Linear Branching Programs and Directional Affine Extractors

Authors Svyatoslav Gryaznov , Pavel Pudlák, Navid Talebanfard

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Svyatoslav Gryaznov
  • Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic
Pavel Pudlák
  • Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic
Navid Talebanfard
  • Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic


We wish to thank Swastik Kopparty for useful correspondence regarding affine extractors.

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Svyatoslav Gryaznov, Pavel Pudlák, and Navid Talebanfard. Linear Branching Programs and Directional Affine Extractors. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


A natural model of read-once linear branching programs is a branching program where queries are 𝔽₂ linear forms, and along each path, the queries are linearly independent. We consider two restrictions of this model, which we call weakly and strongly read-once, both generalizing standard read-once branching programs and parity decision trees. Our main results are as follows. - Average-case complexity. We define a pseudo-random class of functions which we call directional affine extractors, and show that these functions are hard on average for the strongly read-once model. We then present an explicit construction of such function with good parameters. This strengthens the result of Cohen and Shinkar (ITCS'16) who gave such average-case hardness for parity decision trees. Directional affine extractors are stronger than the more familiar class of affine extractors. Given the significance of these functions, we expect that our new class of functions might be of independent interest. - Proof complexity. We also consider the proof system Res[⊕], which is an extension of resolution with linear queries, and define the regular variant of Res[⊕]. A refutation of a CNF in this proof system naturally defines a linear branching program solving the corresponding search problem. If a refutation is regular, we prove that the resulting program is read-once. Conversely, we show that a weakly read-once linear BP solving the search problem can be converted to a regular Res[⊕] refutation with constant blow up, where the regularity condition comes from the definition of weakly read-once BPs, thus obtaining the equivalence between these proof systems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Computational complexity and cryptography
  • Boolean Functions
  • Average-Case Lower Bounds
  • AC0[2]
  • Affine Dispersers
  • Affine Extractors


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