A small minimal aperiodic reversible Turing machine

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Journal of Computer and System Sciences, ISSN: 0022-0000, Vol: 84, Page: 288-301

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Julien Cassaigne; Nicolas Ollinger; Rodrigo Torres-Avilés
Elsevier BV
Mathematics; Computer Science
article description
A simple reversible Turing machine with four states, three symbols and no halting configuration is constructed that has no periodic orbit, simplifying a construction by Blondel, Cassaigne and Nichitiu and positively answering a conjecture by Kari and Ollinger. The constructed machine has other interesting properties: it is symmetric both for space and time and has a topologically minimal associated dynamical system whose column shift is associated to a substitution. Using a particular embedding technique of an arbitrary reversible Turing machine into the one presented, it is proven that the problem of determining if a given reversible Turing machine without halting state has a periodic orbit is undecidable.