Pi-calculus

In theoretical computer science, the π-calculus (or pi-calculus) is a process calculus. The π-calculus allows channel names to be communicated along the channels themselves, and in this way it is able to describe concurrent computations whose network configuration may change during the computation.

The π-calculus is elegantly simple, it has very few terms and so is a very small language [2], yet is very expressive. Functional programs can be encoded into the π-calculus, and the encoding emphasises the dialogue nature of computation, drawing connections with game semantics. Extensions of the π-calculus, such as the spi calculus and applied π, have been successful in reasoning about . Beside the original use in describing concurrent systems, the π-calculus has also been used to reason about business processes and molecular biology.

The π-calculus belongs to the family of process calculi, mathematical formalisms for describing and analyzing properties of concurrent computation. In fact, the π-calculus, like the λ-calculus, is so minimal that it does not contain primitives such as numbers, booleans, data structures, variables, functions, or even the usual control flow statements (such as if-then-else, while).

Central to the π-calculus is the notion of name. The simplicity of the calculus lies in the dual role that names play as communication channels and variables.

The process constructs available in the calculus are the following (a precise definition is given in the following section):

Although the minimalism of the π-calculus prevents us from writing programs in the normal sense, it is easy to extend the calculus. In particular, it is easy to define both control structures such as recursion, loops and sequential composition and datatypes such as first-order functions, truth values, lists and integers. Moreover, extensions of the π-calculus have been proposed which take into account distribution or public-key cryptography. The applied π-calculus due to Abadi and Fournet [3] put these various extensions on a formal footing by extending the π-calculus with arbitrary datatypes.

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