Completeness forMultiplicative Linear Logic without the MIX - Rule ( Working Draft )

We introduce a new category of nite, fair games, and winning strategies, and use it to provide a semantics for the multiplicative fragment of Linear Logic (mll) in which formulae are interpreted as games, and proofs as winning strategies. This interpretation provides a categorical model of mll which satisses the property that every (history-free, uniformly) winning strategy is the denotation of a unique cut-free proof net. Abramsky and Jagadeesan rst proved a result of this kind and they refer to this property as full completeness. Our result diiers from theirs in one important aspect: the mix-rule, which is not part of Girard's Linear Logic, is invalidated in our model. We achieve this sharper characterization by considering fair games. A nite, fair game is speciied by the following data: moves which Player can play, moves which Opponent can play, and a collection of nite sequences of maximal (or terminal) positions of the game which are deemed to be fair. Notably, positions of a game are a derived notion. The maximal positions of a compound game are obtained by an appropriate interleaving of the maximal positions of the respective constituent games. At any position in a nite, fair game, a player can make a move if, and only if, the move can be extended to a maximal position.