## A categorical semantics for polarized mall

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Venue: | Ann. Pure Appl. Logic |

Citations: | 2 - 0 self |

### BibTeX

@ARTICLE{Hamano_acategorical,

author = {Masahiro Hamano and Philip Scott},

title = {A categorical semantics for polarized mall},

journal = {Ann. Pure Appl. Logic},

year = {},

volume = {145},

pages = {2007}

}

### OpenURL

### Abstract

In this paper, we present a categorical model for Multiplicative Additive Polarized Linear Logic MALLP, which is the linear fragment (without structural rules) of Olivier Laurent’s Polarized Linear Logic. Our model is based on an adjunction between reflective/coreflective full subcategories C−/C+ of an ambient ∗-autonomous category C (with products). Similar structures were first introduced by M. Barr in the late 1970’s in abstract duality theory and more recently in work on game semantics for linear logic. The paper has two goals: to discuss concrete models and to present various completeness theorems. As concrete examples, we present (i) a hypercoherence model, using Ehrhard’s hereditary/anti-hereditary objects, (ii) a Chu-space model, (iii) a double gluing model over our categorical framework, and (iv) a model based on iterated double gluing over a ∗-autonomous category. For the multiplicative fragment MLLP of MALLP, we present both weakly full (Läuchli-style) as well as full completeness theorems, using a polarized version of functorial

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Citation Context ...ion is a derived property in the setting of O. Laurent, rather than a primitive notion as it is for us. Let us consider the adjoint equivalence in Lambek-Scott (Proposition 4.2 and Slogan V (p.18) of =-=[33]-=-) when applied to our framework. Remark 3.5 (Adjoint equivalence in Polarized Categories) An adjunction (F, G, η, ɛ) between categories A and B induces an adjoint equivalence Fix η ∼ = Fix ɛ between t... |

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Citation Context ...a “constructive” version of classical logic based on linear logic. These polarities turn out to be related to the notion of focussing in linear logic proof search, a method introduced by J-M Andreoli =-=[4, 5, 38]-=-. In a related direction, many papers in Game Semantics for linear logic have also stressed the notion of polarities, beginning with Lamarche [32] (see also the survey [1]). Olivier Laurent [34, 35] b... |

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Citation Context ...47 10.1 Appendix 1: On Polarized Categories à la M. Barr . . . . . . . . . . . . . . . 47 10.2 Appendix 2: Double Gluing . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1 Introduction Girard =-=[20]-=- introduced the notion of “polarized” (positive and negative) formulas in his study of the theory LC, a “constructive” version of classical logic based on linear logic. These polarities turn out to be... |

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Citation Context ...ucture. More generally still, if we do not assume the ambient category C is ∗-autonomous, we obtain essentially (i.e. with a slight modification) the structures first discussed by M. Barr in his book =-=[7]-=- under the name pre-∗-autonomous situations. These are discussed in Section 3 and Appendix 1 below. Recently, Cockett and Seely [15] have introduced their notion of polarized categories motivated by A... |

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Citation Context ... Chu(Set, K), with composition and identities given pointwise. Chu(Set, K) is a self-dual, complete (thus cocomplete) ∗-autonomous category, with small (co)limits inherited from Set (for details, see =-=[8, 39]-=-). For our purposes, we only need the following properties: Proposition 5.18 Chu(Set, K) is a ∗-autonomous category with products, thus a model of MALL. Proof. Let us sketch the relevant structure. Le... |

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Citation Context ...). A fundamental point of MALLP, is that the polarity-changing operations ↑ and ↓ are more primitive than the full 2sexponentials. On the one hand, O. Laurent shows that Selinger’s Control Categories =-=[40]-=- provide an adequate model for LLP. However, the question of natural categorical models for MALLP remained open. In [35], a notion of linear control category model for MALLP is introduced; however in ... |

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Citation Context ... for polarized logics. The novelties of our paper are in Sections 5 through 7. In Section 5, we present many concrete non-game-theoretic examples of our framework, including Ehrhard’s hypercoherences =-=[18]-=- (studied in our previous MALL full completeness work [10]), Chu spaces, and various models based on double gluing and iterated double gluing. In Sections 6,7 we begin our main focus: a study of full ... |

