We study the emptiness and $\lambda$-reachability problems for unary and binary Probabilistic Finite Automata (PFA) and characterise the complexity of these problems in terms of the degree of ambiguity of the automaton and the size of its alphabet. Our main result is that emptiness and $\lambda$-reachability are solvable in EXPTIME for polynomially ambiguous unary PFA and if, in addition, the transition matrix is binary, we show they are in NP. In contrast to the Skolem-hardness of the $\lambda$-reachability and emptiness problems for exponentially ambiguous unary PFA, we show that these problems are NP-hard even for finitely ambiguous unary PFA. For binary polynomially ambiguous PFA with fixed and commuting transition matrices, we prove NP-hardness of the $\lambda$-reachability (dimension 9), nonstrict emptiness (dimension 37) and strict emptiness (dimension 40) problems.

翻译：我们研究一元和二元概率有限自动机(PFA)的空性问题与λ可达性问题，并根据自动机的模糊程度及其字母表规模刻画了这些问题的复杂度。主要结论是：多项式模糊一元PFA的空性与λ可达性问题可在EXPTIME内求解；若其转移矩阵为二元的，则问题属于NP。与指数模糊一元PFA的λ可达性和空性问题的Skolem难度相反，我们证明即便对于有限模糊一元PFA，这些问题也是NP难的。对于具有固定交换转移矩阵的二元多项式模糊PFA，我们证明了λ可达性问题(维度9)、非严格空性问题(维度37)及严格空性问题(维度40)的NP难度。