How to simulate natural number in Pi Calculus

Question

I have some questions about Pi Calculus:

How to simulate the natural numbers only with its primitives?

It's possible simulate the if else clauses without the sum (+)?

A minimal and standart example would be interesting.

Answer 1

You can encode natural numbers in π-calculus in a way that is a variant on encodings of natural numbers in untyped λ-calculus, see (1). Milner, Parrow & Walker explain how to do this in the paper (2) that introduced π-calculus. Note that such encodings are not efficient (they are essentally unary encodings of numbers, so give rise to large terms). You only do such encodings to gauge expressive power of the untyped π-calculus. In any real implementation, you'd add numbers as a basic data-type.

---

1. Wikipedia, Church encoding.

2. R. Milner, J. Parrow, D. Walker, A Calculus of Mobile Processes, Part I.