License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2021.44
URN: urn:nbn:de:0030-drops-141135
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14113/
Carmosino, Marco ;
Hoover, Kenneth ;
Impagliazzo, Russell ;
Kabanets, Valentine ;
Kolokolova, Antonina
Lifting for Constant-Depth Circuits and Applications to MCSP
Abstract
Lifting arguments show that the complexity of a function in one model is essentially that of a related function (often the composition of the original function with a small function called a gadget) in a more powerful model. Lifting has been used to prove strong lower bounds in communication complexity, proof complexity, circuit complexity and many other areas.
We present a lifting construction for constant depth unbounded fan-in circuits. Given a function f, we construct a function g, so that the depth d+1 circuit complexity of g, with a certain restriction on bottom fan-in, is controlled by the depth d circuit complexity of f, with the same restriction. The function g is defined as f composed with a parity function. With some quantitative losses, average-case and general depth-d circuit complexity can be reduced to circuit complexity with this bottom fan-in restriction. As a consequence, an algorithm to approximate the depth d (for any d > 3) circuit complexity of given (truth tables of) Boolean functions yields an algorithm for approximating the depth 3 circuit complexity of functions, i.e., there are quasi-polynomial time mapping reductions between various gap-versions of AC⁰-MCSP. Our lifting results rely on a blockwise switching lemma that may be of independent interest.
We also show some barriers on improving the efficiency of our reductions: such improvements would yield either surprisingly efficient algorithms for MCSP or stronger than known AC⁰ circuit lower bounds.
BibTeX - Entry
@InProceedings{carmosino_et_al:LIPIcs.ICALP.2021.44,
author = {Carmosino, Marco and Hoover, Kenneth and Impagliazzo, Russell and Kabanets, Valentine and Kolokolova, Antonina},
title = {{Lifting for Constant-Depth Circuits and Applications to MCSP}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {44:1--44:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-195-5},
ISSN = {1868-8969},
year = {2021},
volume = {198},
editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14113},
URN = {urn:nbn:de:0030-drops-141135},
doi = {10.4230/LIPIcs.ICALP.2021.44},
annote = {Keywords: circuit complexity, constant-depth circuits, lifting theorems, Minimum Circuit Size Problem, reductions, Switching Lemma}
}
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Keywords: |
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circuit complexity, constant-depth circuits, lifting theorems, Minimum Circuit Size Problem, reductions, Switching Lemma |
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Collection: |
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48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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02.07.2021 |
