Nonuniform families of polynomial-size finite automata, which are series of indexed finite automata having polynomially many inner states, are used in the past literature to solve nonuniform families of promise decision problems. Among such nonuniform families of finite automata, we focus our attention, in particular, on the variants of nondeterministic finite automata, which have at most "one" (unambiguous), "polynomially many" (few) accepting computation paths, or unambiguous/few computation paths leading to each fixed configuration. When such machines are limited to make only one-way head moves, we can prove with no unproven hardness assumptions that some of these variants are different in computational power from each other. As for two-way machines restricted to instances of polynomially-bounded length, families of two-way polynomial-size nondeterministic finite automata are equivalent in power to families of polynomial-size unambiguous finite automata.

翻译：过去文献中采用非均匀多项式规模有限自动机族（即一系列具有多项式多个内部状态的索引化有限自动机）来解决非均匀承诺判定问题族。在非均匀有限自动机族中，我们特别关注非确定性有限自动机的变体，这些变体至多包含"一条"（唯一性）、"多项式多条"（少量）可接受计算路径，或通向每个固定配置的唯一/少量计算路径。当此类机器仅允许单向磁头移动时，我们无需依赖未证明的硬度假设即可证明其中某些变体在计算能力上彼此不同。对于限制为多项式有界长度的双向机器，双向多项式规模非确定性有限自动机族与多项式规模唯一性有限自动机族的能力等价。