Chernoff-Hoeffding Bounds for Applications with Limited Independence
SIAM Journal on Discrete Mathematics
Testing Basic Boolean Formulae
SIAM Journal on Discrete Mathematics
Testing Random Variables for Independence and Identity
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Functions that have read-twice constant width branching programs are not necessarily testable
Random Structures & Algorithms
Testing Polynomials over General Fields
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Testing Low-Degree Polynomials over Prime Fields
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Set Systems with Restricted Cross-Intersections and the Minimum Rank of Inclusion Matrices
SIAM Journal on Discrete Mathematics
Testing for Concise Representations
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
SIAM Journal on Computing
Testing juntas nearly optimally
Proceedings of the forty-first annual ACM symposium on Theory of computing
Random Low Degree Polynomials are Hard to Approximate
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
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
Efficient sample extractors for juntas with applications
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Information Processing Letters
Isomorphism testing of boolean functions computable by constant-depth circuits
LATA'12 Proceedings of the 6th international conference on Language and Automata Theory and Applications
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We study the problem of testing isomorphism (equivalence up to relabelling of the variables) of two Boolean functions f,g: {0, 1}n → {0, 1}. Our main focus is on the most studied case, where one of the functions is given (explicitly) and the other function may be queried. We prove that for every k ≤ n, the worst-case query complexity of testing isomorphism to a given k-junta is Ω(k) and O(k log k). Consequently, the query complexity of testing function isomorphism is θ(n). Prior to this work, only lower bounds of Ω(log k) queries were known, for limited ranges of k, proved by Fischer et al. (FOCS 2002), Blais and O'Donnell (CCC 2010), and recently by Alon and Blais (RANDOM 2010). The nearly tight O(k log k) upper bound improves on the Õ(k4) upper bound from Fischer et al. (FOCS 2002). Extending the lower bound proof, we also show polynomial query-complexity lower bounds for the problems of testing whether a function can be computed by a circuit of size ≤ s, and testing whether the Fourier degree of a function is ≤ d. This answers questions posed by Diakonikolas et al. (FOCS 2007). We also address two closely related problems -- 1. Testing isomorphism to a k-junta with one-sided error: we prove that for any 1 k n − 1, the query complexity is Ω(log (nk)), which is almost optimal. This lower bound is a consequence of a proof that the query complexity of testing, with one-sided error, whether a function is a k-parity is Θ(log (nk). 2. Testing isomorphism between two unknown functions that can be queried: we prove that the query complexity in this setting is Ω(√2n) and O(√2nn log n).