On the presence of periodic configurations in Turing machines and in counter machines

Periodicity and Immortality in Reversible Computing

“…Theorem 5 ( [10]). It is undecidable whether a given complete 2-DCM admits a periodic configuration.…”

“…".) To prevent such infinite searches we follow the idea of [7], also employed in [10]. Instead of a straightforward search using a loop, the search is done by performing a recursive call to the counter machine from its initial configuration (s 0 , 0, 0).…”

“…In [10] a complete DTM over the binary tape alphabet was provided that does not have any periodic configurations. This easily gives an analogous DTM for any bigger tape alphabet.…”

“…The existence of periodic orbits in Turing machines and counter machines have been investigated before in [9,10]. Article [9] formulated a conjecture that every complete Turing machine (under the moving tape mode) has at least one periodic orbit, while [10] refuted the conjecture by providing an explicit counter example. The counter example followed the general idea of [7] in that recursive calls were used to prevent unbounded searches.…”

“…The counter example followed the general idea of [7] in that recursive calls were used to prevent unbounded searches. In [10] is was also shown that it is undecidable if a given complete counter machine has a periodic orbit. We show that this is the case even under the additional constraint of reversibility (Theorem 6).…”