A weighted finite automaton has the finite-image property if the image of the weighted language associated with it is finite. We show two undecidability results concerning the finite-image property of weighted finite automata over semirings, respectively strong bimonoids. Firstly, we give a computable idempotent commutative past-finite ordered semiring such that it is undecidable, for an arbitrary deterministic weighted finite automaton ${𝒜}$ over that semiring, whether ${𝒜}$ has the finite-image property. Secondly, we give a computable commutative past-finite monotonic ordered strong bimonoid such that it is undecidable, for an arbitrary weighted finite automaton ${𝒜}$ over that strong bimonoid, whether ${𝒜}$ has the finite-image property. This shows that recent decidability results for suitable weighted finite automata over past-finite monotonic strong bimonoids cannot be extended to natural classes of ordered semirings and ordered strong bimonoids without further assumptions.

如果与其相关的加权语言的图像是有限的，则加权有限自动机具有有限图像属性。我们展示了关于半环（分别是强双幺半群）上加权有限自动机的有限图像性质的两个不可判定性结果。首先，我们给出一个可计算的幂等交换过去有限有序半环，对于任意确定性加权有限自动机，它是不可判定的${𝒜}$在那个半环上，是否${𝒜}$具有有限图像性质。其次，我们给出一个可计算交换的过去有限单调有序强双幺半群，对于任意加权的有限自动机，它是不可判定的${𝒜}$超过那个强双歧杆菌，是否${𝒜}$具有有限图像性质。这表明，在没有进一步假设的情况下，过去有限单调强双幺半群上的适当加权有限自动机的最新可判定性结果不能扩展到有序半环和有序强双幺半群的自然类。