# A composition theorem for randomized query complexity via max conflict complexity

Let R_ϵ(·) stand for the bounded-error randomized query complexity with error ϵ > 0. For any relation f ⊆{0,1}^n × S and partial Boolean function g ⊆{0,1}^m ×{0,1}, we show that R_1/3(f ∘ g^n) ∈Ω(R_4/9(f) ·√(R_1/3(g))), where f ∘ g^n ⊆ ({0,1}^m)^n × S is the composition of f and g. We give an example of a relation f and partial Boolean function g for which this lower bound is tight. We prove our composition theorem by introducing a new complexity measure, the max conflict complexity χ̅(g) of a partial Boolean function g. We show χ̅(g) ∈Ω(√(R_1/3(g))) for any (partial) function g and R_1/3(f ∘ g^n) ∈Ω(R_4/9(f) ·χ̅(g)); these two bounds imply our composition result. We further show that χ̅(g) is always at least as large as the sabotage complexity of g, introduced by Ben-David and Kothari.

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