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From proof nets to the free * autonomous category
 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
, 2005
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On the axiomatisation of boolean categories with and without medial
 THEORY APPL. CATEG
, 2007
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Variable binding, symmetric monoidal closed theories, and bigraphs
 In Bravetti and Zavattaro [2
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A categorical semantics for polarized mall
 Ann. Pure Appl. Logic
"... 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 ..."
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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/antihereditary objects, (ii) a Chuspace 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 (Läuchlistyle) as well as full completeness theorems, using a polarized version of functorial
Binding bigraphs as symmetric monoidal closed theories
, 810
"... Abstract. Milner’s bigraphs [1] are a general framework for reasoning about distributed and concurrent programming languages. Notably, it has been designed to encompass both the πcalculus [2] and the Ambient calculus [3]. This paper is only concerned with bigraphical syntax: given what we here call ..."
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Abstract. Milner’s bigraphs [1] are a general framework for reasoning about distributed and concurrent programming languages. Notably, it has been designed to encompass both the πcalculus [2] and the Ambient calculus [3]. This paper is only concerned with bigraphical syntax: given what we here call a bigraphical signature K, Milner constructs a (pre) category of bigraphs Bbg(K), whose main features are (1) the presence of relative pushouts (RPOs), which makes them wellbehaved w.r.t. bisimulations, and that (2) the socalled structural equations become equalities. Examples of the latter are, e.g., in π and Ambients, renaming of bound variables, associativity and commutativity of parallel composition, or scope extrusion for νbound names. Also, bigraphs follow a scoping discipline ensuring that, roughly, bound variables never escape their scope. Here, we reconstruct bigraphs using a standard categorical tool: symmetric monoidal closed (smc) theories. Our theory enforces the same scoping discipline as bigraphs, as a direct property of smc structure. Furthermore, it elucidates the slightly mysterious status of socalled edges in
Proof nets and free semi⋆autonomous categories
, 2013
"... In this paper it is proved that Girard’s proof nets for multiplicative linear logic characterise free semi⋆autonomouscategories. 1 ..."
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In this paper it is proved that Girard’s proof nets for multiplicative linear logic characterise free semi⋆autonomouscategories. 1
Proof Nets and the Identity of Proofs
, 2006
"... These are the notes for a 5lecturecourse given at ESSLLI 2006 in Malaga, Spain. The URL ..."
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These are the notes for a 5lecturecourse given at ESSLLI 2006 in Malaga, Spain. The URL
Some Observations on the Proof Theory of Second Order Propositional Multiplicative Linear Logic (Extended Abstract)
, 2007
"... We present two new aspects of the proof theory of MLL2. First, we will give a novel proof system in the framework of the calculus of structures. The main feature of the new system is the consequent use of deep inference. Due to the new freedom of permuting inference rules, we are able to observe a d ..."
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We present two new aspects of the proof theory of MLL2. First, we will give a novel proof system in the framework of the calculus of structures. The main feature of the new system is the consequent use of deep inference. Due to the new freedom of permuting inference rules, we are able to observe a decomposition theorem, which is not visible in the sequent calculus. Second, we show a new notion of (boxfree) proof nets which is inspired by the deep inference proof system. Nonetheless, the proof nets are independent from the deductive system. We have “sequentialisation” into the calculus of structures as well as into the sequent calculus. We present a notion of cut elimination which is terminating and confluent, and thus gives us a category of proof nets.
No proof nets for MLL with units Proof equivalence in MLL is PSPACEcomplete
"... [Analysis of algorithms and problem complexity]: Nonnumerical algorithms and problems—Complexity of proof procedures Keywords linear logic, proof equivalence, proof nets, constraint logic, PSPACEcompleteness MLL proof equivalence is the problem of deciding whether two proofs in multiplicative linea ..."
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[Analysis of algorithms and problem complexity]: Nonnumerical algorithms and problems—Complexity of proof procedures Keywords linear logic, proof equivalence, proof nets, constraint logic, PSPACEcompleteness MLL proof equivalence is the problem of deciding whether two proofs in multiplicative linear logic are related by a series of inference permutations. It is also known as the word problem for ∗autonomous categories. Previous work has shown the problem to be equivalent to a rewiring problem on proof nets, which are not canonical for full MLL due to the presence of the two units. Drawing from recent work on reconfiguration problems, in this paper it is shown that MLL proof equivalence is PSPACEcomplete, using a reduction from Nondeterministic Constraint Logic. An important consequence of the result is that the existence of a satisfactory notion of proof nets for MLL with units is ruled out (under current complexity assumptions). 1.
Under consideration for publication in Math. Struct. in Comp. Science Proof nets and semi⋆autonomous categories
, 2014
"... In this paper it is proved that Girard’s proof nets for multiplicative linear logic characterise free semi⋆autonomouscategories. 1. ..."
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In this paper it is proved that Girard’s proof nets for multiplicative linear logic characterise free semi⋆autonomouscategories. 1.