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 $g:\{0,1\}^m \rightarrow \{0,1\}$, $\R_{1/3}(f\circ g^n) = \Omega(\R_{4/9}(f)\cdot\R_{1/2-1/n^4}(g))$, where $f \circ g^n$ is the relation obtained by composing $f$ and $g$. We also show using an XOR lemma 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 $g$.

Corrected typos

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 $g:\{0,1\}^m \rightarrow \{0,1\}$, $\R_{1/3}(f\circ g^n) = \Omega(\R_{4/9}(f)\cdot\R_{1/2-1/n^4}(g))$, 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 $g^t$.