Undecidability of the Surjectivity of the Subshift Associated to a Turing Machine

Torres, Rodrigo; Ollinger, Nicolas; Gajardo, Anahí

doi:10.1007/978-3-642-36315-3_4

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…A transitive system needs to be surjective, thus if the TMT or the TMH models are transitive, the machine needs to be reversible. Note this is not the case for trace-shift, as there exist non surjective Turing machines with surjective trace-shift [14].…”

Section: Characterizing Transitivitiesmentioning

confidence: 99%

The Transitivity Problem of Turing Machines

Gajardo

Ollinger

Torres-Avilés

2015

Mathematical Foundations of Computer Science 2015

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A Turing machine is topologically transitive if every partial configuration-that is a state, a head position, plus a finite portion of the tape-can reach any other partial configuration, provided that it is completed into a proper configuration. We characterize topological transitivity and study its computational complexity in the dynamical system models of Turing machines with moving head, moving tape and for the trace-shift. We further study minimality, the property of every configuration reaching every partial configuration.

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Section: Characterizing Transitivitiesmentioning

confidence: 99%

The Transitivity Problem of Turing Machines

Gajardo

Ollinger

Torres-Avilés

2015

Mathematical Foundations of Computer Science 2015

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“…Reversibility is equivalent to surjectivity in TMs (as proven in [2]), which is kept in TMT and TMH dynamical models, and therefore, it is a necessary condition for transitivity. Nevertheless, the trace-shift can be surjective even if the machine is not, but this scenario can occur only when the machine has what we call blocking words [24]. Both properties are strongly related to transitivity and minimality.…”

Section: Introductionmentioning

confidence: 97%

“…This one-way subshift consists of the sequence of states and symbols that the head reads throughout time. Previous works considered properties such as soficity, complexity, transitivity, periodicity and surjectivity [8,9,11,23,24].…”

Section: Introductionmentioning

confidence: 99%

On relations between properties in transitive Turing machines

Torres-Avilés,

Gajardo,

Ollinger

2023

Nonlinearity

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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.

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Undecidability of the speed positiveness problem in reversible and complete Turing machines

Torres-Avilés

2021

Journal of Computer and System Sciences

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Undecidability of the Surjectivity of the Subshift Associated to a Turing Machine

Cited by 4 publications

References 8 publications

The Transitivity Problem of Turing Machines

The Transitivity Problem of Turing Machines

On relations between properties in transitive Turing machines

Undecidability of the speed positiveness problem in reversible and complete Turing machines

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