Testing Isomorphism of Boolean Functions over Finite Abelian Groups

2025-07-10

AI Summary

This paper studies tolerant isomorphism testing of Boolean functions over finite abelian groups: given query access to a function $f$ and a known function $g$, determine whether there exists a group automorphism $sigma$ such that the Hamming distance between $f$ and $g circ sigma$ is at most $varepsilon$, or whether this distance exceeds $varepsilon + au$ for all $sigma$. We present the first efficient tolerant algorithm based on spectral norm, leveraging Pontryagin duality, subgroup annihilators, and pseudo-inner-product structures to reduce the problem to spectral distance testing in the Fourier domain. Our method achieves query complexity $O(mathrm{poly}(s, 1/ au))$, where $s$ is the Fourier spectral norm of $g$; for $k$-sparse $g$ in the Fourier domain, complexity improves to $mathrm{poly}(k, 1/ au)$. This work is the first to systematically integrate duality and annihilator theory into tolerant function isomorphism testing, thereby extending both the theoretical scope and technical toolkit of property testing.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization
• Machine Learning: Kernel Methods

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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 $εgeq 0$ and $τ> 0$, we seek to determine, with minimal queries, whether there exists an automorphism $σ$ of $mathcal{G}$ such that the fractional Hamming distance between $f circ σ$ and $g$ is at most $ε$, or whether for all automorphisms $σ$, the distance is at least $ε+ τ$. We design an efficient tolerant testing algorithm for this problem, with query complexity $mathrm{poly}left( s, 1/τ ight)$, 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.

Problem

• Testing isomorphism of Boolean functions over finite Abelian groups
• Determining fractional Hamming distance with minimal queries
• Designing efficient tolerant testing algorithms for Fourier-sparse functions

Innovation

• Efficient tolerant testing algorithm with poly query complexity
• Improved algorithm for Fourier sparse functions
• Uses Abelian group theory and Fourier analysis