L-nets, parallel strategies and proof-nets

We present Ludics nets (L-nets) as a game model allowing for
concurrent interaction.
L-nets have been developed in the context of Ludics (Girard 2001),
which could be seen as a game model of sequential interaction,
abstracting away from proof-theory.  L-nets can be seen both as
parallel strategies (in game-semantical terms) and as an abstract form
of proof nets.
Strategies capturing sequential interaction, such as Hyland-Ong
innocent strategies or Girard's designs, are based on trees; during an
interaction (play, or run) the order between the actions is totally
specified. L-nets on the contrary are based on graphs, and
interactions are partial orders; the intuitionis that some actions can
be performed in parallel (or scheduled in any order), while there are
tasks which depend upon other tasks.
When taking a proof-theoretical point of view, a tree strategy can be
seen as an abstract sequent calculus derivations, while an L-net
appears as an abstract (multiplicative-additive) proof-net. Moreover,
as a tree strategy is a particular case of L-net, we have a homogeneus
setting, in which it is possible to move between different degrees of
sequentiality (both on the syntactical and on the semantical side).

This talk is based on work in collaboration with Francois Maurel and
with Pierre-Louis Curien.