### Abstract

Full Intuitionistic Linear Logic (FILL) was first introduced by Hyland and de Paiva, and went against current beliefs that it was not possible to incorporate all of the linear connectives, e.g. tensor, par, and implication, into an intuitionistic linear logic. It was shown that their formalization of FILL did not enjoy cut-elimination by Bierman, but Bellin proposed a change to the definition of FILL in the hope to regain cut-elimination. In this note we adopt Bellin’s proposed change and give a direct proof of cut-elimination. Then we show that a categorical model of FILL in the basic dialectica category is also a LNL model of Benton and a full tensor model of Melli`es’ and Tabareau’s tensorial logic. Lastly, we give a double-negation translation of linear logic into FILL that explicitly uses par in addition to tensor.

Original language | English (US) |
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Title of host publication | Logical Foundations of Computer Science - International Symposium, LFCS 2016, Proceedings |

Editors | Anil Nerode, Sergei Artemov |

Publisher | Springer Verlag |

Pages | 90-105 |

Number of pages | 16 |

ISBN (Print) | 9783319276823 |

DOIs | |

State | Published - Jan 1 2016 |

Event | International Symposium on Logical Foundations of Computer Science, LFCS 2016 - Deerfield Beach, United States Duration: Jan 4 2016 → Jan 7 2016 |

### 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 | 9537 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | International Symposium on Logical Foundations of Computer Science, LFCS 2016 |
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Country | United States |

City | Deerfield Beach |

Period | 1/4/16 → 1/7/16 |

### Fingerprint

### Keywords

- Categorical model
- Classical linear logic
- Cut-elimination
- Dialectica category
- Full intuitionistic linear logic
- Linear/non-linear models
- Par
- Proof theory
- Tensorial logic

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Logical Foundations of Computer Science - International Symposium, LFCS 2016, Proceedings*(pp. 90-105). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9537). Springer Verlag. https://doi.org/10.1007/978-3-319-27683-0_7

**Multiple conclusion linear logic : Cut elimination and more.** / Eades, Harley; de Paiva, Valeria.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Logical Foundations of Computer Science - International Symposium, LFCS 2016, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9537, Springer Verlag, pp. 90-105, International Symposium on Logical Foundations of Computer Science, LFCS 2016, Deerfield Beach, United States, 1/4/16. https://doi.org/10.1007/978-3-319-27683-0_7

}

TY - GEN

T1 - Multiple conclusion linear logic

T2 - Cut elimination and more

AU - Eades, Harley

AU - de Paiva, Valeria

PY - 2016/1/1

Y1 - 2016/1/1

N2 - Full Intuitionistic Linear Logic (FILL) was first introduced by Hyland and de Paiva, and went against current beliefs that it was not possible to incorporate all of the linear connectives, e.g. tensor, par, and implication, into an intuitionistic linear logic. It was shown that their formalization of FILL did not enjoy cut-elimination by Bierman, but Bellin proposed a change to the definition of FILL in the hope to regain cut-elimination. In this note we adopt Bellin’s proposed change and give a direct proof of cut-elimination. Then we show that a categorical model of FILL in the basic dialectica category is also a LNL model of Benton and a full tensor model of Melli`es’ and Tabareau’s tensorial logic. Lastly, we give a double-negation translation of linear logic into FILL that explicitly uses par in addition to tensor.

AB - Full Intuitionistic Linear Logic (FILL) was first introduced by Hyland and de Paiva, and went against current beliefs that it was not possible to incorporate all of the linear connectives, e.g. tensor, par, and implication, into an intuitionistic linear logic. It was shown that their formalization of FILL did not enjoy cut-elimination by Bierman, but Bellin proposed a change to the definition of FILL in the hope to regain cut-elimination. In this note we adopt Bellin’s proposed change and give a direct proof of cut-elimination. Then we show that a categorical model of FILL in the basic dialectica category is also a LNL model of Benton and a full tensor model of Melli`es’ and Tabareau’s tensorial logic. Lastly, we give a double-negation translation of linear logic into FILL that explicitly uses par in addition to tensor.

KW - Categorical model

KW - Classical linear logic

KW - Cut-elimination

KW - Dialectica category

KW - Full intuitionistic linear logic

KW - Linear/non-linear models

KW - Par

KW - Proof theory

KW - Tensorial logic

UR - http://www.scopus.com/inward/record.url?scp=84952815116&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84952815116&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-27683-0_7

DO - 10.1007/978-3-319-27683-0_7

M3 - Conference contribution

AN - SCOPUS:84952815116

SN - 9783319276823

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 90

EP - 105

BT - Logical Foundations of Computer Science - International Symposium, LFCS 2016, Proceedings

A2 - Nerode, Anil

A2 - Artemov, Sergei

PB - Springer Verlag

ER -