## Advances in Linear LogicJean-Yves Girard, Yves Lafont, Laurent Regnier Linear logic, introduced in 1986 by J.-Y. Girard, is based upon a fine grain analysis of the main proof-theoretical notions of logic. The subject develops along the lines of denotational semantics, proof nets and the geometry of interaction. Its basic dynamical nature has attracted computer scientists, and various promising connections have been made in the areas of optimal program execution, interaction nets and knowledge representation. This book is the refereed proceedings of the first international meeting on linear logic held at Cornell University, in June 1993. Survey papers devoted to specific areas of linear logic, as well as an extensive general introduction to the subject by J.-Y. Girard, have been added, so as to make this book a valuable tool both for the beginner and for the advanced researcher. |

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### Contents

its syntax and semantics | 1 |

Part I Categories and semantics | 43 |

Part II Complexity and expressivity | 109 |

Part III Proof theory | 197 |

Part IV Proof nets | 225 |

Part V Geometry of interaction | 307 |

### Common terms and phrases

algebra atoms bimodules cell classical logic coherent space commutative proof compatible nodes conclusion conﬁguration connectives constant-only construction corresponding cut-elimination cut-free cut-free proof Danos deﬁned deﬁnition denote derivation edge encoding equivalent example exponential fact ﬁnd ﬁnite ﬁrst order formula occurrences fragment of linear functor game semantics geometry of interaction Girard graph HCohL hence Hilbert space hypercoherences induction hypothesis inﬁnite interpretation intuitionistic logic isometry isomorphism Jean-Yves Girard labelled Lemma linear logic linear negation MALL means MNLL modalities modiﬁcation morphisms multiset natural deduction nets nilpotent non-empty noncommutative proof structure notion NP-complete obtained occurs operators paths permutation polynomial premise problem proof-nets propositional linear logic provable prove qualitative domain quantiﬁers quasi-atomic reduction replace resp result satisﬁes sequent calculus simple sequent speciﬁc structural rules subset switching theorem theory VALUE(P variables veriﬁer vertex wire