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Published online by Cambridge University Press: 29 May 2025
9.1 Using types to obtain encapsulation
It is desirable that a language, whether sequential or parallel, should have facilities for encapsulation, that is, features that make it possible to constrain access to components such as data and resources. The need for encapsulation led to the development of abstract data types. Encapsulation is also a key feature of objects in object-oriented languages.
In CCS, encapsulation is achieved by the restriction operator. Restricting a name x on a process P, written (using π-calculus notation) vx P, guarantees that interactions along x between subcomponents of P occur without interference from outside. For instance, suppose we have two 1-place buffers, Buf1 and Buf2. Suppose Buf1 receives values along a name x and resends them along y, whereas Buf2 receives at y and resends at z. They can be composed into a 2-place buffer that receives at x and resends at z thus: vy (Buf1 | Buf2). Here, the restriction ensures that actions at y from Buf1 and Buf2 are not consumed by the external environment. With the formal definitions of Buf1 and Buf2 at hand, one can indeed prove that the system vy (Buf1 | Buf2) is behaviourally equivalent to a 2-place buffer.
The restriction operator provides a quite satisfactory level of protection in CCS, where the visibility of names in processes is fixed. By contrast, restriction alone is very often not satisfactory in the π-calculus, where the visibility of names can change dynamically. Here are two examples.
Example 9.1.1 (A printer with mobile ownership) Consider a situation in which several client processes cooperate in the use of a shared resource such as a printer. Data are sent for printing by the client processes along a name p. Clients can also communicate name p so that new clients can get access to the printer. Suppose that initially there are two clients
and therefore, writing P for the printer process, the initial system is
One might wish to prove that when Cl's print jobs, represented by j1 and j2, are delivered, they are received and processed in that order by the printer.
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