INTRODUCTION Non-commutative 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 non-commutative logic, with proof nets and a sequent calculus based on the structure of order varieties, and a sequentialization theorem. Here we consider full propositional non-commutative logic. Non-commutative 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 LIENS-CNRS, Ecole Normale Superieure (Paris), at McGill University

Abstract. We consider the setting of L-nets, recently introduced by Faggian and Maurel as a game model of concurrent interaction and based on Girard’s Ludics. We show how L-nets satisfying an additional condition, which we call logical L-nets, can be sequentialized into traditional tree-like strategies, and vice-versa. 1

Abstract. 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 meta-theory 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 programming-style 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, semi-automated proof of cut-elimination for Girard’s Logic of Unicity (LU). 1

We present a sequent calculus for non-commutative 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 Non-commutative 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 non-commutative logic. 1.1 The property of focalization Rules of the sequent calculus for linear logic [6] are well-known to split into two c...

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

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/anti-hereditary objects, (ii) a Chu-space 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 (Läuchli-style) as well as full completeness theorems, using a polarized version of functorial

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 two-player game in which each player follows the same rules. A winning strategy translates to a proof of the formula and a counter-winning 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).

Abstract. We introduce necessary and sufficient conditions for a (single-conclusion) sequent calculus to admit (reductive) cut-elimination. Our conditions are formulated both syntactically and semantically.

. Games comprise a widely used method in a broad intellectual realm, and game-related 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 game-theoretic 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...

We prove a full completeness theorem for multiplicative-additive linear logic (i.e. MALL) using a double gluing construction applied to Ehrhard’s ∗-autonomous category of hypercoherences. This is the first non-game-theoretic full completeness theorem for this fragment. Our main result is that every dinatural transformation between definable functors arises from the denotation of a cut-free MALL proof. Our proof consists of three steps. We show: • Dinatural transformations on this category satisfy Joyal’s softness property for products and coproducts. • Softness, together with multiplicative full completeness, guarantees that every dinatural transformation corresponds to a Girard MALL proof-structure. • The proof-structure associated to any dinatural transformation is a MALL proofnet, hence a denotation of a proof. This last step involves a detailed study of cycles in additive proof structures.