Sharper sample complexity for truncation-based identity testing of rational stochastic languages

Determine sharper sample complexity bounds than N = \widetilde{\Theta}(\sqrt{k}/\varepsilon^2 + k/\log k) for the truncation-based ℓ1 identity tester (Algorithm 1, ℓ1-IdentityTester) applied to rational stochastic languages represented via stochastic regular expressions or weighted automata, by exploiting deeper structural properties of the underlying automata or distributions.

Background

The paper introduces an identity testing framework for stochastic languages over strings, proposing a truncation-based reduction to finite support followed by a finite-domain tolerant tester. The declared sample complexity in the paper depends on the size k of the truncated support: N = \widetilde{\Theta}(\sqrt{k}/\varepsilon^2 + k/\log k).

In the Future Work section, the authors state a conjecture that tighter bounds may be achievable by leveraging deeper structural properties of the underlying rational representations (weighted automata or stochastic regular expressions), indicating a concrete open direction to improve the current complexity guarantees.