Let R_eps denote randomized query complexity for error probability eps, and R:=R_{1/3}. In this work we investigate whether a perfect composition theorem R(f o g^n)=Omega(R(f).R(g)) holds for a relation f in {0,1}^n * S and a total inner function g:{0,1}^m \to {0, 1}. Let D^(prod) denote the maximum distributional query complexity with respect to any product (over variables) distribution. In this work we show the composition theorem R(f o g^n)=Omega(R(f).D^{prod}(g)) up to logarithmic factors. In light of the minimax theorem which states that R(g) is the maximum distributional complexity of g over any distribution, our result makes progress towards answering the composition question. We prove our result by means of a complexity measure R^(prod)_(eps) that we define for total Boolean functions. We show it to be equivalent (up to logarithmic factors) to the sabotage complexity measure RS() defined by Ben-David and Kothari (ICALP 2019): RS(g) = Theta(R^(prod)_(1/3)(g)) (up to log factors). We ask if our bound RS(g) = Omega(D^(prod)(g)) (up to log factors) is tight. We answer this question in the negative, by showing that for the NAND tree function, sabotage complexity is polynomially larger than D^(prod). Our proof yields an alternative and different derivation of the tight lower bound on the bounded error randomized query complexity of the NAND tree function (originally proved by Santha in 1985), which may be of independent interest. Our result gives an explicit polynomial separation between R and D^(prod) which, to our knowledge, was not known prior to our work.

翻译：设R_eps表示误差概率为eps的随机化查询复杂度，并记R:=R_{1/3}。本文研究对于关系f∈{0,1}^n * S与完全内函数g:{0,1}^m → {0,1}，是否成立完美组合定理R(f o g^n)=Ω(R(f)·R(g))。令D^(prod)表示关于任意（变量间）乘积分布的最大分布查询复杂度。本文证明组合定理R(f o g^n)=Ω(R(f)·D^(prod)(g))成立（忽略对数因子）。根据极大极小定理——即R(g)是g关于任意分布的最大分布复杂度——我们的结果推进了对组合问题的解答。我们通过为完全布尔函数定义复杂度度量R^(prod)_(eps)来证明该结果，并证明该度量与Ben-David和Kothari（ICALP 2019）提出的破坏复杂度度量RS()等价（忽略对数因子）：RS(g)=Θ(R^(prod)_(1/3)(g))（忽略对数因子）。我们提出疑问：边界RS(g)=Ω(D^(prod)(g))（忽略对数因子）是否紧？通过证明对于NAND树函数，破坏复杂度在多项式意义上大于D^(prod)，我们给出了否定答案。我们的证明提供了NAND树函数有界误差随机化查询复杂度紧下界（最初由Santha于1985年证明）的另一种推导方法，该方法可能具有独立意义。我们的结果显式给出了R与D^(prod)之间的多项式分离，据我们所知，此前尚无此结论。