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.

中文翻译：

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硬普适性和空性问题的改进随机逼近

我们以多项式随机逼近 (PRAX) 算法的最新研究为基础，解决 NFA 普适性和 NFA 等价性的难题。宽松地说，PRAX 算法使用任意所需精度 $\delta$ 内的无限域采样。本着实验数学的精神，我们扩展了 PRAX 算法的概念，使其适用于任何实例承认本文定义的易处理分布的领域中的空性和普遍性问题。这里的技术结果是线性（wrt $1/\delta$）样本数量就足够了，而不是以前论文中的二次样本数量。我们展示了改进和广义的 PRAX 算法如何应用于各个领域的普遍性和空性问题：普通自动机、命题同义反复检验、二维自动机以及某些丢番图方程的解集。