A Composition Theorem for Randomized Query Complexity

Let $R_\epsilon(\cdot)$ stand for the bounded-error randomized query complexity with error $\epsilon > 0$. For any relation $f \subseteq \{0,1\}^n \times S$ and partial Boolean function $g \subseteq \{0,1\}^m \times \{0,1\}$, we show that $R_{1/3}(f \circ g^n) \in \Omega(R_{4/9}(f) \cdot \sqrt{R_{1/3}(g)})$, where $f \circ g^n \subseteq (\{0,1\}^m)^n \times S$ is the composition of $f$ and $g$. We give an example of a relation $f$ and partial Boolean function $g$ for which this lower bound is tight. We prove our composition theorem by introducing a new complexity measure, the max conflict complexity $\bar \chi(g)$ of a partial Boolean function $g$. We show $\bar \chi(g) \in \Omega(\sqrt{R_{1/3}(g)})$ for any (partial) function $g$ and $R_{1/3}(f \circ g^n) \in \Omega(R_{4/9}(f) \cdot \bar \chi(g))$; these two bounds imply our composition result. We further show that $\bar \chi(g)$ is always at least as large as the sabotage complexity of $g$, introduced by Ben-David and Kothari.