## Non-commutative logic II: sequent calculus and phase semantics (1998)

Citations: | 25 - 6 self |

### BibTeX

@MISC{Ruet98non-commutativelogic,

author = {Paul Ruet},

title = {Non-commutative logic II: sequent calculus and phase semantics},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

INTRODUCTION Non-commutative logic is a unication of : | commutative linear logic (Girard 1987) and | cyclic linear logic (Girard 1989; Yetter 1990), a classical conservative extension of the Lambek calculus (Lambek 1958). In a previous paper with Abrusci (Abrusci and Ruet 1999) we presented the multiplicative fragment of non-commutative logic, with proof nets and a sequent calculus based on the structure of order varieties, and a sequentialization theorem. Here we consider full propositional non-commutative logic. Non-commutative logic. Let us rst review the basic ideas. Consider the purely noncommutative fragment of linear logic, obtained by removing the exchange rule entirely : ` ; ; ; , ` ; ; ; y This work has been partly carried out at LIENS-CNRS, Ecole Normale Superieure (Paris), at McGill University

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Citation Context ... x 2 ?, thus x 2 A ? rB = (B ? \Delta A) ? . Conversely, assume x 2 A ? rB, and take a 2 A. For all y 2 B ? , y \Delta a \Delta x 2 ?, thus a \Delta x 2 B ?? = B, whence x 2 A \Gamma ffl B. \Xi As in =-=[7, 9]-=-, we extend the semantics to exponential connectives. If P is a phase space, define J(P ) = fx 2 1 j x 2 fx?xg ?? g. Note that x 2 J(P ) ) x 2 fx \Delta xg ?? , because fx ? xg ?? ` fx \Delta xg ?? . ... |

130 |
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Citation Context ... ?. Examples. I If (P; ?; 1; ?) is a commutative phase space in the sense of [5], then (P; ?; ?; 1; =; ?) is a phase space. I Let (P; \Delta; 1; ; ; ) be a lattice-ordered monoid (a good reference is =-=[4]-=-), i.e., a monoid (P; \Delta; 1) together with a lattice structure (; ; ) compatible with multiplication. The product x ? y = xysyx is obviously commutative with unit 1, and satisfies x ? ysxy. If for... |

100 |
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Citation Context .... . 32 5.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6 Cut elimination 34 1 Introduction Non-commutative logic unifies commutative linear logic [5] and cyclic linear logic =-=[6, 14]-=-. In a previous paper with Abrusci [1] we presented the multiplicative fragment of non-commutative logic, with proof nets and a sequent calculus based on the structure of order varieties, and a sequen... |

83 |
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Citation Context .... . 32 5.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6 Cut elimination 34 1 Introduction Non-commutative logic unifies commutative linear logic [5] and cyclic linear logic =-=[6, 14]-=-. In a previous paper with Abrusci [1] we presented the multiplicative fragment of non-commutative logic, with proof nets and a sequent calculus based on the structure of order varieties, and a sequen... |

56 |
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Citation Context ...fl \Gamma A = BrA ? . 2.2 Rules Definition 2.4 Sequents are of the form ` \Gamma, where \Gamma is a parially ordered set of occurrences of formulas. In practice the order will be series-parallel: see =-=[10]-=- for a survey on series-parallel orders. We just recall the definition of serial and parallel compositions of orders: let ! 1 and ! 2 be orders on disjoint sets E and F respectively; their serial and ... |

33 | Non-commutative Logic I: the multiplicative fragment
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(Show Context)
Citation Context .... . . . . . . . . . . . . . . . . . 34 6 Cut elimination 34 1 Introduction Non-commutative logic unifies commutative linear logic [5] and cyclic linear logic [6, 14]. In a previous paper with Abrusci =-=[1]-=- we presented the multiplicative fragment of non-commutative logic, with proof nets and a sequent calculus based on the structure of order varieties, and a sequentialization theorem. Here we consider ... |

28 | Partially commutative linear logic: sequent calculus and phase semantics - Groote - 1996 |

25 | The finite model property for various fragments of linear logic, manuscript
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(Show Context)
Citation Context ... x 2 ?, thus x 2 A ? rB = (B ? \Delta A) ? . Conversely, assume x 2 A ? rB, and take a 2 A. For all y 2 B ? , y \Delta a \Delta x 2 ?, thus a \Delta x 2 B ?? = B, whence x 2 A \Gamma ffl B. \Xi As in =-=[7, 9]-=-, we extend the semantics to exponential connectives. If P is a phase space, define J(P ) = fx 2 1 j x 2 fx?xg ?? g. Note that x 2 J(P ) ) x 2 fx \Delta xg ?? , because fx ? xg ?? ` fx \Delta xg ?? . ... |

22 | On the meaning of logical rules I: syntax vs. semantics - Girard - 1999 |

21 |
Logique non-commutative et programmation concurrente par contraintes
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Citation Context ... and a non-commutative one : : : Order varieties. The solution is based on: -- a syntactic idea: the seesaw rule, and -- its semantic counterpart: the structure of order variety. Order varieties (see =-=[13, 1]-=- and section 3) are structures that can be presented by partial orders in several ways, a good analogy being the oriented circle which becomes a total order as soon as an origin is fixed. An essential... |

13 | A complete axiomatisation of the inclusion of series-parallel partial orders - Bechet, Groote, et al. - 1997 |

9 |
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Citation Context ...; z)sff(x; y; t). An order variety ff on E is said total when 8x; y; z 2 E, x 6= y 6= z 6= x ) ff(x; y; z)sff(z; y; x). Ternary relations satisfying the first three axioms have been studied by Nov'ak =-=[11] and -=-called cyclic orders. A few elementary properties and examples of order varieties (for proofs, see [1]): Remarks. I As expected, if ff is a total order variety, ff(x; y; z) can be read as "y is b... |

8 |
Girard’s phase semantics and a higher-order cut-elimination proof
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(Show Context)
Citation Context ...y in section 5, we give a phase semantics. At this stage, we think it lacks a simple and natural construction; still it enables to prove (weak) cut elimination (section 6), using a technique of Okada =-=[12]-=-. 2 Sequent calculus : first version 2.1 Language Definition 2.1 The formulas (of NL) are built from atoms p; q; : : : , p ? ; q ? ; : : : , constants 1, ? (multiplicative), ?, 0 (additive), and the f... |

3 | Logiques Linéaires Hybrides et leurs Modalités: Théories et Applications - Demaille - 1999 |

2 |
On the meaning of logical rules I : syntax vs. semantics. Pr'epublication, Institut de Math'ematiques de Luminy
- Girard
- 1998
(Show Context)
Citation Context ...junction on a partial order: A times B can have therefore three different meanings, depending on the order between A and B. We are then faced with the following problems: 1 See Girard's Appendix F in =-=[8]-=- for the associativity problems in absence of cyclicity. 1. Entropy. We must be able ro replace a partial order by a weaker one (or a stronger one, depending on the connective considered), for instanc... |

1 |
Commutativity of the exponentials in mixed linear logics
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(Show Context)
Citation Context ...ices: 1. ?ed formulas do not commute in a non-commutative situation: from ` \Gamma[?A; B], we may not infer ` \Gamma[B; ?A], even if B is itself a ?ed formula. This has been considered by Demaille in =-=[3]-=-. 2. Bags of ?ed formulas commute. This has been considered in [13], and it is consistent with the intuition that there is basically a single par --- and a single tensor --- and the isomorphisms: (?A;... |

1 | The model property for various fragments of linear logic - Ruet - 1997 |