We study the composition question for bounded-error randomized query complexity: Is R(f∘g)=Ω(R(f)R(g)) for all Boolean functions f and g? We show that inserting a simple Boolean function h, whose query complexity is only Θ(log R(g)), in between f and g allows us to prove R(f∘h∘g)=Ω(R(f)R(h)R(g)).
We prove this using a new lower bound measure for randomized query complexity we call randomized sabotage complexity, 𝖱𝖲(f). Randomized sabotage complexity has several desirable properties, such as a perfect composition theorem, 𝖱𝖲(f∘g)≥𝖱𝖲(f)𝖱𝖲(g), and a composition theorem with randomized query complexity, R(f∘g)=Ω(R(f)𝖱𝖲(g)). It is also a quadratically tight lower bound for total functions and can be quadratically superior to the partition bound, the best known general lower bound for randomized query complexity.
Using this technique we also show implications for lifting theorems in communication complexity. We show that a general lifting theorem for zero-error randomized protocols implies a general lifting theorem for bounded-error protocols.