Algorithms for Boolean Function Query Properties

TL;DR

This paper introduces efficient algorithms for computing key properties of Boolean functions, including block sensitivity, tree decomposition, and quasisymmetry, along with a subexponential quantum query complexity algorithm, based on new structural insights.

Contribution

It provides novel algorithms with improved complexity for analyzing Boolean functions and develops a theory for quantum query complexity with limited-precision unitary representations.

Findings

O(N^2.322 log N) algorithm for block sensitivity

O(N^1.585 log N) algorithm for tree decomposition

O(N) algorithm for quasisymmetry

Abstract

We present new algorithms to compute fundamental properties of a Boolean function given in truth-table form. Specifically, we give an O(N^2.322 log N) algorithm for block sensitivity, an O(N^1.585 log N) algorithm for `tree decomposition,' and an O(N) algorithm for `quasisymmetry.' These algorithms are based on new insights into the structure of Boolean functions that may be of independent interest. We also give a subexponential-time algorithm for the space-bounded quantum query complexity of a Boolean function. To prove this algorithm correct, we develop a theory of limited-precision representation of unitary operators, building on work of Bernstein and Vazirani.