Linear logic has been introduced by Girard as a logic of actions that seems well suited for concurrent computation. This paper surveys recent work on the applications of linear logic to concurrency, with special emphasis on Petri nets and on the use of categorical models. In particular, we present a synthesis of our previous work on the systematic correspondence between Petri nets, linear logic theories, and linear categories, and explain its relationships to work by many other authors. Throughout, we discuss the computational interpretation of the linear logic connectives and illustrate the ideas with examples. Categories play an important role in this survey. On the one hand, from a computational perspective, they are interpreted as concurrent systems whose objects are states, and whose morphisms are transitions; on the other hand, when a model-theoretic perspective is adopted, they provide a very flexible conceptual framework within which the relationships among quite different models already proposed for linear logic can be better understood; this framework also suggests the study of new models and an axiomatic treatment of classes of models. Our categorical semantics for linear logic is based on dualizing objects and permits a very simple presentation of ideas requiring a more complicated treatment in the language of *-autonomous categories.
A Categorical Semantics for Hierarchical Petri Nets
