All Classical Adversary Methods are Equivalent for Total Functions

Abstract

We show that all known classical adversary lower bounds on randomized query complexity are equivalent for total functions, and are equal to the fractional block sensitivity $\text{fbs}(f)$. That includes the Kolmogorov complexity bound of Laplante and Magniez and the earlier relational adversary bound of Aaronson. This equivalence also implies that for total functions, the relational adversary is equivalent to a simpler lower bound, which we call rank-1 relational adversary. For partial functions, we show unbounded separations between $\text{fbs}(f)$ and other adversary bounds, as well as between the adversary bounds themselves. We also show that, for partial functions, fractional block sensitivity cannot give lower bounds larger than $\sqrt{n \cdot \text{bs}(f)}$, where $n$ is the number of variables and $\text{bs}(f)$ is the block sensitivity. Then we exhibit a partial function $f$ that matches this upper bound, $\text{fbs}(f) = \Omega(\sqrt{n \cdot \text{bs}(f)})$.