Abstract
A configuration of a Turing machine is given by a tape content together with a particular state of the machine. Petr Kůrka has conjectured that every Turing machine — when seen as a dynamical system on the space of its configurations — has at least one periodic orbit. In this paper, we provide an explicit counter-example to this conjecture. We also consider counter machines and prove that, in this case, the problem of determining if a given machine has a periodic orbit in configuration space is undecidable.
This research was supported by NATO under grant CRG-961115, by the European Community Framework IV program through the research network ALAPEDES, by the Belgian program on interuniversity attraction poles IUAP P4-02, and by the French Ministry of Education and Research.
Acknowledgment
The authors are grateful to Professor Maurice Margenstern for his suggestions of improvement.
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Blondel, V.D., Cassaigne, J., Nichitiu, C. (2001). On a Conjecture of Kůrka. A Turing Machine with No Periodic Configurations. In: Margenstern, M., Rogozhin, Y. (eds) Machines, Computations, and Universality. MCU 2001. Lecture Notes in Computer Science, vol 2055. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45132-3_10
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DOI: https://doi.org/10.1007/3-540-45132-3_10
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