### Abstract

It is well-known that every proof net of a non-commutative version of MLL (Multiplicative fragment of Commutative Linear Logic) can be drawn as a plane Danos-Regnier graph (drawing) satisfying the switching condition of Danos-Regnier [3]. In this paper, we study the reverse direction; we introduce a system MNCLL logically equivalent to the multiplicative fragment of Cyclic Linear Logic introduced by Yetter [9], and show that any plane Danos-Regnier graph drawing satisfying the switching condition represents a unique non-commutative proof net (i.e., a proof net of MNCLL) modulo cyclic shifts. In the course of proving this, we also give the characterization of the non-commutative proof nets by means of the notion of strong planity, as well as the notion of a certain long-trip condition, called the stack-condition, of a Danos-Regnier graph, the latter of which is related to Abrusci balanced long-trip condition [2].

Original language | English |
---|---|

Pages (from-to) | 153 |

Number of pages | 1 |

Journal | Electronic Notes in Theoretical Computer Science |

Volume | 3 |

Issue number | C |

DOIs | |

Publication status | Published - 1996 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)
- Computer Science (miscellaneous)

### Cite this

**A Graph-Theoretic Characterization Theorem for Multiplicative Fragment of Non-Commutative Linear Logic (Extended Abstract).** / Nagayama, Misao; Okada, Mitsuhiro.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - A Graph-Theoretic Characterization Theorem for Multiplicative Fragment of Non-Commutative Linear Logic (Extended Abstract)

AU - Nagayama, Misao

AU - Okada, Mitsuhiro

PY - 1996

Y1 - 1996

N2 - It is well-known that every proof net of a non-commutative version of MLL (Multiplicative fragment of Commutative Linear Logic) can be drawn as a plane Danos-Regnier graph (drawing) satisfying the switching condition of Danos-Regnier [3]. In this paper, we study the reverse direction; we introduce a system MNCLL logically equivalent to the multiplicative fragment of Cyclic Linear Logic introduced by Yetter [9], and show that any plane Danos-Regnier graph drawing satisfying the switching condition represents a unique non-commutative proof net (i.e., a proof net of MNCLL) modulo cyclic shifts. In the course of proving this, we also give the characterization of the non-commutative proof nets by means of the notion of strong planity, as well as the notion of a certain long-trip condition, called the stack-condition, of a Danos-Regnier graph, the latter of which is related to Abrusci balanced long-trip condition [2].

AB - It is well-known that every proof net of a non-commutative version of MLL (Multiplicative fragment of Commutative Linear Logic) can be drawn as a plane Danos-Regnier graph (drawing) satisfying the switching condition of Danos-Regnier [3]. In this paper, we study the reverse direction; we introduce a system MNCLL logically equivalent to the multiplicative fragment of Cyclic Linear Logic introduced by Yetter [9], and show that any plane Danos-Regnier graph drawing satisfying the switching condition represents a unique non-commutative proof net (i.e., a proof net of MNCLL) modulo cyclic shifts. In the course of proving this, we also give the characterization of the non-commutative proof nets by means of the notion of strong planity, as well as the notion of a certain long-trip condition, called the stack-condition, of a Danos-Regnier graph, the latter of which is related to Abrusci balanced long-trip condition [2].

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U2 - 10.1016/S1571-0661(05)82518-6

DO - 10.1016/S1571-0661(05)82518-6

M3 - Article

VL - 3

SP - 153

JO - Electronic Notes in Theoretical Computer Science

JF - Electronic Notes in Theoretical Computer Science

SN - 1571-0661

IS - C

ER -