Constant depth circuits, Fourier transform, and learnability
Journal of the ACM (JACM)
Hypergraph isomorphism and structural equivalence of Boolean functions
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
The Formula Isomorphism Problem
SIAM Journal on Computing
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Isomorhism of Hypergraphs of Low Rank in Moderately Exponential Time
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Lower Bounds for Testing Function Isomorphism
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
Testing Boolean function isomorphism
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Nearly tight bounds for testing function isomorphism
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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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. 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 ε 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.