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Citation Context ...oof search, a method introduced by J-M Andreoli [4, 5, 38]. In a related direction, many papers in Game Semantics for linear logic have also stressed the notion of polarities, beginning with Lamarche =-=[32]-=- (see also the survey [1]). Olivier Laurent [34, 35] began a systematic study of polarized versions of linear logic. In his thesis he introduced polarized proof structures and nets which are simpler t... |

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Citation Context ...eral one: some connections with their work will be discussed below. Also we have recently become aware that structures similar to ours have arisen in (mostly unpublished) work of Melliès and Selinger =-=[36, 37]-=- again inspired by game semantics. Thus the kinds of structures we deal with here, which essentially go back to M. Barr in a totally different setting, seem to be a natural framework for polarized log... |

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Citation Context ... proof theory of linear logic. For example, in the case of multiplicative-additive linear logic MALL, we can divide the connectives according to whether their introduction rules are reversible or not =-=[21, 1]-=-. Those connectives which are reversible are called negative; those which are not are called positive. As we discussed above, positive connectives are the foundation of Andreoli’s influential notion o... |

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Citation Context ...egories permits us to reduce the full completeness problem for our polarized hypercoherence model for MLLP to the ordinary multiplicative full completeness problem for Coh, which was solved by A. Tan =-=[41]-=-. In Subsection 7.1 we prepare the background by introducing a polarized (MLLP) version of Joyal’s softness (with respect to removability of ↑). In Subsection 7.2 we prove a version of full completene... |

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Citation Context ...ur framework of positive/negative subcategories C+/C− of a ∗-autonomous category C, the condition of polarized n-ary softness can be characterized by means of an n-dimensional weak pushout (cf. Joyal =-=[31]-=-). E.g., when n = 3 the condition is equivalent to the fact that the following cube is a 3-dimensional weak pushout, where each D denotes the functor Ci×Cj ×Ck → Set for appropriate i, j, k ∈ {+, −} d... |

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Citation Context ...ely fruitful for building models of linear logic, as well as in applications to theoretical computer science and in mathematical studies of duality theories. For detailed discussions and history, see =-=[7, 8, 39]-=-. There are many categories of Chu spaces, depending on the underlying symmetric monoidal closed category. We now briefly describe the category Chu(Set, K), a particularly simple one. Let K be a set. ... |

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Citation Context ...f additive proof nets. These have been looked at in various works of Cockett and Seely [9, 15] as well as recent work on multiplicative unitless categories of proof nets of Hughes, Houston and Schalk =-=[27]-=- (and the literature cited there) aiming towards the additive case. See also the recent book of Doˇsen and Petrić ([17], Chapter 4). At the additive level, the advantage of MALLP to nonpolarized MALL ... |

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Citation Context ...le negation ( ) ⊥⊥ is strictly involutive, rather than up-to-isomorphism (see also Proposition 3.3 below). This causes no problem, by recent coherence theorems for ∗-autonomous categories with units (=-=[14, 28, 16]-=-). Moreover, as pointed out in Proposition 5.35 below, all the models in this paper actually satisfy more: namely that the entire polarized structure is strict. Remark 3.2 (On Bimodules) Although this... |

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Citation Context ...P proofs. In what sense do proofs form (the morphisms of) a category? Let us consider the problem of categories of additive proof nets. These have been looked at in various works of Cockett and Seely =-=[9, 15]-=- as well as recent work on multiplicative unitless categories of proof nets of Hughes, Houston and Schalk [27] (and the literature cited there) aiming towards the additive case. See also the recent bo... |

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Citation Context ...ntly designed for the syntax of O. Laurent’s MALLP (for discussion of the literature, see Remark 3.2 and Appendix 1 below.) Recall, if C is a ∗-autonomous category with products (i.e. a model of MALL =-=[8, 12]-=-), then tensor and cotensor (par) are functors ⊗, . : C × C → C, along with the dualizing functor (−) ⊥ : C op → C. We will use the same notation for the connectives of the syntax of MALLP and their d... |

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Citation Context ...duced by J-M Andreoli [4, 5, 38]. In a related direction, many papers in Game Semantics for linear logic have also stressed the notion of polarities, beginning with Lamarche [32] (see also the survey =-=[1]-=-). Olivier Laurent [34, 35] began a systematic study of polarized versions of linear logic. In his thesis he introduced polarized proof structures and nets which are simpler than the nonpolarized orig... |

