IntroductionThe sequent calculus [5] is very appropriate for classical logic, but it has some problems in dealing with more refined logics like linear logic [6]. Observing certain logical relations in the sequent calculus might be impossible. In this paper we show a calculus, called the calculus of structures, which is able to overcome those difficulties. We call calculus a framework, like natural deduction or the sequent calculus, for specifying logical systems. We say formal system to indicate a collection of inference rules in a given calculus. A derivation is a composition of instances of inference rules, a proof is a derivation free from hypotheses.A proof in the sequent calculus is a tree, and branching occurs when two-premise rules are used. The two branches are statements that proofs exist for both premises. At the meta level, we say that the left branch is a proof and the right branch is a proof. In classical logic, this 'and' corresponds to the 'and' at the object level. This is not the case in other logics, like in linear logic.Another founding property of the sequent calculus is the pivotal rôle of main connectives. Given a main connective in the conclusion, a rule gives meaning to it by saying that the conclusion is provable if subformulae obtained by removing the connective are in turn provable.These two properties together have remarkable success in making the study of systems independent of their semantics, which is important if a semantics is incomplete, missing or still under development, as often happens in computer science. The problem is that the sequent calculus is unnecessarily rigid for some logics. We can relax the 'and' branching between premise trees, and abandon the decomposing of the conclusion around the main connective of one of its formulae. The question is whether we can do so while keeping the good properties, cut elimination especially.The calculus of structures draws from a very simple principle, which is very dangerous if not realised with care. The inference rules are of the kind S{T } ρ S{R} , where premise and conclusion are structures, i.e., formulae subject to certain equivalences

Abstract. We present cut-free deductive systems without labels for the intuitionistic variants of the modal logics obtained by extending IK with a subset of the axioms d, t, b, 4, and 5. For this, we use the formalism of nested sequents, which allows us to give a uniform cut elimination argument for all 15 logic in the intuitionistic S5 cube.

We introduce the calculus of structures: it is more general than the sequent calculus and it allows for cut elimination and the subformula property. We show a simple extension of multiplicative linear logic, by a self-dual noncommutative operator inspired by CCS, that seems not to be expressible in the sequent calculus. Then we show that multiplicative exponential linear logic benefits from its presentation in the calculus of structures, especially because we can replace the ordinary, global promotion rule by a local version. These formal systems, for which we prove cut elimination, outline a range of techniques and properties that were not previously available. Contrarily to what happens in the sequent calculus, the cut elimination proof is modular.

No abstract

Abstract. In the first part of this paper we present a theory of proof nets for full multiplicative linear logic, including the two units. It naturally extends the well-known theory of unit-free multiplicative proof nets. A linking is no longer a set of axiom links but a tree in which the axiom links are subtrees. These trees will be identified according to an equivalence relation based on a simple form of graph rewriting. We show the standard results of sequentialization and strong normalization of cut elimination. In the second part of the paper we show that the identifications enforced on proofs are such that the class of two-conclusion proof nets defines the free * -autonomous category.

No abstract

No abstract

No abstract