We introduce a new category of finite, 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 satisfies the property that every (history-free, uniformly) winning strategy is the denotation of a unique cut-free proof net. Abramsky and Jagadeesan first proved a result of this kind and they refer to this property as full completeness. Our result differs 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 finite, fair game is specified by the following data: ffl moves which Player can play, ffl moves which Opponent can play, and ffl a collection of finite sequences of maximal (or terminal) positions of the game which are deemed to be fair. N...

We show that two models M and N of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories C and D are related by a pair of monoidal functors F : C D : G and transformations Id C ) GF and Id D ) FG, and (2) their exponentials ! are related by distributive laws % : ! : ! M G ) G ! N commuting to the promotion rule. The key ingredient of the proof is a notion of back-and-forth translation between the hierarchies of types induced by M and N. We apply this result to compare (1) the qualitative and the quantitative hierarchies induced by the coherence (or hypercoherence) space model, (2) several paradigms of games semantics: error-free vs. error-aware, alternated vs. non-alternated, backtracking vs. repetitive, uniform vs. non-uniform.

In this note, we reformulate Hyland-Ong-Nickau pointer game semantics for PCF, as a more concurrent style of games, called 2-dimensional games. The main novelty of these games is that they are played on concurrent graphs, instead of trees. This enables to permute moves, and to decompose the innocence constraint on strategies, as local permutation properties.