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Citation Context ...ized version of functorial polymorphism [6, 11] on our hypercoherence model. This framework is developed in Section 6. We distinguish between full and weakly full completeness as in Läuchli semantics =-=[26, 11]-=-. The main point of the category HCoh of hypercoherences is that, unlike coherence spaces, there are nontrivial natural polarized subcategories HCoh− and HCoh+ with an adjunction between them. The cat... |

5 |
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Citation Context ...e name pre-∗-autonomous situations. These are discussed in Section 3 and Appendix 1 below. Recently, Cockett and Seely [15] have introduced their notion of polarized categories motivated by AJ games (=-=[2]-=-), with an elaborate theory of focalized syntax, (two-sided) proof nets, and abstract games based on categorical proof theory. Our work was begun independently of theirs, as an attempt to directly mod... |

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Citation Context ...le negation ( ) ⊥⊥ is strictly involutive, rather than up-to-isomorphism (see also Proposition 3.3 below). This causes no problem, by recent coherence theorems for ∗-autonomous categories with units (=-=[14, 28, 16]-=-). Moreover, as pointed out in Proposition 5.35 below, all the models in this paper actually satisfy more: namely that the entire polarized structure is strict. Remark 3.2 (On Bimodules) Although this... |

4 |
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Citation Context ... introduce the notion of restricting dinaturals to a subcategory by allowing variables and morphisms in multivariant functors to range over the subcategory (see [6]). This framework is used in Hamano =-=[23]-=- for multiplicative full completeness. In the polarized case, we show this provides an important bridge between the polarized and the usual notion of dinatural. Definition 6.2 (DinatD-C) Let C be a ∗-... |

3 |
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Citation Context ...s of ↓-boxes (hence it is removable). Moreover this polarized softness is still quite trivial compared to usual additive ⊕-softness for nonpolarized MALL proof structures, which was studied in Hamano =-=[24]-=-. In the next subsection we study an appropriate multiplicative version of polarized softness in the concrete framework of pDinat-HCoh+,−. This is done by making a connection between this property and... |

2 |
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Citation Context ...a “constructive” version of classical logic based on linear logic. These polarities turn out to be related to the notion of focussing in linear logic proof search, a method introduced by J-M Andreoli =-=[4, 5, 38]-=-. In a related direction, many papers in Game Semantics for linear logic have also stressed the notion of polarities, beginning with Lamarche [32] (see also the survey [1]). Olivier Laurent [34, 35] b... |

1 |
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Citation Context ...a “constructive” version of classical logic based on linear logic. These polarities turn out to be related to the notion of focussing in linear logic proof search, a method introduced by J-M Andreoli =-=[4, 5, 38]-=-. In a related direction, many papers in Game Semantics for linear logic have also stressed the notion of polarities, beginning with Lamarche [32] (see also the survey [1]). Olivier Laurent [34, 35] b... |

1 |
Softness of Hypercoherences and MALLFull Completeness
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Citation Context ...ections 5 through 7. In Section 5, we present many concrete non-game-theoretic examples of our framework, including Ehrhard’s hypercoherences [18] (studied in our previous MALL full completeness work =-=[10]-=-), Chu spaces, and various models based on double gluing and iterated double gluing. In Sections 6,7 we begin our main focus: a study of full completeness theorems (as in our [10]) for polarized logic... |

1 |
Handbook of Categorical Algebra 1, Camb
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Citation Context ...ed Categories [15] (as they remark in their Example 4.4.3 ). In particular, to compare with their work, note that diagram (7) above determines a profunctor (also called a distributor or bimodule, see =-=[13]-=-) of a particular kind (since the two subcategories C− and C+ are respectively reflective and coreflective subcategories of C). Profunctors are genuine functors of the form φ : C op + × C− → Set. We t... |

1 | Coherence for star-autonomous categories Ann - Doˇsen, Petrić |

1 |
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Citation Context ...[4, 5, 38]. In a related direction, many papers in Game Semantics for linear logic have also stressed the notion of polarities, beginning with Lamarche [32] (see also the survey [1]). Olivier Laurent =-=[34, 35]-=- began a systematic study of polarized versions of linear logic. In his thesis he introduced polarized proof structures and nets which are simpler than the nonpolarized original versions. He introduce... |