Periodicity and Immortality in Reversible Computing

Abstract. We investigate the decidability of the periodicity and the immortality problems in three models of reversible computation: re-versible counter machines, reversible Turing machines and reversible one-dimensional cellular automata. Immortality and periodicity are proper-ties that describe the behavior of the model starting from arbitrary ini-tial configurations: immortality is the property of having at least one non-halting orbit, while periodicity is the property of always eventually returning back to the starting configuration. It turns out that periodicity and immortality problems are both undecidable in all three models. We also show that it is undecidable whether a (not-necessarily reversible) Turing machine with moving tape has a periodic orbit