On the Composition of Randomized Query Complexity and Approximate Degree
share
For any Boolean functions f and g, the question whether R(f∘ g) = Θ̃(R(f)R(g)), is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether deg(f∘ g) = Θ̃(deg(f)·deg(g)). These questions are two of the most important and well-studied problems, 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 R and deg compose. A recent landmark result (Ben-David and Blais, 2020) showed that R(f ∘ g) = Ω(noisyR(f)· R(g)). This implies that composition holds whenever noisyR(f) = Θ(R(f)). We show two results: (1)When R(f) = Θ(n), then noisyR(f) = Θ(R(f)). (2) If R composes with respect to an outer function, then noisyR also composes with respect to the same outer function. On the other hand, no result of the type deg(f ∘ g) = Ω(M(f) ·deg(g)) (for some non-trivial complexity measure M(·)) was known to the best of our knowledge. We prove that deg(f∘ g) = Ω(√(bs(f))·deg(g)), where bs(f) is the block sensitivity of f. This implies that deg composes when deg(f) is asymptotically equal to √(bs(f)). It is already known that both R and deg compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function.
READ FULL TEXT