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.

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