Computability of the entropy of one-tape Turing Machines

TL;DR

This paper demonstrates that the maximum speed and entropy of one-tape Turing machines are computable and can be approximated to any precision, challenging the common belief about undecidability of dynamical properties.

Contribution

It proves the computability of speed and entropy for one-tape Turing machines, a result not applicable to multi-tape machines, using crossing sequences methodology.

Findings

Maximum speed is computable and approximable.

Entropy of one-tape Turing machines is computable.

Contradicts the belief that all dynamical properties are undecidable.

Abstract

We prove that the maximum speed and the entropy of a one-tape Turing machine are computable, in the sense that we can approximate them to any given precision $\epsilon$. This is contrary to popular belief, as all dynamical properties are usually undecidable for Turing machines. The result is quite specific to one-tape Turing machines, as it is not true anymore for two-tape Turing machines by the results of Blondel et al., and uses the approach of crossing sequences introduced by Hennie.