Abstract

The founding idea of linear logic is the duality between A and A, with values in . This idea is at work in the original denotational semantics of linear logic, coherent spaces, but also in the phase semantics of linear logic, where the “bilinear form” which induces the duality is nothing but the product in a monoid being an arbitrary subset of . The rather crude phase semantics has the advantage of being complete, and against all predictions, this kind of semantics had some applications. Coherent semantics is not complete for an obvious reason, namely that the coherent space interpreting is too small (one point), hence the duality between A and A expressed by the cut-rule cannot be informative enough. But is indeed the simplest case of a Par-monoid, i.e. the dual of a comonoid, and it is tempting to replace with any commutative Par-monoid . Now we can replace coherent spaces with “free -modules over ”, linear maps with “ -linear maps”, with the essential result that all usual constructions remain unchanged: technically speaking cliques are replaced with -cliques and that is it. The essential intuition behind is that it accounts for arbitrary contexts: instead of dealing with Γ,A, one deals with A, but a clique of Γ,A can be seen as a -clique in A. In particular, all logical rules are now defined only on the main formulas of rules, as operations on -cliques. The duality between A and A yields a -clique in , i.e. a clique in ; strangely enough, one must keep the phase layer, i.e. a monoid (useful in the degenerated case), and the result of the duality is a -clique. We specify an arbitrary set of such cliques as the interpretation of . Soundness and completeness are then easily established for closed Π¹-formulas, i.e. second-order propositional formulas without existential quantifiers. We must however find the equivalent of (which is the condition for being a “provable fact”): a -clique is essential when it does not make use of and , i.e. when it is induced by a clique in A^•. We can now state the theorem:

Let A be a closed Π¹ formula, and let a be a clique in the (usual) coherent interpretation A^• of A, which is the interpretation of a proof of A; then a (as an essential clique), belongs to the “denotational fact” A° interpreting A for all and . Conversely any essential clique with this property comes from a proof of A.