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We study the composition question for bounded-error randomized querycomplexity: Is R(f o g) = Omega(R(f) R(g)) for all Boolean functions f and g?We show that inserting a simple Boolean function h, whose query complexity isonly Theta(log R(g)), in between f and g allows us to prove R(f o h o g) =Omega(R(f) R(h) R(g)).  We prove this using a new lower bound measure for randomized query complexitywe call randomized sabotage complexity, RS(f). Randomized sabotage complexityhas several desirable properties, such as a perfect composition theorem, RS(f og) >= RS(f) RS(g), and a composition theorem with randomized query complexity,R(f o g) = Omega(R(f) RS(g)). It is also a quadratically tight lower bound fortotal functions and can be quadratically superior to the partition bound, thebest known general lower bound for randomized query complexity.  Using this technique we also show implications for lifting theorems incommunication complexity. We show that a general lifting theorem for zero-errorrandomized protocols implies a general lifting theorem for bounded-errorprotocols.

Randomized query complexity of sabotaged and composed functions