Note on Finite-Automata Bernoulli Factories for Rational Functions
This paper identifies a technical oversight in Mossel and Peres (2005) theorem on designing Bernoulli factories for multivariable functions and provides a counterexample.
Identified a limitation in the application of Pólya's Theorem to multivariable rational functions and provided a counterexample.
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Abstract
More Like ThisMossel 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.