Unboundedness Problems for Machines with Reversal-Bounded Counters

We consider a general class of decision problems concerning formal languages, called (one-dimensional) unboundedness predicates, for automata that feature reversal-bounded counters (RBCA). We show that each problem in this class reduces-non-deterministically in polynomial time to the same problem for just nite automata. We also show an analogous reduction for automata that have access to both a push- down stack and reversal-bounded counters (PRBCA). This allows us to answer several open questions: For example, we settle the complexity of deciding whether a given (P)RBCA language L is bounded, meaning whether there exist words w1, . . . , wn with L ⊆ w1∗ · · · wn∗ . For PRBCA, even decidability was open. Our methods also show that there is no language of a (P)RBCA of intermediate growth. Part of our proof is likely of independent interest: We show that one can translate an RBCA into a machine with Z-counters in logarithmic space.