We investigate the minimal cases for realtime probabilistic machines that can define uncountably many languages with bounded error. We show that logarithmic space is enough for realtime PTMs on unary languages. On non-unary case, we obtain the same result for double logarithmic space, which is also tight. When replacing the work tape with a few counters, we can still achieve similar results for unary linear-space two-counter automata, unary sublinear-space three-counter automata, and non-unary sublinear-space two-counter automata. We also show how to slightly improve the sublinear-space constructions by using more counters.

中文翻译：

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不可数的实时概率类

我们研究了实时概率机器的最小情况，这些机器可以定义无数种有界错误的语言。我们展示了对数空间对于一元语言上的实时 PTM 来说已经足够了。在非一元情况下，我们对双对数空间得到相同的结果，这也是紧的。当用几个计数器替换工作磁带时，对于一元线性空间二计数器自动机、一元次线性空间三计数器自动机和非一元次线性空间二计数器自动机，我们仍然可以得到类似的结果。我们还展示了如何通过使用更多计数器来稍微改进次线性空间结构。