In this thesis I study several deductive systems for linear logic, its fragments, and some noncommutative extensions. All systems will be designed within the calculus of structures, which is a proof theoretical formalism for specifying logical systems, in the tradition of Hilbert's formalism, natural deduction, and the sequent calculus. Systems in the calculus of structures are based on two simple principles: deep inference and top-down symmetry. Together they have remarkable consequences for the properties of the logical systems. For example, for linear logic it is possible to design a deductive system, in which all rules are local. In particular, the contraction rule is reduced to an atomic version, and there is no global promotion rule. I will also show an extension of multiplicative exponential linear logic by a noncommutative, self-dual connective which is not representable in the sequent calculus. All systems enjoy the cut elimination property. Moreover, this can be proved independently from the sequent calculus via techniques that are based on the new top-down symmetry. Furthermore, for all systems, I will present several decomposition theorems which constitute a new type of normal form for derivations.

Cette recherche vise à établir les propriétés fondamentales d'un système formel aux preuves circulaires introduit par Santocanale, auquel on a rajouté la règle de coupure. On démontre, dans un premier temps, qu'il y a une pleine correspondance entre les preuves circulaires et les flèches issues des catégories dites µ-bicomplètes. Ces flèches sont celles que l'on peut définir purement à partir des outils suivants: les produits et coproduits finis, les algèbres initiales et les coalgèbres finales. Dans la catégorie des ensembles, les preuves circulaires dénotent donc les fonctions qu'on peut définir en utilisant les produits cartésiens finis, les unions disjointes finies, l'induction et la coinduction. On décrit également une procédure d'élimination des coupures qui produit, à partir d'une preuve circulaire finie, une preuve sans cycles et sans coupures, mais possiblement infinie. On démontre que l'élimination des coupures fournit une sémantique opérationnelle aux preuves circulaires, c'est-à-dire qu'elle permet de calculer les fonctions dénotées par celles-ci, par le moyen d'une sorte d'automate avec mémoire. Enfin, on s'intéresse au problème de la puissance expressive de cet éliminateur de coupures, c'est-à-dire à la question de caractériser la classe des expressions qu'il peut calculer. On démontre, par une simulation, que l'éliminateur des coupures est strictement plus expressif que les automates à pile d'ordre supérieur. / This research aims at establishing the fundamental properties of a formal system with circular proofs introduced by Santocanale, to which we added the cut rule. We first show that there is a full correspondence between circular proofs and arrows from the so-called µ-bicomplete categories. These arrows are those that can be defined purely from the following tools: finite products and coproducts, initial algebras and final coalgebras. In the category of sets, circular proofs denote functions that one can define by using finite cartesian products, finite disjoint unions, induction and coinduction. We also describe a cut-elimination procedure that produces, from a given finite circular proof, a proof without cycles and cuts, but which may be infinite. We prove that cut-elimination gives an operational semantics to circular proofs, which is to say that they allow to compute the functions denoted by them, by using a sort of automaton with memory. Finally, we are interested in finding the expressive power of that cut-eliminating automaton. In other words, we want to characterize the class of functions that it can compute. We show, through a simulation, that the cut-eliminating automaton is strictly more expressive than higher-order pushdown automata.

In this thesis I study several deductive systems for linear logic, its fragments, and some noncommutative extensions. All systems will be designed within the calculus of structures, which is a proof theoretical formalism for specifying logical systems, in the tradition of Hilbert's formalism, natural deduction, and the sequent calculus. Systems in the calculus of structures are based on two simple principles: deep inference and top-down symmetry. Together they have remarkable consequences for the properties of the logical systems. For example, for linear logic it is possible to design a deductive system, in which all rules are local. In particular, the contraction rule is reduced to an atomic version, and there is no global promotion rule. I will also show an extension of multiplicative exponential linear logic by a noncommutative, self-dual connective which is not representable in the sequent calculus. All systems enjoy the cut elimination property. Moreover, this can be proved independently from the sequent calculus via techniques that are based on the new top-down symmetry. Furthermore, for all systems, I will present several decomposition theorems which constitute a new type of normal form for derivations.