A Composition Theorem for Randomized Query Complexity

Let the randomized query complexity of a relation for error probability $\epsilon$ be denoted by $R\epsilon(\cdot)$. We prove that for any relation $f \subseteq {0,1}ⁿ \times \mathcal{R}$ and Boolean function $g:{0,1}^m \rightarrow {0,1}$, $R{1/3}(f\circ gⁿ⁾ = \Omega(R{4/9}(f)\cdot R{1/2-1/n^4}(g))$, where $f \circ g^n$ is the relation obtained by composing $f$ and $g$. We also show that $R{1/3}\left(f \circ \left(g^\oplus{O(\log n)}\right)^{n\right)=\Omega(\log} n \cdot R{4/9}(f) \cdot R{1/3}(g))$, where $g^{\oplus_{O(\log} n)}$ is the function obtained by composing the xor function on $O(\log n)$ bits and $g^t$.