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 ε-approximate isomorphism between them. An ε-approximate isomorphism is a permutation π of the n   Boolean variables such that f(x1,x2,…,xn)f(x1,x2,…,xn) and g(xπ(1),xπ(2),…,xπ(n))g(xπ(1),xπ(2),…,xπ(n)) differ on at most an ε   fraction of all Boolean inputs {0,1}n{0,1}n. We give a randomized 2O(nlog⁡(n/ε)O(d)) time algorithm that computes an ε-approximate isomorphism between two isomorphic Boolean functions f and g that are given by depth d   circuits of poly(n)poly(n) size, where d is a constant independent of n, for any positive ε. In contrast, the best known algorithm for computing an exact isomorphism between n  -ary Boolean functions has running time 2O(n)2O(n)[12] even for functions computed by poly(n)poly(n) size DNF formulas. Our algorithm is based on a result for hypergraph isomorphism with bounded edge size [4] and the classical Linial–Mansour–Nisan result on approximating small depth and size Boolean circuits by small degree polynomials using Fourier analysis [11].