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A Lifting Theorem with Applications to Symmetric Functions

Authors Arkadev Chattopadhyay, Nikhil S. Mande

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Arkadev Chattopadhyay
Nikhil S. Mande

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Arkadev Chattopadhyay and Nikhil S. Mande. A Lifting Theorem with Applications to Symmetric Functions. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.FSTTCS.2017.23

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]
Keywords
  • Symmetric functions
  • lifting
  • circuit complexity
  • communication com- plexity

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