From Proof Nets to Games (Extended Abstract)

We give a class of proof nets for Intuitionistic Linear Logic with the connectives (;!, prove a correctness criterion for them and show that a games semantics can be directly derived from these nets, along with a full completeness theorem. It is well-known that games semantics is intimately connected to linear logic, but there is an important example of games semantics where the connection is far from clear, namely Hyland and Ong's 9] for the simply typed lambda-calculus and PCF. Although in this semantics the construction of the function space (intuitionistic implication) depends quite explicitly on the standard decomposition X) Y = !X (Y , it is not clear at all how one would be able to describe the semantics of these two linear operators independently. In particular if one naively follows the spirit of the constructions given in that paper, it seems one would get that the natural morphism !X ! !!X (comul-tiplication in the comonad) is an isomorphism. It follows from the theory of (co)monads that the two possible natural morphisms !!X ! !X would be identical, which means an enormous loss of information in such a semantics, something quite unlike what we would expect from games semantics. In this paper we give a games semantics for propositional intuitionistic linear logic with the connectors (and !, along with a full completeness theorem of the likes of 9,2]. We should mention that apart from Hyland and Ong's, the other games semantics with a full completeness theorem for the lambda-calculus, Abramsky-Jagadeesan-Malacaria's 2], is easier to relate to linear logic, and this is due to its close relation to the Geometry of Interaction 6]. We will see that this jump from the simply typed lambda-calculus to the smallest fragment of linear logic able to express it is rather nontrivial, due to the appearance of new subtleties. Our approach will start by giving a theory of proof nets for this logical fragment, with the usual correctnes criterion. We call these nets semantical nets. Given a proof net, we then show how a strategy for a two-person game can be constructed from it, as a set of paths in the proof net. The correctness condition that describes the strategies that come from proofs can then be readily deduced. Hence semantical nets c 1996 Elsevier Science B. V. Lamarche can be thought of as an intermediary step between the sequent calculus and the games, …