The word problem distinguishes counter languages
  Counter automata are more powerful versions of finite-state automata where addition and subtraction operations are permitted on a set of n integer registers, called counters. We show that the word problem of $\Z^n$ is accepted by a nondeterministic $m$-counter automaton if and only if $m \geq n$. 
Sean Cleary, Murray Elder, and Gretchen Ostheimer