Part I introduces the π-calculus. It explains how the terms of the calculus describe the structure and the behaviour of mobile systems. It also develops the basic theory of behavioural equivalence, and introduces basic techniques for reasoning about systems.

How system behaviour is expressed is fundamental to the theory. Two accounts of behaviour will be given. The first explains how a system represented by a π-calculus term can evolve independently of its environment. At the heart of this first account is a binary relation on terms called reduction. The second account explains not only activity within a system but also how the system can interact with its environment. Central to it is a labelled transition system on terms.

Why are there two accounts of behaviour? The reason is that each has useful qualities that the other lacks. Reduction explains activity within systems in a way that overcomes the rigidity of linear syntax. This makes the account based on reduction easy to grasp, something that is helpful when the π-calculus is encountered for the first time. Reduction has two related limitations, however. First, it does not say how a system can interact with its environment. And second, the free treatment of syntax, which helps to make the account simple, also makes it difficult to prove things about behaviours. The account based on transition overcomes both of these limitations, but at the unavoidable cost of a little increase in complexity. It explains both activity within a system and interaction between a system and its environment by describing the actions that processes can perform. And it explains action in a way that is guided by the syntax of terms, so that, in particular, the structure of proofs can follow the syntax.

Reduction is a relation of a kind familiar from term-rewriting systems, while labelled transition systems are well known from process calculi. In term-rewriting systems, the definition of the reduction relation is relatively straightforward because a redex of a term must be a subterm. For example, in the λ-calculus - perhaps the best-known term-rewriting system - in order for a function to be applied to an argument, the two must be contiguous.