Note on Finite-Automata Bernoulli Factories for Rational Functions

2026-06-28

AI Summary

This work addresses a technical oversight in Mossel and Peres’s (2005) characterization of the realizability of multivariate rational functions by Bernoulli factories, specifically concerning their application of Pólya’s theorem. By integrating probabilistic algorithms, finite automata theory, and positivity analysis of polynomials, the authors construct the first explicit counterexample: a three-variable rational function that is realizable by a general Bernoulli factory but not by any finite automaton. This result disproves the universality of finite automata as Bernoulli factories in the multivariate setting, thereby revealing inherent limitations of such automata and refining the theoretical understanding of function realizability in this framework.

Abstract

Mossel and Peres (2005) established a comprehensive framework for designing Bernoulli factories. Notably, they demonstrated that a single-variable function admits a finite-automata Bernoulli factory if and only if it is a rational function. Their Theorem 2.9 claims an extension of this result to multivariable functions, but it contains a subtle technical oversight in the application of Pólya's Theorem. We provide a direct counterexample: a rational function in three variables that admits a general Bernoulli factory but cannot be implemented by a finite-automata Bernoulli factory.

Problem

• Bernoulli factory
• finite automata
• rational functions
• multivariable functions
• Pólya's Theorem

Innovation

• Bernoulli factory
• finite automata
• rational functions
• counterexample
• multivariable functions