Natural deduction and coherence for weakly distributive categories

Blute, Richard; Cockett, J. R. B.; Seely, R. A. G.; Trimble, Todd

doi:10.1016/0022-4049(95)00159-x

Cited by 94 publications

(253 citation statements)

References 13 publications

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“…This is done in [BCST96], where an equivalence class is also used, built on the ability to slide the constants up and down the branches.…”

Section: Proof Nets For Multiplicative Linear Logicmentioning

confidence: 99%

“…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 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 proof objects, which are ordinary terms with binders.…”

Section: Introductionmentioning

confidence: 99%

“…If we now add axioms of the form a • , a • then the main system in [BCST96] is exactly equivalent to multiplicative linear logic with the usual rules, but where the introduction of connectives have to obey the polarity restrictions above. A polarized one-sided sequent of the form ⊢ A • 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…Later in [BCST96], a side-switching negation connective is introduced, as is also done in [Pui01], along with non-logical axioms for it, and the larger system is equivalent to classical multiplicative linear logic.…”

Section: Introductionmentioning

confidence: 99%

“…In the next section we will say how our approach to proof nets differs from the one which is used in [BCST96]. It should be obvious eventually that what we propose is considerably simpler.…”

Section: Introductionmentioning

confidence: 99%

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From Proof Nets to the Free *-Autonomous Category

Lamarche¹,

Straßburger²

2006

Logical Methods in Computer Science

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Abstract. In the first part of this paper we present a theory of proof nets for full multiplicative linear logic, including the two units. It naturally extends the well-known theory of unit-free multiplicative proof nets. A linking is no longer a set of axiom links but a tree in which the axiom links are subtrees. These trees will be identified according to an equivalence relation based on a simple form of graph rewriting. We show the standard results of sequentialization and strong normalization of cut elimination. In the second part of the paper we show that the identifications enforced on proofs are such that the class of two-conclusion proof nets defines the free * -autonomous category.

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“…This is done in [BCST96], where an equivalence class is also used, built on the ability to slide the constants up and down the branches.…”

Section: Proof Nets For Multiplicative Linear Logicmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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From Proof Nets to the Free *-Autonomous Category

Lamarche¹,

Straßburger²

2006

Logical Methods in Computer Science

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Functorial Boxes in String Diagrams

Melliès

2006

Computer Science Logic

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String diagrams were introduced by Roger Penrose as a handy notation to manipulate morphisms in a monoidal category. In principle, this graphical notation should encompass the various pictorial systems introduced in proof-theory (like Jean-Yves Girard's proof-nets) and in concurrency theory (like Robin Milner's bigraphs). This is not the case however, at least because string diagrams do not accomodate boxes-a key ingredient in these pictorial systems. In this short tutorial, based on our accidental rediscovery of an idea by Robin Cockett and Robert Seely, we explain how string diagrams may be extended with a notion of functorial box to depict a functor separating an inside world (its source category) from an outside world (its target category). We expose two elementary applications of the notation: first, we characterize graphically when a faithful balanced monoidal functor F : C −→ D transports a trace operator from the category D to the category C, and we then exploit this to construct well-behaved fixpoint operators in cartesian closed categories generated by models of linear logic; second, we explain how the categorical semantics of linear logic induces that the exponential box of proof-nets decomposes as two enshrined functorial boxes.

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Partially Additive Categories and Fully Complete Models of Linear Logic

Haghverdi

2001

Lecture Notes in Computer Science

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Natural deduction and coherence for weakly distributive categories

Cited by 94 publications

References 13 publications

From Proof Nets to the Free *-Autonomous Category

From Proof Nets to the Free *-Autonomous Category

Functorial Boxes in String Diagrams

Partially Additive Categories and Fully Complete Models of Linear Logic

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