# The Polyadic π-Calculus: a Tutorial

@inproceedings{Milner1993TheP,
title={The Polyadic $\pi$-Calculus: a Tutorial},
author={Robin Milner},
year={1993}
}
The π-calculus is a model of concurrent computation based upon the notion of naming. [] Key Method Semantics is done in terms of both a reduction system and a version of labelled transitions called commitment; the known algebraic axiomatization of strong bisimilarity is given in the new setting, and so also is a characterization in modal logic. Some theorems about the replication operator are proved.
1,126 Citations
A formalisation of π-calculus in the Coq system is presented. Based on a de Bruijn notation for names, our implementation exploits the mechanisation of some proof techniques described by Sangiorgi in
• Computer Science
TLCA
• 2001
The result shows how a typed π-calculus can be used as a descriptive tool for a significant class of programming languages without losing the latter's semantic properties.
Preliminary work on a definitional formal theory of the π-calculus in higher order logic using the HOL theorem prover is described, with the ultimate goal of providing practical mechanized support for reasoning with the ρ-Calculus about applications.
The rules for type-checking and for evaluation and formalize a proof of type preservation in the Coq system are given and the specification for the pi-calculus formalizes communication by means of function application.
• Computer Science
ICALP
• 2008
A general framework for extending the pi-calculus with data terms using a single untyped notion of agent, name and scope, an operational semantics without structural equivalence and a simple definition of bisimilarity is demonstrated.
• Computer Science
• 1996
A translation of the π-calculus into the MONSTR graph rewriting language is described and proved correct, illustrating many features typically encountered in reasoning about graph rewriting systems, and particularly how serialisation techniques can be used to reorder an arbitrary execution into one having stated desirable properties.
• Computer Science
• 1994
The aim of this work is to describe the prototypical mobility expressed by the π-calculus within a CCS-like approach to process algebras by introducing suitable constructors for both the explicit handling of name substitutions and the explicit instantiation of names.
The characterization supports a bisimulation-like proof technique which avoids explicit case analysis by taking a dynamic point of view of actions a process may perform, thus providing a new way of proving bisimulations congruence.

## References

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This paper exhibits accurate encodings of the λ-calculus in the π-calculus. The former is canonical for calculation with functions, while the latter is a recent step (Milner et al. 1989) towards a
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A general method for proving/deciding equivalences between omega-regular languages, whose recognizers are modified forms of Buchi or Muller-McNaughton automata, derived from Milner's notion of “simulation” is obtained.
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TAPSOFT, Vol.1
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It is shown how to derive a family of calculi, suitable for the specification of concurrent systems and languages, that include functional abstraction and application, and can be higher order calculi with behaviours as first class objects.
Semantics for a pair of parallel object-oriented programming languages are presented by translation into the π-calculus, a foundation for the study of computational systems with evolving communication structure.
A structural operational semantics for the typed λ-calculus is developed and it is proved that the operational semantics preserves the types and is used to give examples of ‘errors’ that cannot arise for well-typed programs.
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TAPSOFT, Vol.1
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This work introduces a calculus for concurrent and communicating processes, which is a direct and simple extension of the λ-calculus, and shows that the ε-abstraction is a particular case of reception (on a port named λ), and application is a specific case of cooperation.
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The paper demonstrates, for a sequence of simple languages expressing finite behaviors, that in each case observation congruence can be axiomatized algebraically and the algebraic language described here becomes a calculus for writing and specifying concurrent programs and for proving their properties.
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