We build on recent research on polynomial randomized approximation (PRAX) algorithms for the hard problems of NFA universality and NFA equivalence. Loosely speaking, PRAX algorithms use sampling of infinite domains within any desired accuracy $\delta$. In the spirit of experimental mathematics, we extend the concept of PRAX algorithms to be applicable to the emptiness and universality problems in any domain whose instances admit a tractable distribution as defined in this paper. A technical result here is that a linear (w.r.t. $1/\delta$) number of samples is sufficient, as opposed to the quadratic number of samples in previous papers. We show how the improved and generalized PRAX algorithms apply to universality and emptiness problems in various domains: ordinary automata, tautology testing of propositions, 2D automata, and to solution sets of certain Diophantine equations.

翻译：我们基于近期的多项式随机近似（PRAX）算法研究，针对NFA通用性问题与NFA等价性问题的困难情形展开工作。广义而言，PRAX算法通过给定精度$\delta$对无限域进行采样。延续实验数学的思想，我们将PRAX算法概念拓展至任意具备本文定义的可处理分布的实例域，使其可应用于该域中的空性与通用性问题。本文取得的技术成果表明：相较于先前文献所需的二次采样次数，仅需线性于$1/\delta$的采样次数即足够。我们展示了改进型广义PRAX算法如何适用于各类域中的通用性与空性问题：包括常规自动机、命题永真性检验、二维自动机以及特定丢番图方程的解集。