Query Complexity of Recursively Composed Functions

Abstract

In this work, we explore two well-studied notions of randomized query complexity; bounded-error randomized ($\R(f)$), and zero-error randomized ($\R_0(f)$). These have their natural analogues from the classical model of computation, $\R$ corresponding to BPP or ``Monte Carlo" algorithms and $\R_0$ to ZPP or ``Las Vegas" algorithms.

For a query complexity measure $M$, one can define the composition limit of $M$ on $f$ by $M^*(f) = \lim_{k \to \infty} \sqrt[k]{M(f^k)}$. The composition limit is a useful way to understand the asymptotic complexity of a function with respect to a specific measure (e.g. if $M(f) = O(1)M(g)$, then $M^*(f) = M^*(g)$).

We show that under the composition limit, Las Vegas algorithms can be reduced to Monte Carlo algorithms in the query complexity world. Specifically, $\R_0^*(f) = \max(\C^*(f), \R^*(f))$ for all possibly-partial boolean functions $f$. This has wide-reaching implications for the classical query complexity of boolean functions that are still open. For example, this result implies that any bounded-error algorithm for recursive 3-majority can be converted into a zero-error algorithm with no additional cost (i.e. $R^*(\text{3-MAJ}) = R_0^*(\text{3-MAJ})$.

Furthermore, we explore one possible generalization of the recursive 3-majority problem itself, by analyzing 3-majority as a special case of a combinatorial game we call Denial Nim.