Testing Isomorphism of Boolean Functions over Finite Abelian Groups

TL;DR

This paper develops efficient algorithms for testing whether two Boolean functions over finite Abelian groups are isomorphic under automorphisms, using Fourier analysis and group theory techniques.

Contribution

It introduces the first polynomial-query complexity tolerant testing algorithms for Boolean function isomorphism over finite Abelian groups, with improvements for Fourier sparse functions.

Findings

Query complexity is polynomial in spectral norm bounds.

Algorithms work efficiently for functions with Fourier sparsity.

Techniques involve advanced Abelian group theory and Fourier analysis.

Abstract

Let $f$ and $g$ be Boolean functions over a finite Abelian group $\mathcal{G}$, where $g$ is fully known, and we have {\em query access} to $f$, that is, given any $x \in \mathcal{G}$ we can get the value $f(x)$. We study the tolerant isomorphism testing problem: given $\epsilon \geq 0$ and $\tau > 0$, we seek to determine, with minimal queries, whether there exists an automorphism $\sigma$ of $\mathcal{G}$ such that the fractional Hamming distance between $f \circ \sigma$ and $g$ is at most $\epsilon$, or whether for all automorphisms $\sigma$, the distance is at least $\epsilon + \tau$. We design an efficient tolerant testing algorithm for this problem, with query complexity $\mathrm{poly}\left( s, 1/\tau \right)$, where $s$ bounds the spectral norm of $g$. Additionally, we present an improved algorithm when $g$ is Fourier sparse. Our approach uses key concepts from Abelian group theory and Fourier analysis, including the annihilator of a subgroup, Pontryagin duality, and a pseudo inner-product for finite Abelian groups. We believe these techniques will find further applications in property testing.