Linearity in Process Languages

The meaning and mathematical consequences of linearity (managing without a presumed ability to copy) are studied for a path-based model of processes which is also a model of affine-linear logic. This connection yields an affine-linear language for processes, automatically respecting open-map bisimulation, in which a range of process operations can be expressed. An operational semantics is provided for the tensor fragment of the language. Different ways to make assemblies of processes lead to different choices of exponential, some of which respect bisimulation. 1 Path-based models of processes In distributed computation it can be hard or impossible for a process to copy a process, while it is generally easy for a process to ignore another process. For this reason an operation on processes associated with distributed computation often has the following property: a computation path of the process arising from the application of the operation to an input process has resulted from at most one computation path of the input process. As we will see, this property expresses that the operation is an affine linear map within a model of linear logic, a fact which carries many consequences. This article presents a model of processes based on computation paths and so can make precise the sense in which many process operations are associated with linear maps, investigates the consequences of linearity for the important equivalence of bisimulation, and delineates the boundaries of linearity with respect to one, fairly broad, mathematical model, in which processes are represented as presheaves. ∗Basic Research in Computer Science (www.brics.dk), funded by the Danish National Research Foundation.