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 finite automata. We also show an analogous reduction for automata that have access to both a pushdown stack and reversal-bounded counters (PRBCA). This allows us to answer several open questions: For example, we show that it is coNP-complete to decide whether a given (P)RBCA language \(L\) is bounded, meaning whether there exist words \(w_1,\ldots,w_n\) with \( L \subseteq w^*_1 \cdots w^*_n \). For PRBCA, even decidability was open. Our methods also show that there is no language of a (P)RBCA of intermediate growth. This means, the number of words of each length grows either polynomially or exponentially. Part of our proof is likely of independent interest: We show that one can translate an RBCA into a machine with \(\mathbb{Z}\)-counters in logarithmic space, while preserving the accepted language.