The emptiness and containment problems for probabilistic automata are natural quantitative generalisations of the classical language emptiness and inclusion problems for Boolean automata. It is known that both problems are undecidable. We provide a more refined view of these problems in terms of the degree of ambiguity of probabilistic automata. We show that a gap version of the emptiness problem (known to be undecidable in general) becomes decidable for automata of polynomial ambiguity. We complement this positive result by showing that emptiness remains undecidable when restricted to automata of linear ambiguity. We then turn to finitely ambiguous automata and give a conditional decidability proof for containment in case one of the automata is assumed to be unambiguous. Part of our proof relies on the decidability of the theory of real exponentiation, proved, subject to Schanuel's Conjecture, by Macintyre and Wilkie.

概率自动机的空性和包容性问题是布尔自动机的经典语言空性和包含性问题的自然定量概括。众所周知，这两个问题都是无法确定的。我们从概率自动机的歧义度的角度，对这些问题提供了更为精确的观点。我们表明，空缺问题的缺口版本（通常无法确定）对于多项式歧义自动机是可确定的。我们通过证明当局限于线性歧义自动机时，空度仍然是不确定的，从而补充了这一积极结果。然后，我们转向有限模棱两可的自动机，并在假定其中一个自动机是明确的情况下，给出了包含条件的可判定性证明。