To show that a formula A is not provable in propositional classical logic, it suffices to exhibit a finite boolean model which does not satisfy A. A similar property holds in the intuitionistic case, with Kripke models instead of boolean models (see for instance [11]). One says that the propositional classical logic and the propositional intuitionistic logic satisfy a finite model property. In particular, they are decidable: there is a semi-algorithm for provability (proof search) and a semi-algorithm for non provability (model search). For that reason, a logic which is undecidable, such as first order logic, cannot satisfy a finite model property. The case of linear logic is more complicated. The full propositional fragment LL has a complete semantics in terms of phase spaces [2, 3], but it is undecidable [9]. The multiplicative additive fragment MALL is decidable, in fact PSPACE-complete [9], but the decidability of the multiplicative exponential fragment MELL is still an open problem. For affine logic, that is, linear logic with weakening, the situation is somewhat better: the full propositional fragment LLW is decidable [5]. Here, we show that the finite phase semantics is complete for MALL and for LLW, but not for MELL. In particular, this gives a new proof of the decidability of LLW. The noncommutative case is mentioned, but not handled in detail.

The multiplicative fragment of second order propositional linear logic is shown to be undecidable.

Abstract Recently, Lincoln, Scedrov and Shankar showed that the multiplicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicative-additive fragment, but it does not work in the classical case, because second order classical logic is decidable. Here we show that the multiplicative-additive fragment of second order classical linear logic is also undecidable, using an encoding of two-counter machines originally due to Kanovich. The faithfulness of this encoding is proved by means of the phase semantics.

An attempt is made to relate various notions of duality used in mathematics with the denotational semantics of linear logic. The author proposes a naive semantics for linear logic that, in a certain sense, generalizes various notions such as finite-dimensional vector spaces, topological spaces, and J.-Y. Girard's (1987) coherence spaces. A game consists of a set of vectors (or strategies), a set of forms (or co-strategies) and an evaluation bracket. This is enough to interpret the connectives of full propositional linear logic, including exponentials.>

We present a result of C.C. Squier relating two topics: — the canonical rewriting systems, which have been widely studied by computer scientists as a tool for solving word problems, — the homology of groups (more generally of monoids), which belongs to the core of pure mathematics, on the borders of algebra and topology.

We propose a new kind of programming language, with the following features:a simple graph rewriting semantics,a complete symmetry between constructors and destructors,a type discipline for deterministic and deadlock-free (microscopic) parallelism.Interaction nets generalize Girard's proof nets of linear logic and illustrate the advantage of an integrated logic approach, as opposed to the external one. In other words, we did not try to design a logic describing the behaviour of some given computational system, but a programming language for which the type discipline is already (almost) a logic.In fact, we shall scarcely refer to logic, because we adopt a naive and pragmatic style. A typical application we have in mind for this language is the design of interactive softwares such as editors or window managers.