The relational model is injective for multiplicative exponential linear logic (without weakenings)

Carvalho, Daniel de; Falco, Lorenzo Tortora de

doi:10.1016/j.apal.2012.01.004

Cited by 14 publications

(19 citation statements)

References 23 publications

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“…For instance, while it is well-known that the coherent model of second order linear logic [13] is not dinatural ( [11]), it can be easily seen that it satisfies the Yoneda isomorphism. Hence it can be conjectured that the model is injective (in the sense of [7]) with respect to ∃-linkings for MLL2 Y .…”

Section: Discussionmentioning

confidence: 99%

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Proof Nets, Coends and the Yoneda Isomorphism

Pistone¹

2019

Electron. Proc. Theor. Comput. Sci.

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Proof nets provide permutation-independent representations of proofs and are used to investigate coherence problems for monoidal categories. We investigate a coherence problem concerning Second Order Multiplicative Linear Logic (MLL2), that is, the one of characterizing the equivalence over proofs generated by the interpretation of quantifiers by means of ends and coends.We provide a compact representation of proof nets for a fragment of MLL2 related to the Yoneda isomorphism. By adapting the "rewiring approach" used in coherence results for * -autonomous categories, we define an equivalence relation over proof nets called "rewitnessing". We prove that this relation characterizes, in this fragment, the equivalence generated by coends. IntroductionProof nets are usually investigated as canonical representations of proofs. For the proof-theorist, the adjective "canonical" indicates a representation of proofs insensitive to admissible permutations of rules; for the category-theorist, it indicates a faithful representation of arrows in free monoidal categories (e.g.* -autonomous categories), by which coherence results can be obtained.This twofold approach has been developed extensively in the case of Multiplicative Linear Logic (see for instance [5,6]). The use of MLL proof nets to investigate coherence problems relies on the correspondence between proof nets and a particular class of dinatural transformations (see [5]). As dinatural transformations provide a well-known interpretation of parametric polymorphism (see [1,16]), it is natural to consider the extension of this correspondence to second order Multiplicative Linear Logic MLL2. This means investigating the "coherence problem" generated by the interpretation of quantifiers as ends/coends, that is, to look for a faithful proof net representation of coends over a * -autonomous category.The main difficulty of this extension is that, as is well-known, dinaturality does not scale to second order (e.g. System F, see [26]): the dinatural interpretation of proofs generates an equivalence over proofs which strictly extends the equivalence generated by β and η conversions. In particular, coends induce "generalized permutations" of rules ([36]) to which neither System F proofs nor standard proof nets for MLL2 are insensitive. For instance, the interpretation of quantifiers as ends/coends (whose definition is recalled in appendix A) equates the distinct System F derivations in fig. 1a as well as the distinct proof nets in fig. 1b. From these examples it can be seen that such generalized permutations do not preserve the witnesses of existential quantification (or, equivalently, of the elimination of universal quantification).Several well-known issues in the System F representation of categorial structures can be related to this phenomenon. For instance, the failure of universality for the "Russell-Prawitz" translation of connectives (e.g. the failure of the isomorphism A ⊗ B ≃ ∀X ((A ⊸ B ⊸ X ) ⊸ X )), and the failure of initiality for the System F representation of initial...

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Section: Discussionmentioning

confidence: 99%

“…We give here a functorial definition of ends and coends which can be easily deduced from the usual definition (see[28]). 7 We will abbreviate δ x 1 ,...,x n ,a and ω x 1 ,...,x n ,a simply as δ a and ω a , respectively.…”

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confidence: 99%

Proof Nets, Coends and the Yoneda Isomorphism

Pistone¹

2019

Electron. Proc. Theor. Comput. Sci.

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“…box : |R| → A is a morphism of oriented graphs, 6 the box-function of R, such that box F induces a partial bijection from…”

Section: Dill Proof-structuresmentioning

confidence: 99%

“…A MELL proof-structure is an oriented graph together with some additional information to identify the content and the border of each box. This additional information can be provided either informally, just drawing the border of each box in the graph [14,10,19], but then the definition of MELL proof-structure is not rigorous; or in a more formal way [6,15,7], but then the definition is highly technical and ad hoc; 2. A MELL proof-structure is an inductive oriented graph [17,22,26,8], i.e.…”

Section: Introductionmentioning

confidence: 99%

Proof-Net as Graph, Taylor Expansion as Pullback

Guerrieri

Pellissier

Falco

2019

Logic, Language, Information, and Computation

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“…This investigation has been extended to concurrent strategies by Mimram and Melliès [25,26]. De Carvalho showed that the relational model is injective for MELL [11]. In another direction, [4] provides a fully complete model for MALL without game semantics, by using a glueing construction on the model of hypercoherences.…”

Section: Extensions and Related Workmentioning

confidence: 99%

Causality in Linear Logic

Castellan

Yoshida

2019

Lecture Notes in Computer Science

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Commuting conversions of Linear Logic induce a notion of dependency between rules inside a proof derivation: a rule depends on a previous rule when they cannot be permuted using the conversions. We propose a new interpretation of proofs of Linear Logic as causal invariants which captures exactly this dependency. We represent causal invariants using game semantics based on general event structures, carving out, inside the model of [6], a submodel of causal invariants. This submodel supports an interpretation of unit-free Multiplicative Additive Linear Logic with MIX (MALL − ) which is (1) fully complete: every element of the model is the denotation of a proof and (2) injective: equality in the model characterises exactly commuting conversions of MALL − . This improves over the standard fully complete game semantics model of MALL − .

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The relational model is injective for multiplicative exponential linear logic (without weakenings)

Abstract: We prove a completeness result for Multiplicative Exponential Linear Logic (MELL): we show that the relational model is injective for MELL proof-nets, i.e. the equality between MELL proof-nets in the relational model is exactly axiomatized by cut-elimination.

Cited by 14 publications

References 23 publications

Proof Nets, Coends and the Yoneda Isomorphism

Proof Nets, Coends and the Yoneda Isomorphism

Proof-Net as Graph, Taylor Expansion as Pullback

Causality in Linear Logic

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