  Nonuniform families of polynomial-size finite automata and pushdown automata
respectively have strong connections to nonuniform-NL and nonuniform-LOGCFL. We
examine the behaviors of unambiguous and co-nondeterministic computations
produced by such families of automata operating multiple counters. As its
consequences, we obtain various collapses of the complexity classes of families
of promise problems solvable by finite and pushdown automata families when all
valid instances are limited to either polynomially long strings or unary
strings. A key technical ingredient of our proofs is an inductive counting of
reachable vertices of each computation graph of finite and pushdown automata
that operate multiple counters simultaneously.
