Periodicity and Immortality in Reversible Computing

Kari, Jarkko; Ollinger, Nicolas

doi:10.1007/978-3-540-85238-4_34

Cited by 60 publications

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“…To do this, we will start from the proof of the undecidability of the immortality problem by Kari and Ollinger [11], and explain how to modify it for our purposes. In particular, a thorough examination of [11] by the reader is encouraged.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…In particular, a thorough examination of [11] by the reader is encouraged. We will try as much as possible to use the same notations.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…It's important to note that the proof of Hooper and subsequent proofs [11] do not give any insights about the structure of I(M ). Indeed, in these proofs, I(M ), if not empty, always contains a finite configuration (ie every cell of the tape, except a finite number, contains the same symbol).…”

Section: Definitionsmentioning

confidence: 99%

“…Informally, a DCM M is reversible (RCM) if there exists a DCM M so that for every oracle w, (s, c) (s , c ) by M iff (s , c ) (s, c) by M , see [11] for a syntactical definition. Any DCM can be extended into a RCM by using two additional counters to store the previous instructions, so that we have Lemma 3.…”

Section: Informally An Oracle Counter Machines Thus Contain Three Tymentioning

confidence: 99%

“…The main ingredient is a new proof of Hooper's theorem by Kari and Ollinger [11] combined with an encoding into subshifts of an effective set with no recursive points. We first define all relevant vocabulary in the next section, then proceed to the proof.…”

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confidence: 99%

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On Immortal Configurations in Turing Machines

Jeandel

2012

Lecture Notes in Computer Science

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In this paper we investigate the behaviour of Turing machines seen as dynamical systems. In this context, a Turing machine does not start from a specified initial state and tape, but from any configuration. In this context, we call a configuration immortal if the machine runs forever starting from it. The seminal article of Hooper [8] proves that we cannot decide whether a Turing machine has an immortal configuration (is immortal). However, the result does not say anything about what the immortal configurations look like. In fact, in the construction by Hooper, if the Turing machine has an immortal configuration, then it has one where the tape is almost completely empty.A few results give some structure on the set of immortal configurations: Kurka [13] asked whether there always exists a (temporally) periodic configuration, and was refuted by Blondel et al. [1]. Delvenne and Blondel [6] proved that the set of immortal configurations, if nonempty, always contains quasi-periodic configurations.In this article, we go further, and give some news results on the set of immortal configurations. We prove in particular in Theorem 2 that there exists a Turing machine for which the set of immortal configurations is nonempty and contains no computable configurations.The main ingredient is a new proof of Hooper's theorem by Kari and Ollinger [11] combined with an encoding into subshifts of an effective set with no recursive points. We first define all relevant vocabulary in the next section, then proceed to the proof.

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Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…In particular, a thorough examination of [11] by the reader is encouraged. We will try as much as possible to use the same notations.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Section: Definitionsmentioning

confidence: 99%

Section: Informally An Oracle Counter Machines Thus Contain Three Tymentioning

confidence: 99%

mentioning

confidence: 99%