Abstract Games for Linear Logic Extended Abstract

We draw attention to a number of constructions which lie behind many concrete models for linear logic; we develop an abstract context for these and describe their general theory. Using these constructions we give a model of classical linear logic based on an abstract notion of game. We derive this not from a category with built-in computational content but from the simple category of sets and relations. To demonstrate the computational content of the resulting model we make comparisons at each stage of the construction with a standard very simple notion of game. Our model provides motivation for a less familiar category of games (played on directed graphs) which is closely reflected by our notion of abstract game. We briefly indicate a number of variations on this theme and sketch how the abstract concept of game may be refined further. It will be clear that we have been influenced in a general way by colleagues who have worked on models for linear logic. Many have given readings of their models as abstract games. However, we would like to mention that a specific precursor of the work described here is a joint project with Robin Cockett on the analysis of sequentiality in the context of yet other categories of abstract games. We hope to report on this in the fullness of time.