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14
Noncommutative logic II: sequent calculus and phase semantics
, 1998
"... INTRODUCTION Noncommutative logic is a unication of :  commutative linear logic (Girard 1987) and  cyclic linear logic (Girard 1989; Yetter 1990), a classical conservative extension of the Lambek calculus (Lambek 1958). In a previous paper with Abrusci (Abrusci and Ruet 1999) we presented the mu ..."
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Cited by 25 (6 self)
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INTRODUCTION Noncommutative logic is a unication of :  commutative linear logic (Girard 1987) and  cyclic linear logic (Girard 1989; Yetter 1990), a classical conservative extension of the Lambek calculus (Lambek 1958). In a previous paper with Abrusci (Abrusci and Ruet 1999) we presented the multiplicative fragment of noncommutative logic, with proof nets and a sequent calculus based on the structure of order varieties, and a sequentialization theorem. Here we consider full propositional noncommutative logic. Noncommutative logic. Let us rst review the basic ideas. Consider the purely noncommutative fragment of linear logic, obtained by removing the exchange rule entirely : ` ; ; ; , ` ; ; ; y This work has been partly carried out at LIENSCNRS, Ecole Normale Superieure (Paris), at McGill University
Lnets, strategies and proofnets
 In CSL 05 (Computer Science Logic), LNCS
, 2005
"... Abstract. We consider the setting of Lnets, recently introduced by Faggian and Maurel as a game model of concurrent interaction and based on Girard’s Ludics. We show how Lnets satisfying an additional condition, which we call logical Lnets, can be sequentialized into traditional treelike strateg ..."
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Cited by 13 (4 self)
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Abstract. We consider the setting of Lnets, recently introduced by Faggian and Maurel as a game model of concurrent interaction and based on Girard’s Ludics. We show how Lnets satisfying an additional condition, which we call logical Lnets, can be sequentialized into traditional treelike strategies, and viceversa. 1
On the specification of sequent systems
 IN LPAR 2005: 12TH INTERNATIONAL CONFERENCE ON LOGIC FOR PROGRAMMING, ARTIFICIAL INTELLIGENCE AND REASONING, NUMBER 3835 IN LNAI
, 2005
"... Recently, linear Logic has been used to specify sequent calculus proof systems in such a way that the proof search in linear logic can yield proof search in the specified logic. Furthermore, the metatheory of linear logic can be used to draw conclusions about the specified sequent calculus. For e ..."
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Cited by 13 (6 self)
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Recently, linear Logic has been used to specify sequent calculus proof systems in such a way that the proof search in linear logic can yield proof search in the specified logic. Furthermore, the metatheory of linear logic can be used to draw conclusions about the specified sequent calculus. For example, derivability of one proof system from another can be decided by a simple procedure that is implemented via bounded logic programmingstyle search. Also, simple and decidable conditions on the linear logic presentation of inference rules, called homogeneous and coherence, can be used to infer that the initial rules can be restricted to atoms and that cuts can be eliminated. In the present paper we introduce Llinda, a logical framework based on linear logic augmented with inference rules for definition (fixed points) and induction. In this way, the above properties can be proved entirely inside the framework. To further illustrate the power of Llinda, we extend the definition of coherence and provide a new, semiautomated proof of cutelimination for Girard’s Logic of Unicity (LU).
Which Structural Rules Admit Cut Elimination? — An Algebraic Criterion
 Journal of Symbolic Logic
"... Consider a general class of structural inference rules such as exchange, weakening, contraction and their generalizations. Among them, some are harmless but others do harm to cut elimination. Hence it is natural to ask under which condition cut elimination is preserved when a set of structural rules ..."
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Cited by 7 (4 self)
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Consider a general class of structural inference rules such as exchange, weakening, contraction and their generalizations. Among them, some are harmless but others do harm to cut elimination. Hence it is natural to ask under which condition cut elimination is preserved when a set of structural rules is added to a structurefree logic. The aim of this work is to give such a condition by using algebraic semantics. We consider full Lambek calculus (FL), i.e., intuitionistic logic without any structural rules, as our basic framework. Residuated lattices are the algebraic structures corresponding to FL. In this setting, we introduce a criterion, called the propagation property, that can be stated both in syntactic and algebraic terminologies. We then show that, for any set R of structural rules, the cut elimination theorem holds for FL enriched with R if and only if R satisfies the propagation property. As an application, we show that any set R of structural rules can be ”completed” into another set R ⋆ , so that the cut elimination theorem holds for FL enriched with R ⋆ , while the provability remains the same. 1
Noncommutative logic III: focusing proofs
 Information and Computation
, 2000
"... We present a sequent calculus for noncommutative logic which enjoys the focalization property. In the multiplicative case, we give a focalized sequentialization theorem, and in the general case, we show that our focalized sequent calculus is equivalent to the original one by studying the permut ..."
