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 algorithm that computes a 1/2 poly log(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. © 2012 Springer-Verlag.