On relations between properties in transitive Turing machines

Abstract

For over two decades, Turing machines (TMs) have been studied as dynamical systems. Several results related to topological properties were established, such as equicontinuity, periodicity, mortality, and entropy. There are two main topological models for TMs, and these properties strongly depend on the considered model. Here, we focus on transitivity, minimality and other related properties. In the context of TMs, transitivity refers to the existence of a configuration whose evolution contains every possible pattern over any finite window. Minimality means that every configuration fulfills the aforementioned statement, which strongly restricts TM behaviour. This paper establishes relations between the following properties: transitivity, minimality, the existence of blocking words, aperiodicity and reversibility. It also explores some properties of the embedding technique, which combines two TMs to produce a third. This technique has been used in previous works to prove the undecidability of several dynamical properties. Here, we demonstrate its power and versatility and how the produced machine, under a few particular conditions, will inherit the properties of one of the original machines.