For a (possibly partial) Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ as well as a query complexity measure $M$ which maps Boolean functions to real numbers, define the composition limit of $M$ on $f$ by $M^*(f)=\lim_{k\to\infty} M(f^k)^{1/k}$. We study the composition limits of general measures in query complexity. We show this limit converges under reasonable assumptions about the measure. We then give a surprising result regarding the composition limit of randomized query complexity: we show $R_0^*(f)=\max\{R^*(f),C^*(f)\}$. Among other things, this implies that any bounded-error randomized algorithm for recursive 3-majority can be turned into a zero-error randomized algorithm for the same task. Our result extends also to quantum algorithms: on recursively composed functions, a bounded-error quantum algorithm can be converted into a quantum algorithm that finds a certificate with high probability. Along the way, we prove various combinatorial properties of measures and composition limits.

翻译：对于（可能为部分的）布尔函数 $f\colon\{0,1\}^n\to\{0,1\}$ 以及将布尔函数映射到实数的查询复杂度度量 $M$，定义 $M$ 在 $f$ 上的组合极限为 $M^*(f)=\lim_{k\to\infty} M(f^k)^{1/k}$。我们研究了查询复杂度中一般度量的组合极限。我们证明在关于该度量的合理假设下，该极限是收敛的。随后我们给出了关于随机查询复杂度组合极限的一个令人意外的结果：我们证明 $R_0^*(f)=\max\{R^*(f),C^*(f)\}$。特别地，这意味着任何用于递归三多数函数的有限错误随机算法都可以转化为针对同一任务的零错误随机算法。我们的结果也适用于量子算法：对于递归组合函数，有限错误量子算法可以转化为能以高概率找到证书的量子算法。在此过程中，我们证明了度量与组合极限的若干组合性质。