On a Conjecture of Kurka. A Turing Machine with No Periodic Configurations

Authors:
Vincent D. Blondel;Julien Cassaigne;Codrin Nichitiu
Affiliations:
-;-;-
Venue:
MCU '01 Proceedings of the Third International Conference on Machines, Computations, and Universality
Year:
2001

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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.