Reversible Computations of One-Way Counter Automata

Abstract

Deterministic one-way time-bounded multi-counter automata are studied with respect to their ability to perform reversible computations, which means that the automata are also backward deterministic and, thus, are able to uniquely step the computation back and forth. We study the computational capacity of such devices and obtain separation results between irreversible and reversible k-counter automata for polynomial and superpolynomial time. For polynomial time and exponential time we obtain moreover infinite and tight hierarchies with respect to the number of counters. These hierarchies are shown with Kolmogorov complexity and incompressibility arguments. In this way, on passing we can prove these hierarchies also for ordinary counter automata. This improves the known hierarchies for ordinary counter automata in the sense that we consider here a weaker acceptance condition. Then, it turns out that k +1 reversible counters are not better than k ordinary counters and vice versa. Finally, almost all usually studied decidability questions turn out to be undecidable and not even semidecidable for reversible multi-counter automata, if at least two counters are provided. In case of reversible one-counter automata, it is possible to show that the inclusion problem is also not even semidecidable.