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

Unambiguity and Fewness for Nonuniform Families of Polynomial-Size Nondeterministic Finite Automata