License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2017.23
URN: urn:nbn:de:0030-drops-83839
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8383/
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Chattopadhyay, Arkadev ; Mande, Nikhil S.

A Lifting Theorem with Applications to Symmetric Functions

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Abstract

We use a technique of “lifting” functions introduced by Krause and Pudlak [Theor. Comput. Sci., 1997], to amplify
degree-hardness measures of a function to corresponding monomial-hardness properties of the
lifted function. We then show that any symmetric function F projects onto a “lift” of another
suitable symmetric function f . These two key results enable us to prove several results on the
complexity of symmetric functions in various models, as given below:

1. We provide a characterization of the approximate spectral norm of symmetric functions in
terms of the spectrum of the underlying predicate, affirming a conjecture of Ada et al. [APPROX-RANDOM, 2012]
which has several consequences.

2. We characterize symmetric functions computable by quasi-polynomial sized Threshold
of Parity circuits.

3. We show that the approximate spectral norm of a symmetric function f characterizes the
(quantum and classical) bounded error communication complexity of f o XOR.

4. Finally, we characterize the weakly-unbounded error communication complexity of symmetric
XOR functions, resolving a weak form of a conjecture by Shi and Zhang [Quantum Information & Computation, 2009]

BibTeX - Entry

@InProceedings{chattopadhyay_et_al:LIPIcs:2018:8383,
  author =	{Arkadev Chattopadhyay and Nikhil S. Mande},
  title =	{{A Lifting Theorem with Applications to Symmetric Functions}},
  booktitle =	{37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)},
  pages =	{23:1--23:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-055-2},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{93},
  editor =	{Satya Lokam and R. Ramanujam},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8383},
  URN =		{urn:nbn:de:0030-drops-83839},
  doi =		{10.4230/LIPIcs.FSTTCS.2017.23},
  annote =	{Keywords: Symmetric functions, lifting, circuit complexity, communication com- plexity}
}

Keywords: Symmetric functions, lifting, circuit complexity, communication com- plexity
Collection: 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)
Issue Date: 2018
Date of publication: 12.02.2018

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