound on the size of depth-cnloglogn Res(λ•) refutations, where n is the number of variables in the refuted formula, and c is a constant. The hard instances in this setting are Tseitin formulas lifted with the Maj₅ gadget. Since even depth-n Res(λ•) captures all possible definitions of regular Res(λ•), our result yields an exponential lower bound for top-regular Res(λ•), resolving an open question posed by Gryaznov, Pudlák, and Talebanfard (CCC 2022).