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DOI: 10.4230/LIPIcs.CCC.2022.13
URN: urn:nbn:de:0030-drops-165755
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16575/
Mihajlin, Ivan ; Sofronova, Anastasia
A Better-Than-3log(n) Depth Lower Bound for De Morgan Formulas with Restrictions on Top Gates
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Abstract
We prove that a modification of Andreev’s function is not computable by (3 + α - ε) log(n) depth De Morgan formula with (2α - ε)log{n} layers of AND gates at the top for any 0 < α < 1/5 and any constant ε > 0. In order to do this, we prove a weak variant of Karchmer-Raz-Wigderson conjecture. To be more precise, we prove the existence of two functions f : {0,1}ⁿ → {0,1} and g : {0,1}ⁿ → {0,1}ⁿ such that f(g(x) ⊕ y) is not computable by depth (1 + α - ε) n formulas with (2 α - ε) n layers of AND gates at the top. We do this by a top-down approach, which was only used before for depth-3 model.Our technical contribution includes combinatorial insights into structure of composition with random boolean function, which led us to introducing a notion of well-mixed sets. A set of functions is well-mixed if, when composed with a random function, it does not have subsets that agree on large fractions of inputs. We use probabilistic method to prove the existence of well-mixed sets.
BibTeX - Entry
@InProceedings{mihajlin_et_al:LIPIcs.CCC.2022.13,
author = {Mihajlin, Ivan and Sofronova, Anastasia},
title = {{A Better-Than-3log(n) Depth Lower Bound for De Morgan Formulas with Restrictions on Top Gates}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {13:1--13:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16575},
URN = {urn:nbn:de:0030-drops-165755},
doi = {10.4230/LIPIcs.CCC.2022.13},
annote = {Keywords: formula complexity, communication complexity, Karchmer-Raz-Wigderson conjecture, De Morgan formulas}
}
| Keywords: | formula complexity, communication complexity, Karchmer-Raz-Wigderson conjecture, De Morgan formulas | |
| Seminar: | 37th Computational Complexity Conference (CCC 2022) | |
| Issue date: | 2022 | |
| Date of publication: | 11.07.2022 |
