## From proof nets to the free *- autonomous category (2005)

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Venue: | Logical Methods in Computer Science, 2(4:3):1–44, 2006. Available from: http://arxiv.org/abs/cs/0605054. [McK05] Richard McKinley. Classical categories and deep inference. In Structures and Deduction 2005 (Satellite Workshop of ICALP’05 |

Citations: | 21 - 7 self |

### BibTeX

@INPROCEEDINGS{Lamarche05fromproof,

author = {François Lamarche and Lutz Strassburger},

title = {From proof nets to the free *- autonomous category},

booktitle = {Logical Methods in Computer Science, 2(4:3):1–44, 2006. Available from: http://arxiv.org/abs/cs/0605054. [McK05] Richard McKinley. Classical categories and deep inference. In Structures and Deduction 2005 (Satellite Workshop of ICALP’05},

year = {2005}

}

### OpenURL

### Abstract

Vol. 2 (4:3) 2006, pp. 1–44 www.lmcs-online.org

### Citations

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Citation Context ... of these operations, the function module operation being a form of implication, whose associated conjunction, tensoring, was not necessarily commutative. Monoidal closed categories first appeared in =-=[EK66]-=-, and they give the necessary axiomatic treatment for Lambek’s implication; as a matter of fact they were formulated in such a way that in some cases an implication/function object operator can be def... |

167 | A new constructive logic: Classical logic - Girard - 1991 |

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Citation Context ...� and � links are used in a particular fashion to deal with the units. (The readers can choose their favorite correctness criterion since they are all equivalent; in this paper we will use the one of =-=[DR89]-=- because of its popularity.) • The equivalence relation we will present is based on a very simple set of rewriting rules on proof graphs. As a matter of fact, there is only one non-trivial rule, since... |

115 | Weakly distributive categories - Cockett, Seely - 1997 |

114 |
On the calculus of syntactic types
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Citation Context ...his additional structure on modules led to the axiomatization of monoidal closed categories. Even before these abstract concepts of monoidal and monoidal closed categories had been formulated, Lambek =-=[Lam61]-=- had noticed the strong logical flavor of these operations, the function module operation being a form of implication, whose associated conjunction, tensoring, was not necessarily commutative. Monoida... |

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93 | Proof-nets: the parallel syntax for proof theory. Logic and Algebra
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Citation Context ...lem of constructing free *-autonomous categories, which improves on its predecessors in that the cut-elimination process can be effected at the level of the representatives of the equivalence classes =-=[Gir96b]-=-. Outline of the paper. This paper consists of two parts. The first one is only concerned with syntax: the sequent calculus and the more modern syntax of proof nets, and our variation on ordinary mult... |

84 | Quantales and (Non-Commutative) Linear Logic - Yetter - 1990 |

66 | Natural deduction and coherence for weakly distributive categories
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Citation Context ...nly two other treatments of multiplicative units that we were aware of. In [KO99], the authors provide an internal language for autonomous and *-autonomous categories based on the λµ-calculus, and in =-=[BCST96]-=-, several classes of free monoidal categories are constructed, by the means of a nonstandard version of two-sided proof nets, with a correctness criterion which is a version of the Danos contractibili... |

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57 | From proof nets to interaction nets
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Citation Context ...L) comes equipped with an extremely successful theory of proof identification: not only do we know exactly when two sequent proofs should be identified (the allowed rule permutations are described in =-=[Laf95]-=-), but there is a class of simple formal objects that precisely represent these equivalence classes of sequent proofs. These objects are called proof nets, and they have a strong geometric character, ... |

57 | Deductive Systems and Categories II: Standard Constructions and Closed Categories - Lambek - 1969 |

55 | Non-commutativity and MELL in the calculus of structures - Guglielmi, Straßburger - 2001 |

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46 | Proof normalization modulo
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Citation Context ...e notoriously suspicious of non-logical axioms. In general they kill cut-elimination/normalization, but there are indeed classes of harmless non-logical axioms. Recently, general theoretical criteria =-=[DW03]-=- have been developed that allow the identification of such harmless classes. Since this manuscript was submitted, yet another approach [Hug05a] has been proposed to the problem of constructing free *-... |

45 |
Autonomous Categories, volume 752
- Barr
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Citation Context ...FROM PROOF NETS TO THE FREE ∗-AUTONOMOUS CATEGORY 3 Moreover proof nets are vindicated by category theory, since the category of two-formula sequents and proof nets is the free *-autonomous category =-=[Bar79]-=- (if we omit units from the definition of a *-autonomous category) on the set of generating atomic formulas. This first appeared in [Blu93], but the problem of precisely defining a *-autonomous catego... |

42 | Proof nets for unit-free multiplicativeadditive linear logic
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Citation Context ...particularly well with the categorical ideal—and can be firmly put in the “successful” class even without appealing to categorical considerations—is the inclusion of additive connectives presented in =-=[HvG03]-=-, in which the additives correspond exactly to categorical product and coproduct. In this paper we give a theory of proof nets for the full multiplicative fragment, that is, including the multiplicati... |

42 | Categories for the Working Mathematician. Number 5 - Lane - 1971 |

30 |
La Logique Linéaire appliquée à l’étude de divers processus de normalisation
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Citation Context ...sses of free monoidal categories are constructed, by the means of a nonstandard version of two-sided proof nets, with a correctness criterion which is a version of the Danos contractibility criterion =-=[Dan90]-=-. Being based on the λµ-calculus, the first of these papers is firmly in the tradition of natural deduction, in which the logical rules (introduction/elimination) mechanically generate the system of p... |

