  We present several new results and connections between various extensions of
finite automata through the study of vector automata and homing vector
automata. We show that homing vector automata outperform extended finite
automata when both are defined over $ 2 \times 2 $ integer matrices. We study
the string separation problem for vector automata and demonstrate that
generalized finite automata with rational entries can separate any pair of
strings using only two states. Investigating stateless homing vector automata,
we prove that a language is recognized by stateless blind deterministic
real-time version of finite automata with multiplication iff it is commutative
and its Parikh image is the set of nonnegative integer solutions to a system of
linear homogeneous Diophantine equations.
