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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
"... Vol. 2 (4:3) 2006, pp. 1–44 www.lmcsonline.org ..."
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On the axiomatisation of boolean categories with and without medial
 THEORY APPL. CATEG
, 2007
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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
Variable binding, symmetric monoidal closed theories, and bigraphs
 In Bravetti and Zavattaro [2
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Proofnets for additive linear logic with units
 In LiCS’11
, 2011
"... Abstract—Additive linear logic, the fragment of linear logic concerning linear implication between strictly additive formulae, coincides with sumproduct logic, the internal language of categories with free finite products and coproducts. Deciding equality of its proof terms, as imposed by the cate ..."
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Abstract—Additive linear logic, the fragment of linear logic concerning linear implication between strictly additive formulae, coincides with sumproduct logic, the internal language of categories with free finite products and coproducts. Deciding equality of its proof terms, as imposed by the categorical laws, is complicated by the presence of the units (the initial and terminal objects of the category) and the fact that in a free setting products and coproducts do not distribute. The best known desicion algorithm, due to Cockett and Santocanale (CSL 2009), is highly involved, requiring an intricate case analysis on the syntax of terms. This paper provides canonical, graphical representations of the categorical morphisms, yielding a novel solution to this decision problem. Starting with (a modification of) existing proof nets, due to Hughes and Van Glabbeek, for additive linear logic without units, canonical forms are obtained by graph rewriting. The rewriting algorithm is remarkably simple. As a decision procedure for term equality it matches the known complexity of the problem. A main technical contribution of the paper is the substantial correctness proof of the algorithm. I.
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.
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.
Binding bigraphs as symmetric monoidal closed theories
, 810
"... Bbg(K) over a signature K as the free (or initial) symmetric monoidal closed (smc) category S(TK) generated by a derived theory TK. The morphisms of S(TK) are essentially proof nets from the Intuitionistic Multiplicative fragment (imll) of Linear Logic [2]. Formally, we construct a faithful, essenti ..."
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Bbg(K) over a signature K as the free (or initial) symmetric monoidal closed (smc) category S(TK) generated by a derived theory TK. The morphisms of S(TK) are essentially proof nets from the Intuitionistic Multiplicative fragment (imll) of Linear Logic [2]. Formally, we construct a faithful, essentially injective on objects functor Bbg(K) → S(TK), which is surjective on closed bigraphs (i.e., bigraphs without free names or sites). The functor is not full, which we view as a gain in modularity: we maintain the scoping discipline for whole programs (bound names never escape their scope) but allow more program fragments, including a large class of binding contexts, thanks to richer interfaces. Possible applications include bigraphical programming languages [3] and Rathke and Sobociński’s derived labelled transition systems [4]. 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
Proof Theory of the Cut Rule
"... The cut rule is a very basic component of any sequentstyle presentation of a logic. This essay starts by describing the categorical proof theory of the cut rule in a calculus which allows sequents to have many formulas on the left but only one on the right of the turnstile. We shall assume a minimu ..."
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The cut rule is a very basic component of any sequentstyle presentation of a logic. This essay starts by describing the categorical proof theory of the cut rule in a calculus which allows sequents to have many formulas on the left but only one on the right of the turnstile. We shall assume a minimum of structural rules and connectives: in fact, we shall start with