Gryaznov, S., Pudlák, P. and Talebanfard, N. (2022) Linear branching programs and directional affine extractors. In: Lovett, S., (ed.) 37th Computational Complexity Conference (CCC 2022). 37th Computational Complexity Conference (CCC 2022), 20-23 Jul 2022, Philadelphia, Pennsylvania, USA. Leibniz International Proceedings in Informatics (LIPIcs), 234 . Schloss Dagstuhl - Leibniz-Zentrum für Informatik , 4:1-4:16. ISBN 9783959772419
Abstract
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.
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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Editors: |
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Copyright, Publisher and Additional Information: | © 2022 Svyatoslav Gryaznov, Pavel Pudlák, and Navid Talebanfard. Licensed under Creative Commons License CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/) |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 18 Oct 2023 15:17 |
Last Modified: | 18 Oct 2023 15:17 |
Status: | Published |
Publisher: | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |
Series Name: | Leibniz International Proceedings in Informatics (LIPIcs) |
Refereed: | Yes |
Identification Number: | 10.4230/LIPIcs.CCC.2022.4 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:204340 |