A Composition Theorem for Randomized Query Complexity

TL;DR

This paper establishes a composition theorem for randomized query complexity, providing lower bounds for the complexity of composed functions and relations, which advances understanding of query complexity in computational theory.

Contribution

It introduces a new composition theorem for randomized query complexity that relates the complexities of composed functions to their components, improving previous bounds.

Findings

Proves a lower bound for the randomized query complexity of composed relations and functions.

Shows the complexity scales with the product of component complexities under composition.

Provides bounds for compositions involving XOR functions on logarithmic bits.

Abstract

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 $g^t$.