Monte Carlo to Las Vegas for Recursively Composed Functions

For a (possibly partial) Boolean function fcolon\{0,1\}ⁿ→\{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→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 R₀^*(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.