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Cited by 5 (2 self)
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We present a sequent calculus for noncommutative logic which enjoys the focalization property. In the multiplicative case, we give a focalized sequentialization theorem, and in the general case, we show that our focalized sequent calculus is equivalent to the original one by studying the permutabilities of rules for NL and showing that all LL permutabilities involved in focalization can be lifted to NL permutabilities. These results are based on a study of the partitions of partially ordered sets modulo entropy. 1 Introduction Noncommutative logic, NL for short, was introduced by Abrusci and the second author in [1, 15] (see also section 3). It unies commutative linear logic [6] and cyclic linear logic [16], a classical conservative extension of the Lambek calculus [10]. The present paper investigates the \focalization" property for noncommutative logic. 1.1 The property of focalization Rules of the sequent calculus for linear logic [6] are wellknown to split into two c...
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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Cited by 2 (0 self)
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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
Towards a Semantic Characterization of
"... Abstract. We introduce necessary and sufficient conditions for a (singleconclusion) sequent calculus to admit (reductive) cutelimination. Our conditions are formulated both syntactically and semantically. ..."
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Cited by 2 (0 self)
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Abstract. We introduce necessary and sufficient conditions for a (singleconclusion) sequent calculus to admit (reductive) cutelimination. Our conditions are formulated both syntactically and semantically.
Proof and refutation in MALL as a game
, 2009
"... We present a setting in which the search for a proof of B or a refutation of B (i.e., a proof of ¬B) can be carried out simultaneously: in contrast, the usual approach in automated deduction views proving B or proving ¬B as two, possibly unrelated, activities. Our approach to proof and refutation is ..."
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Cited by 2 (2 self)
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We present a setting in which the search for a proof of B or a refutation of B (i.e., a proof of ¬B) can be carried out simultaneously: in contrast, the usual approach in automated deduction views proving B or proving ¬B as two, possibly unrelated, activities. Our approach to proof and refutation is described as a twoplayer game in which each player follows the same rules. A winning strategy translates to a proof of the formula and a counterwinning strategy translates to a refutation of the formula. The game is described for multiplicative and additive linear logic (MALL). A game theoretic treatment of the multiplicative connectives is intricate and our approach to it involves two important ingredients. First, labeled graph structures are used to represent positions in a game and, second, the game playing must deal with the failure of a given player and with an appropriate resumption of play. This latter ingredient accounts for the fact that neither player might win (that is, neither B nor ¬B might be provable).
Resourceoriented Programming Based on Linear Logic
"... Abstract: In our research we consider programming as logical reasoning over types. Linear logic with its resourceoriented features yields a proper means for our approach because it enables to consider about resources as in real life: after their use they are exhausted. Computation then can be regar ..."
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Cited by 1 (1 self)
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Abstract: In our research we consider programming as logical reasoning over types. Linear logic with its resourceoriented features yields a proper means for our approach because it enables to consider about resources as in real life: after their use they are exhausted. Computation then can be regarded as proof search. In our paper we present how space and time can be introduced into this logic and we discuss several programming languages based on linear logic.
Cv
, 2000
"... . Games comprise a widely used method in a broad intellectual realm, and gamerelated ideas are distributed across such areas as philosophy, logic, mathematics, cognitive science, articial intelligence, computation, linguistics, and of course economics. Each discipline nonetheless advocates its own ..."
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. Games comprise a widely used method in a broad intellectual realm, and gamerelated ideas are distributed across such areas as philosophy, logic, mathematics, cognitive science, articial intelligence, computation, linguistics, and of course economics. Each discipline nonetheless advocates its own methodologies, and a unied understanding of games is lacking. In this paper, fundamentals of games are approached from the semantic perspective, by arguing that for example gametheoretic semantics, already having an avowed role in logic and linguistics, may provide foundational approach for ways of studying aspects of information ow, catering a common perspective for games in other areas as well. Games as they are used in formal studies are critically expounded, and the doctrine of them as providing possible explanations for logical constants and reasoning is assessed, illustrating the relevance of conceptual analyses for research in branches of cognitive science as well. We end by consi...