Identity Testing for Stochastic Languages

Abstract

Determining whether an unknown distribution matches a known reference is a cornerstone problem in distributional analysis. While classical results establish a rigorous framework in the case of distributions over finite domains, real-world applications in computational linguistics, bioinformatics, and program analysis demand testing over infinite combinatorial structures, particularly strings. In this paper, we initiate the theoretical study of identity testing for stochastic languages, bridging formal language theory with modern distribution property testing. We first propose a polynomial-time algorithm to verify if a finite state machine represents a stochastic language, and then prove that rational stochastic languages can approximate an arbitrary probability distribution. Building on these representations, we develop a truncation-based identity testing algorithm that distinguishes between a known and an unknown distributions with sample complexity $\widetilde{\Theta}\left( \frac{\sqrt{n}}{\varepsilon^2} + \frac{n}{\log n} \right)$ where $n$ is the size of the truncated support. Our approach leverages the exponential decay inherent in rational stochastic languages to bound truncation error, then applies classical finite-domain testers to the restricted problem. This work establishes the first identity testing framework for infinite discrete distributions, opening new directions in probabilistic formal methods and statistical analysis of structured data.