 
Summary: ALGORITHMS FOR BOOLEAN FUNCTION QUERY PROPERTIES
SCOTT AARONSON #
Abstract. We investigate e#cient algorithms for computing Boolean function properties rel
evant to query complexity. Such properties include, for example, deterministic, randomized, and
quantum query complexities; block sensitivity; certificate complexity; and degree as a real polyno
mial. The algorithms compute the properties given an nvariable function's truth table (of size
N = 2 n ) as input.
Our main results are the following:
O(N log 2 3 log N) algorithms for many common properties.
An O(N log 2 5 log N) algorithm for block sensitivity.
An O(N) algorithm for testing `quasisymmetry.'
A notion of a `tree decomposition' of a Boolean function, proof that the decomposition is unique,
and an O(N log 2 3 log N) algorithm for finding it.
A subexponentialtime approximation algorithm for spacebounded quantum query complexity.
To develop this algorithm, we give a new way to search systematically through unitary matrices
using finiteprecision arithmetic.
The algorithms discussed have been implemented in a linkable library.
Key words. algorithm, Boolean function, truth table, query complexity, quantum computation.
AMS subject classifications. 68Q10, 68Q17, 68Q25, 68W01, 81P68.
1. Introduction. The query complexity of Boolean functions, also called black
