Carmosino, Marco ;
Hoover, Kenneth ;
Impagliazzo, Russell ;
Kabanets, Valentine ;
Kolokolova, Antonina
Lifting for ConstantDepth 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 fanin 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 fanin, 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, averagecase and general depthd circuit complexity can be reduced to circuit complexity with this bottom fanin 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 quasipolynomial time mapping reductions between various gapversions of AC⁰MCSP. 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 ConstantDepth Circuits and Applications to MCSP}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {44:144:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771955},
ISSN = {18688969},
year = {2021},
volume = {198},
editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14113},
URN = {urn:nbn:de:0030drops141135},
doi = {10.4230/LIPIcs.ICALP.2021.44},
annote = {Keywords: circuit complexity, constantdepth circuits, lifting theorems, Minimum Circuit Size Problem, reductions, Switching Lemma}
}
02.07.2021
Keywords: 

circuit complexity, constantdepth circuits, lifting theorems, Minimum Circuit Size Problem, reductions, Switching Lemma 
Seminar: 

48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Issue date: 

2021 
Date of publication: 

02.07.2021 