Year: 2026
Type: article
Abstract: For any Boolean functions f and g, the question whether $$\text {R}(f\circ g) = \tilde{\Theta }(\text {R}(f) \cdot \text {R}(g))$$ is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether $$\widetilde{\text {deg}}(f\circ g) = \tilde{\Theta }(\widetilde{\text {deg}}(f)\cdot \widetilde{\text {deg}}(g))$$ . These questions are two of the most important and well-studied problems in the field of analysis of Boolean functions, and yet we are far from answering them satisfactorily. It is known that the measures compose if one assumes various properties of the outer function f (or inner function g). This paper extends the class of outer functions for which $$\text {R}$$ and $$\widetilde{\text {deg}}$$ compose. A recent landmark result (Ben-David and Blais, 2020) showed that $$\text {R}(f \circ g) = \Omega (\text {noisyR}(f)\cdot \text {R}(g))$$ . This implies that composition holds iff $$\text {noisyR}(f) = \tilde{\Theta }(\text {R}(f))$$ . We show two results: On the other hand, we prove that $$ \widetilde{\text {deg}}(f\circ g) = \widetilde{\Omega }(\sqrt{\text {bs}(f)} \cdot \widetilde{\text {deg}}(g)), $$ where $$\text {bs}(f)$$ is the block sensitivity of f. This implies that $$\widetilde{\text {deg}}$$ composes when $$\widetilde{\text {deg}}(f)$$ is asymptotically equal to $$\sqrt{\text {bs}(f)}$$ . It is already known that both $$\text {R}$$ and $$\widetilde{\text {deg}}$$ compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function.
Source: Computational Complexity
Authors: Sourav Chakraborty, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Swagato Sanyal, Nitin Saurabh
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4384111017