arXiv subjects

Christophe Fouqueré

Publications and source records attributed to Christophe Fouqueré.

3 recordsLinked to original sources

arXiv2018

Study of Behaviours via Visitable Paths

Around 2000, J.-Y. Girard developed a logical theory, called Ludics. This theory was a step in his program of Geometry of Interaction, the aim of which being to account for the dynamics of logical proofs. In Ludics, objects called designs keep only what is relevant for the cut elimination process, hence the dynamics of a proof: a design is an abstraction of a formal proof. The notion of behaviour is the counterpart in Ludics of the notion of type or the logical notion of formula. Formally a behaviour is a closed set of designs. Our aim is to explore the constructions of behaviours and to analyse their properties. In this paper a design is viewed as a set of coherent paths. We recall or give variants of properties concerning visitable paths, where a visitable path is a path in a design or a set of designs that may be traversed by interaction with a design of the orthogonal of the set. We are then able to answer the following question: which properties should satisfy a set of paths for being exactly the set of visitable paths of a behaviour? Such a set and its dual should be prefix-closed, daimon-closed and satisfy two saturation properties. This allows us to have a means for defining the whole set of visitable paths of a given set of designs without closing it explicitly, that is without computing the orthogonal of this set of designs. We finally apply all these results for making explicit the structure of a behaviour generated by constants and multiplicative/additive connectives. We end by proposing an oriented tensor for which we give basic properties.

cs.LO↗

arXiv2015

Incarnation in Ludics and maximal cliques of paths

Ludics is a reconstruction of logic with interaction as a primitive notion, in the sense that the primary logical concepts are no more formulas and proofs but cut-elimination interpreted as an interaction between objects called designs. When the interaction between two designs goes well, such two designs are said to be orthogonal. A behaviour is a set of designs closed under bi-orthogonality. Logical formulas are then denoted by behaviours. Finally proofs are interpreted as designs satisfying particular properties. In that way, designs are more general than proofs and we may notice in particular that they are not typed objects. Incarnation is introduced by Girard in Ludics as a characterization of "useful" designs in a behaviour. The incarnation of a design is defined as its subdesign that is the smallest one in the behaviour ordered by inclusion. It is useful in particular because being "incarnated" is one of the conditions for a design to denote a proof of a formula. The computation of incarnation is important also as it gives a minimal denotation for a formula, and more generally for a behaviour. We give here a constructive way to capture the incarnation of the behaviour of a set of designs, without computing the behaviour itself. The method we follow uses an alternative definition of designs: rather than defining them as sets of chronicles, we consider them as sets of paths, a concept very close to that of play in game semantics that allows an easier handling of the interaction: the unfolding of interaction is a path common to two interacting designs.

cs.LO↗

arXiv2007

A Sequent Calculus for Modelling Interferences

A logic calculus is presented that is a conservative extension of linear logic. The motivation beneath this work concerns lazy evaluation, true concurrency and interferences in proof search. The calculus includes two new connectives to deal with multisequent structures and has the cut-elimination property. Extensions are proposed that give first results concerning our objectives.

cs.LO↗