This special issue is devoted to complete and revised versions of papers presented at the Logique et Modèles du Calcul (Logic and Models of Computation) Conference held at CIRM in Marseilles in September 1996.

The conference was organized by Marie-Renée Donnadieu, and the members of the programme committee were Gérard Berry, Jean-Yves Girard, Max Kanovich, Jean-Louis Krivine, Yves Lafont and François Lamarche.

The scope of the conference was quite large, including, for instance, digital circuits, process calculi, abstract machines for lambda-calculus, computational interpretations of linear logic and logical aspects of proof search. Only the last three topics are represented in this special issue of Mathematical Structures in Computer Science, but we think that these papers are quite representative of the liveliness of research in this area on the borderline between mathematics and computing.

We wish to thank Giuseppe Longo, who kindly proposed publishing this issue. For editorial reasons, a few contributions could not be included – they should appear in later issues.

Almost a decade ago, Girard invented linear logic with the notion of a proof-net. Proof-nets are special graphs built from formulas, links and boxes. However, not all nets are proof-nets. First, they must be well constructed (we say that such graphs are proof-structures). Second, a proof-net is a proof-structure that corresponds to a sequential proof. It must satisfy a correctness criterion. One may wonder what this static criterion means for cut-elimination. We prove that every incorrect proof-structure (without cut) can be put in an environment where reductions lead to two kinds of basically wrong configurations: deadlocks and disconnected proof-structures. Thus, this proof says that there does not exist a bigger class of proof-structures than proof-nets where normalization does not lead to obviously bad configurations.

We present a formalism of trees with pointers, called abstract Böhm trees, that provide a suitable abstract framework in which various cut-free proofs or normal terms of several λ-calculus based languages (including PCF and Parigot's λμ-calculus) can be faithfully encoded. A simple abstract machine called the View Abstract Machine (VAM) allows us to compute over abstract Böhm trees. The VAM is closely related to Coquand's interaction sequences and debates. The VAM execution over finite abstract Böhm trees always terminates. We next introduce an abstract notion of type that fits the purpose of guaranteeing that the VAM cannot go into deadlock, i.e., that it always reaches a satisfactory final state. Typed abstract Böhm trees can be turned into a category – more naturally a ‘multi-category’ where the domains of arrows are sets of named objects or records. We then go from the abstract to the concrete by giving examples. Our sets of abstract (typed) Böhm trees are relative to an alphabet and a set of types. By instantiating these two parameter sets appropriately, we recover, successively: (η-long) typed Böhm trees; PCF trees as considered in the game models of Hyland–Ong or of Abramsky–Jagadeesan–Malacaria; a notion of classical Böhm tree due to Herbelin that provides a classical version of PCF trees in the style of λμ-calculus; and, finally, cut-free proofs in Novikov's infinitary propositional logic as investigated by Coquand. In a companion paper, we investigate the operational aspects of (untyped) Böhm trees in more depth.

Interaction nets have proved to be a useful tool for the study of computational aspects of various formalisms (e.g. λ-calculus, term rewriting systems), but they are also a programming paradigm in themselves, and this is actually how they were introduced by Lafont. In this paper we consider semi-simple interaction nets as a programming language, and present a type assignment system using intersection types. First we show that interactions preserve types (i.e., the system enjoys subject reduction), and we compare this type assignment system with the intersection systems for λ-calculus and term rewriting systems. Then we define a recursion scheme that ensures termination of all interaction sequences. By relaxing the scheme and using the type assignment system, we derive another sufficient condition for termination of interaction nets. Finally, we show that although the type system based on general intersection types is not decidable, its restriction to rank 2 types is, and we give an algorithm that computes principal types for nets.

We introduce a natural deduction-like formalisation of Parigot's λμ-calculus. From this, we derive an environment machine that allows any well-typed λμ-term to be reduced to its weak head normal form. The soundness and completeness of the machine is proved.

We give a decision procedure for the extensional equality of total Böhm trees presented by regular systems of recursion equations.

We propose a concurrent process calculus, called Calcul Parallèle Logique (CPL), based on the paradigm of computation as proof net construction in linear logic. CPL uses a fragment of first-order intuitionistic linear logic where formulas represent processes and proof nets represent successful computations. In these computations, communication is expressed in an asynchronous way by means of axiom links. We define testing equivalences for processes, which are based on a concept of interface, and use the power of proof theory in linear logic.