27 |
The mix rule
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Citation Context ...). Then G ′ is called an extended switching of P ⊲ Γ with respect to ⊥i. Observe that, if P ⊲ Γ is correct, then every extended switching is disconnected and consists of two connected components (see =-=[FR94]-=- for a discussion on connected components in switchings). We will use the notation P {Q} ⊲ Γ to distinguish the subtree Q of the linking tree of the graph. Then P { } is the context of Q. 1.2.5. Equiv... |

23 | Proof theory for full intuitionistic linear logic, bilinear logic - Cockett, Seely - 1997 |

23 | Exploring the gap between linear and classical logic - Lamarche - 2007 |

21 | Order-enriched Categorical Models of the Classical Sequent Calculus - Führmann, Pym |

21 | Naming proofs in classical propositional logic - Lamarche, Straßburger - 2005 |

17 | Deductive systems and categories I: Syntactic calculus and residuated categories - Lambek - 1968 |

17 | Constructing free Boolean categories - Lamarche, Straßburger - 2005 |

15 | Coherent Banach spaces: a continuous denotational semantics. Theoretical Computer Science, 227:275 297
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Citation Context ...ns: applying Chu to sets and functions gives the category of Lafont-Streicher games [LS91] (now better known as “Chu spaces”), and applying it to Banach spaces gives Girard’s “coherent Banach spaces” =-=[Gir96a]-=-, thus closing a fifty-year loop. 2.1. Basic definitions and properties. In this section we will recall the definition of a *-autonomous category and show some properties that they have and that we wi... |

15 | Handsome proof-nets: perfect matchings and cographs - Retoré |

14 | Proof nets for the Lambek-calculus — an overview - Lamarche, Retoré - 1996 |

14 | On proof nets for multiplicative linear logic with units - Straßburger, Lamarche - 2004 |

11 | A linear logical view of linear type isomorphisms
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Citation Context ...t simplifies many of the diagrams, avoiding unnecessary crossings, and this as much for proof nets as for commutative diagrams. 6 If Γ consists of only one formula, then we have an object which is in =-=[BC99]-=- called a bipartite proof net. In fact, two proof graphs (in our sense) are equivalent if and only if the two linkings (seen as formulas) are isomorphic (in the sense of [BC99]).12 F. LAMARCHE AND L.... |

11 |
Adjunction of Semifunctors: Categorical Structures
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Citation Context ... with the η-rule, because it makes normalization much harder, if not impossible. But it is not easy at all to construct a denotational semantics that does not obey the η-rule, although it can be done =-=[Hay85]-=-. As another example of tension, if a poset can be used to embody provability—A ≤ B means I can prove B if I assume A—replacing that poset by a category will allow us to name proofs, and to single one... |

10 |
logic, coherence and dinaturality
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(Show Context)
Citation Context ...la sequents and proof nets is the free *-autonomous category [Bar79] (if we omit units from the definition of a *-autonomous category) on the set of generating atomic formulas. This first appeared in =-=[Blu93]-=-, but the problem of precisely defining a *-autonomous category without units has given rise to recent developements [LS05a, HHS05, DP05]. As a matter of fact, axiom links were already visible, under ... |

10 | On the geometry of interaction for classical logic - Führmann, Pym - 2004 |

9 | Modelling linear logic without units (preliminary results). Preprint, available at http://arxiv.org/ abs/math/0504037 - Houston, Hughes, et al. - 2005 |

9 | Infinite dimensional vector spaces - Mackey - 1945 |

9 | Hilbert’s twenty-fourth problem - Thiele, Wos |

8 | Simple free star-autonomous categories and full coherence. Preprint, available at http://arxiv.org/abs/math.CT/0506521
- Hughes
- 2005
(Show Context)
Citation Context ...non-logical axioms. Recently, general theoretical criteria [DW03] have been developed that allow the identification of such harmless classes. Since this manuscript was submitted, yet another approach =-=[Hug05a]-=- has been proposed to the problem of constructing free *-autonomous categories, which improves on its predecessors in that the cut-elimination process can be effected at the level of the representativ... |

7 |
Constructing ∗-autonomous categories
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Citation Context ...sarily the unit to the tensor. He also found a general technique for producing such categories out of any ordinary monoidal closed category that has pullbacks; this is now called the Chu construction =-=[Chu79]-=-. The main inspiration for the Chu construction had been around since Mackey’s thesis of 1942, published in 1945 [Mac45], and can be summarized as follows. In the realm of topological vector spaces, i... |

6 | On the Geometry of Interaction for Classical Logic (Extended Abstract - Fuhrman, Pym - 2004 |

6 | and Valeria de Paiva. Full intuitionistic linear logic (extended abstract - Hyland - 1993 |

5 | Entropic hopf algebras and models of noncommutative linear logic. Theory and Applications of Categories - Blute, Lamarche, et al. - 2002 |

5 |
On the algebra of structural contexts. Accepted at
- Lamarche
- 2001
(Show Context)
Citation Context ...nly sets of objects (atomic variables) as generating sets. It would be easy to extend our work to construct the free *-autonomous category generated by an arbitrary category, or an arbitrary structad =-=[Lam01]-=-, but we are trying very hard to be read by both algebraists and proof theorists, and proof theorists are notoriously suspicious of non-logical axioms. In general they kill cut-elimination/normalizati... |

5 | Homology of proof-nets - Métayer - 1994 |

5 | On the Geometry of Interaction for Classical Logic (Extended Abstract - Führmann, Pym |

4 | Simple multiplicative proof nets with units - Hughes |

3 | Lambda -autonomous categories of bimodules - Rosenthal - 1994 |

3 | Natural deduction and coherence for non-symmetric linearly distributive categories - Schneck - 1999 |