Testing Isomorphism of Boolean Functions over Finite Abelian Groups

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.