Isomorphism testing of Boolean functions computable by constant-depth circuits

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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) and g(xπ(1),xπ(2),…,xπ(n)) differ on at most an ε fraction of all Boolean inputs {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) 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)[12] even for functions computed by 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].