### Abstract

An involutory function, also called involution, is a function that is its own inverse, i.e., holds whenever is defined. This paper presents a computational model of involution as a variant of Turing machines, called an involutory Turing machine. The computational model is shown to be complete in the sense that not only does an involutory Turing machine always compute an involution but also every involutory computable function can be computed by an involutory Turing machine. As any involution is injective (hence reversible), any involutory Turing machine forms a standard reversible Turing machine that is backward deterministic. Furthermore, the existence of a universal involutory Turing machine is shown under an appropriate redefinition of universality given by Axelsen and Glück for reversible Turing machines. This work is motivated by characterizing bidirectional transformation languages.

Original language | English |
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Title of host publication | Reversible Computation - 12th International Conference, RC 2020, Proceedings |

Editors | Ivan Lanese, Mariusz Rawski |

Publisher | Springer |

Pages | 54-70 |

Number of pages | 17 |

ISBN (Print) | 9783030524814 |

DOIs | |

Publication status | Published - 2020 |

Event | 12th International Conference on Reversible Computation,RC 2020 - Oslo, Norway Duration: 2020 Jul 9 → 2020 Jul 10 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 12227 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 12th International Conference on Reversible Computation,RC 2020 |
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Country | Norway |

City | Oslo |

Period | 20/7/9 → 20/7/10 |

### Keywords

- Bidirectional transformation language
- Involution
- Reversible Turing machine
- Time-symmetric machine
- Universal Turing machine

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Reversible Computation - 12th International Conference, RC 2020, Proceedings*(pp. 54-70). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12227 LNCS). Springer. https://doi.org/10.1007/978-3-030-52482-1_3