Lower Bounds for Regular Resolution over Parities

Klim Efremenko, Michal Garlík, Dmitry Itsykson

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


The proof system resolution over parities (Res(⊕)) operates with disjunctions of linear equations (linear clauses) over GF(2); it extends the resolution proof system by incorporating linear algebra over GF(2). Over the years, several exponential lower bounds on the size of tree-like refutations have been established. However, proving a superpolynomial lower bound on the size of dag-like Res(⊕) refutations remains a highly challenging open question. We prove an exponential lower bound for regular Res(⊕). Regular Res(⊕) is a subsystem of dag-like Res(⊕) that naturally extends regular resolution. This is the first known superpolynomial lower bound for a fragment of dag-like Res(⊕) which is exponentially stronger than tree-like Res(⊕). In the regular regime, resolving linear clauses C1 and C2 on a linear form f is permitted only if, for both i∈ {1,2}, the linear form f does not lie within the linear span of all linear forms that were used in resolution rules during the derivation of Ci. Namely, we show that the size of any regular Res(⊕) refutation of the binary pigeonhole principle BPHPnn+1 is at least 2Ω(3√n/logn). A corollary of our result is an exponential lower bound on the size of a strongly read-once linear branching program solving a search problem. This resolves an open question raised by Gryaznov, Pudlak, and Talebanfard (CCC 2022). As a byproduct of our technique, we prove that the size of any tree-like Res(⊕) refutation of the weak binary pigeonhole principle BPHPnm is at least 2Ω(n) using Prover-Delayer games. We also give a direct proof of a width lower bound: we show that any dag-like Res(⊕) refutation of BPHPnm contains a linear clause C with Ω(n) linearly independent equations.

Original languageAmerican English
Title of host publicationSTOC 2024 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing
EditorsBojan Mohar, Igor Shinkar, Ryan O�Donnell
Number of pages12
ISBN (Electronic)9798400703836
StatePublished - 10 Jun 2024
Event56th Annual ACM Symposium on Theory of Computing, STOC 2024 - Vancouver, Canada
Duration: 24 Jun 202428 Jun 2024

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing


Conference56th Annual ACM Symposium on Theory of Computing, STOC 2024


  • binary pigeonhole principle
  • lower bounds
  • proof complexity
  • regular resolution
  • resolution over linear equations

All Science Journal Classification (ASJC) codes

  • Software


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