Some undecidable problems about the trace-subshift associated to a Turing machine

Abstract

We consider three problems related to dynamics of one-tape Turing machines: Existence of blocking configurations, surjectivity in the trace, and entropy positiveness. In order to address them, a reversible two-counter machine is simulated by a reversible Turing machine on the right side of its tape. By completing the machine in different ways, we prove that none of the former problems is decidable. In particular, the problems about blocking configurations and entropy are shown to be undecidable for the class of reversible Turing machines.

Keywords

• turing machine
• discrete-time dynamical system
• subshift
• formal language
• entropy
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]