8 - Advanced Type Systems

Summary

8.1 Linearity

The purpose of the type systems for linearity is to control the number of times a resource is used. In the specific case of the π-calculus the resources are the names.

The linear types for the π-calculus are a refinement of the i/o types that allow us to impose not only polarity, but also multiplicity constraints: we can say that a name should be used in input or in output only once. A linear type can take three forms: ℓ_o T, ℓ_i Tand ℓ_# T. They are, respectively, the linear versions of the types o T, i T, and # T. A type ℓ_o T for a name gives the capability of transmitting a value of type T along that name once. Dually, ℓ_i T gives the capability of receiving a value of type T once. Finally ℓ_# T is the type of a name that can be used once for transmitting and once for receiving a value of type T; type ℓ_# T is the union of the capabilities offered by ℓ_i T and ℓ_o T. (By contrast with _# T, the linear type ℓ_# T does not give the right to test the identity of a name; a name that is used both for a communication and for a test is not linear, because it is used twice.) For instance, the processes

are well typed in {a : ℓ_o T, b : T, c : # ℓ_o T}. Note that the process ca does not itself exercise the output capability on a, but passes it to another process. By contrast, the same typing is not respected by the processes

We add linear types to a simplv-tvped π-calculus with i/o types; is the resulting calculus. Types, and related operations, are presented in Table 8.1. Moving from to all typing rules for processes and values, except that for restriction, that for matching, and that for subsumption, have to be modified.