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Approximate Degree Composition for Recursive Functions

Authors Sourav Chakraborty , Chandrima Kayal, Rajat Mittal, Manaswi Paraashar , Nitin Saurabh

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Author Details

Sourav Chakraborty
  • Indian Statistical Institute, Kolkata, India
Chandrima Kayal
  • Indian Statistical Institute, Kolkata, India
Rajat Mittal
  • Indian Institute of Technology Kanpur, India
Manaswi Paraashar
  • University of Copenhagen, Denmark
Nitin Saurabh
  • Indian Institute of Technology Hyderabad, India

Funding

  • Chakraborty, Sourav: supported by the Science & Engineering Research Board of the DST, India, through the MATRICS grant MTR/2021/000318.
  • Paraashar, Manaswi: supported by ERC grant (QInteract, Grant Agreement No 101078107).
  • Saurabh, Nitin: supported by the seed grant (SG/IITH/F285/2022-23/SG-123) from IIT Hyderabad.

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Sourav Chakraborty, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, and Nitin Saurabh. Approximate Degree Composition for Recursive Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 71:1-71:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.71

Abstract

Determining the approximate degree composition for Boolean functions remains a significant unsolved problem in Boolean function complexity. In recent decades, researchers have concentrated on proving that approximate degree composes for special types of inner and outer functions. An important and extensively studied class of functions are the recursive functions, i.e. functions obtained by composing a base function with itself a number of times. Let h^d denote the standard d-fold composition of the base function h. The main result of this work is to show that the approximate degree composes if either of the following conditions holds: - The outer function f:{0,1}ⁿ → {0,1} is a recursive function of the form h^d, with h being any base function and d = Ω(log log n). - The inner function is a recursive function of the form h^d, with h being any constant arity base function (other than AND and OR) and d = Ω(log log n), where n is the arity of the outer function. In terms of proof techniques, we first observe that the lower bound for composition can be obtained by introducing majority in between the inner and the outer functions. We then show that majority can be efficiently eliminated if the inner or outer function is a recursive function.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Approximate degree
  • Boolean function
  • Composition theorem

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