Algorithms for Boolean Function Query Properties

We investigate efficient algorithms for computing Boolean function properties relevant to query complexity. Such properties include, for example, deterministic, randomized, and quantum query complexities; block sensitivity; certificate complexity; and degree as a real polynomial. The algorithms compute the properties, given an n-variable function's truth table (of size N=2n) as input. Our main results are the following: O(N log2 3 log N) algorithms for many common properties, an O(N log2 5 log N) algorithm for block sensitivity, an O(N) algorithm for testing "quasi symmetry," a notion of a "tree decomposition" of a Boolean function, proof that the decomposition is unique, and an O(Nlog2 3 log N) algorithm for finding it, a subexponential-time approximation algorithm for space-bounded quantum query complexity. To develop this algorithm, we give a new way to search systematically through unitary matrices using finite-precision arithmetic. The algorithms discussed have been implemented in a linkable library.

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