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Testing Isomorphism of Boolean Functions over Finite Abelian Groups

Authors Swarnalipa Datta , Arijit Ghosh , Chandrima Kayal , Manaswi Paraashar , Manmatha Roy

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Subject Classification

ACM Subject Classification
  • Theory of computation → Boolean function learning
Keywords
  • Analysis of Boolean functions
  • Abelian groups
  • Automorphism group
  • Function isomorphism
  • Spectral norm

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Abstract

Let f and g be Boolean functions over a finite Abelian group 𝒢, where g is fully known and f is accessible via queries; that is, given any x ∈ 𝒢, we can obtain the value f(x). We study the problem of tolerant isomorphism testing: given parameters ε ≥ 0 and τ > 0, the goal is to determine, using as few queries as possible, whether there exists an automorphism σ of 𝒢 such that the fractional Hamming distance between f∘σ and g is at most ε, or whether for every automorphism σ, the distance is at least ε + τ.
We design an efficient tolerant property testing algorithm for this problem over finite Abelian groups with constant exponent. The exponent of a finite group refers to the largest order of any element in the group. The query complexity of our algorithm is polynomial in s and 1/τ, where s bounds the spectral norm of the function g, and τ is the tolerance parameter. In addition, we present an improved algorithm in the case where g is Fourier sparse, meaning that its Fourier expansion contains only a small number of nonzero coefficients.
Our approach draws on key ideas from Abelian group theory and Fourier analysis, including the annihilator of a subgroup, Pontryagin duality, and a pseudo inner product defined over finite Abelian groups. We believe that these techniques will be useful more broadly in the design of property testing algorithms.

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Swarnalipa Datta, Arijit Ghosh, Chandrima Kayal, Manaswi Paraashar, and Manmatha Roy. Testing Isomorphism of Boolean Functions over Finite Abelian Groups. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 66:1-66:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.66

Author Details

Swarnalipa Datta
  • Indian Statistical Institute, Kolkata, India
Arijit Ghosh
  • Indian Statistical Institute, Kolkata, India
Chandrima Kayal
  • Université Paris Cité, CNRS, IRIF, France
Manaswi Paraashar
  • University of Copenhagen, Denmark
Manmatha Roy
  • Indian Statistical Institute, Kolkata, India

Funding

  • Ghosh, Arijit: A. G. is partially supported by the Science & Engineering Research Board of the DST, India, through the MATRICS grant MTR/2023/001527.
  • Kayal, Chandrima: C.K. is supported by French PEPR integrated project EPiQ (ANR-22-PETQ-0007).
  • Paraashar, Manaswi: M.P. is supported by ERC grant (QInteract, Grant Agreement No 101078107).

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