Isomorphism Testing of Boolean Functions Computable by Constant-Depth Circuits

Given two n-variable Boolean functions f and g, we study the problem of computing an e-approximate isomorphism between them. I.e. a permutation π of the n variables such that f(x1,x2,…,xn) and g(xπ(1),xπ(2),…,xπ(n)) differ on at most an e fraction of all Boolean inputs {0,1}n. We give a randomized 2O(√n polylog(n)) algorithm that computes a 1/{2polylog(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 2O(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.