It is widely known that emptiness of counter automata is undecidable since two counters are enough to simulate a Turing machine (see the classic book from Hopcroft and Ullman, for example).
However, what happens if we put a bound $k$ on the values stored by the counters? In this restricted model, the counters cannot be incremented more than $k$.
I think this makes the problem decidable, is this correct? And in this case, which is the complexity? Is there any reference about this problem?