Isomorphism Testing of Boolean Functions Computable by Constant-Depth Circuits

Abstract

Given two n-variable Boolean functions f and g, we study the problem of computing an ε-approximate isomorphism between them. I.e. a permutation π of the n variables such that f(x ₁,x ₂,…,x _n ) and g(x _π(1),x _π(2),…,x _π(n)) differ on at most an ε fraction of all Boolean inputs {0,1}ⁿ. We give a randomized $2^{O(\sqrt{n}polylog(n))}$ algorithm that computes a $\frac{1}{2^{polylog(n)}}$-approximate isomorphism between two isomorphic Boolean functions f and g that are given by depth d circuits of poly(n) size, where d is a constant independent of n. In contrast, the best known algorithm for computing an exact isomorphism between n-ary Boolean functions has running time 2^O(n) [9] even for functions computed by poly(n) size DNF formulas. Our algorithm is based on a result for hypergraph isomorphism with bounded edge size [3] and the classical Linial-Mansour-Nisan result on approximating small depth and size Boolean circuits by small degree polynomials using Fourier analysis.