Monte Carlo to Las Vegas for Recursively Composed Functions

2026-01-12

AI Summary

This study investigates the compositional limits of query complexity under recursively composed Boolean functions, with a focus on the feasibility of transforming bounded-error randomized (Monte Carlo) algorithms into zero-error (Las Vegas) ones. By integrating tools from combinatorics, query complexity theory, and recursive function analysis within both classical and quantum computational models, the work establishes—for the first time—that, under reasonable assumptions, the compositional limit of randomized query complexity equals the maximum of the compositional limits of its bounded-error variant and certificate complexity. As an application, it demonstrates that for recursively defined functions such as the 3-majority function, any bounded-error randomized or quantum algorithm can be converted into a zero-error algorithm, thereby transcending established theoretical barriers in the Monte Carlo to Las Vegas conversion paradigm.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

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.

Problem

• query complexity
• composition limit
• randomized algorithms
• Boolean functions
• recursively composed functions

Innovation

• composition limit
• randomized query complexity
• zero-error algorithms
• recursively composed functions
• quantum query complexity