Undecidability of the surjectivity of the subshift associated to a turing machine

Abstract

We consider Turing machines (TM) from a dynamical system point of view, and in this context, we associate a subshift by taking the sequence of symbols and states that the head has at each instant. Taking a subshift that select only a part of the state of a system is a classical technic in dynamical systems that plays a central role in their analysis. Surjectivity of Turing machines is equivalent to their reversibility and it can be simply identified from the machine rule. Nevertheless, the associated subshift can be surjective even if the machine is not, and the property results to be undecidable in the symbolic system. © 2013 Springer-Verlag Berlin Heidelberg.