13.1 Compiling higher order into first order

Passing processes looks like a drastic departure from passing names. We show in this section and the next, however, that process-passing can be faithfully compiled down to name-passing. We present two main compilations. The second is an optimization of the first. Moreover, the second, unlike the first, is also defined on calculi with recursive types. However, the first compilation is mathematically easier to work with.

We begin with the first compilation, which we call D. The communication of a higher-order value v is translated as the communication of a private name that acts as a pointer to (the translation of) v and that the recipient can use to trigger a copy of (the translation of) v, with appropriate arguments. For instance a process is translated to the process a recipient of the pointer y can use it to activate as many copies of as needed, with appropriate arguments. The compilation separates the acts of copying and of activating the value (x).R; copying is rendered by the replication, and activation by communications along the pointer y. Here are some simple examples of how the compilation works. We omit, for now, all type annotations.

In the example above, it is important that the trigger name y is private: if it were not, a process in the surrounding environment could interfere with the second communication step.

The examples above show translations of applications vw in which v is a name; in this case the application is translated in the same way as an output vw. Here is an example in which v is an abstraction (it is similar to Example 13.1.1, except that (z). R is the argument of an application, rather than the value being output):

The compilation modifies types: a name used in HOπ to exchange processes becomes, in π-calculus, a name used for exchanging other names. The definition of the compilation on types, type environments, values, and terms is given in Table 13.1.