Periodicity and Immortality in Reversible Computing

We investigate the decidability of the periodicity and the immortality problems in three models of reversible computation: reversible counter machines, reversible Turing machines and reversible one-dimensional cellular automata. Immortality and periodicity are properties that describe the behavior of the model starting from arbitrary initial 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.

[1]  Vincent D. Blondel,et al.  On the presence of periodic configurations in Turing machines and in counter machines , 2002, Theoretical Computer Science.

[2]  J.H. van Schuppen,et al.  Observability of hybrid systems and turing machines , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[3]  Arto Salomaa,et al.  Algorithmic Bioprocesses , 2009, Natural Computing Series.

[4]  Jarkko Kari,et al.  Theory of cellular automata: A survey , 2005, Theor. Comput. Sci..

[5]  Petr Kurka,et al.  On Topological Dynamics of Turing Machines , 1997, Theor. Comput. Sci..

[6]  R. Landauer,et al.  Irreversibility and heat generation in the computing process , 1961, IBM J. Res. Dev..

[7]  Charles H. Bennett,et al.  Logical reversibility of computation , 1973 .

[8]  Masato Ishikawa,et al.  43rd IEEE Conference on Decision and Control , 2005 .

[9]  Kenichi Morita,et al.  Universality of a Reversible Two-Counter Machine , 1996, Theor. Comput. Sci..

[10]  Philip K. Hooper.  The undecidability of the Turing machine immortality problem , 1966, Journal of Symbolic Logic.

[11]  Jarkko Kari,et al.  Some Undecidable Dynamical Properties for One-Dimensional Reversible Cellular Automata , 2009, Algorithmic Bioprocesses.