Non uniform (hyper/multi)coherence spaces

In (hyper)coherence semantics, proofs/terms are cliques in (hyper)graphs. Intuitively, vertices represent results of computations and the edge relation witnesses the ability of being assembled into a same piece of data or a same (strongly) stable function, at arrow types. In (hyper)coherence semantics, the argument of a (strongly) stable functional is always a (strongly) stable function. As a consequence, comparatively to the relational semantics, where there is no edge relation, some vertices are missing. Recovering these vertices is essential for the purpose of reconstructing proofs/terms from their interpretations. It shall also be useful for the comparison with other semantics, like game semantics. In [BE01], Bucciarelli and Ehrhard introduced a so called non uniform coherence space semantics where no vertex is missing. By constructing the co-free exponential we set a new version of this last semantics, together with non uniform versions of hypercoherences and multicoherences, a new semantics where an edge is a finite multiset. Thanks to the co-free construction, these non uniform semantics are deterministic in the sense that the intersection of a clique and of an anti-clique contains at most one vertex, a result of interaction, and extensionally collapse onto the corresponding uniform semantics.Comment: 32 page

Categorical models of Linear Logic with fixed points of formulas

We develop a denotational semantics of muLL, a version of propositional Linear Logic with least and greatest fixed points extending David Baelde's propositional muMALL with exponentials. Our general categorical setting is based on the notion of Seely category and on strong functors acting on them. We exhibit two simple instances of this setting. In the first one, which is based on the category of sets and relations, least and greatest fixed points are interpreted in the same way. In the second one, based on a category of sets equipped with a notion of totality (non-uniform totality spaces) and relations preserving them, least and greatest fixed points have distinct interpretations. This latter model shows that muLL enjoys a denotational form of normalization of proofs.Comment: arXiv admin note: text overlap with arXiv:1906.0559

On denotational completeness (Extended 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 A bilinear form B which induces the duality is nothing but the product in a monoid M , ? being an arbitrary subset B of M . 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 P. Now we can replace coherent spaces with A free P-modules over PB, linear maps with A P-linear maps B, with the ..