  We study the reachability problem for continuous one-counter automata, COCA
for short. In such automata, transitions are guarded by upper and lower bound
tests against the counter value. Additionally, the counter updates associated
with taking transitions can be (non-deterministically) scaled down by a nonzero
factor between zero and one. Our three main results are as follows: (1) We
prove that the reachability problem for COCA with global upper and lower bound
tests is in NC2; (2) that, in general, the problem is decidable in polynomial
time; and (3) that it is decidable in the polynomial hierarchy for COCA with
parametric counter updates and bound tests.
