# 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 Full Formalisation of pi-Calculus Theory in the Calculus of Constructions

- Computer ScienceTPHOLs
- 1997

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…

### Sequentiality and the pi-Calculus

- Computer ScienceTLCA
- 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.

### A mechanized theory of the π-calculus in HOL

- Computer Science, Mathematics
- 1992

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.

### A Higher-Order Specification of the pi-Calculus

- Computer Science, MathematicsIFIP TCS
- 2000

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.

### Extended pi-Calculi

- Computer ScienceICALP
- 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.

### A Translation of the Pi-Calculus Into MONSTR

- 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.

### The Calculus of Explicit Substitutions

- 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.

### Characterizing Bisimulation Congruence in the pi-Calculus (Extended Abstract)

- MathematicsCONCUR
- 1994

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.

### A Pi-Calculus with Explicit Substitutions

- Computer ScienceTheor. Comput. Sci.
- 1996

## References

SHOWING 1-10 OF 25 REFERENCES

### Functions as processes

- MathematicsMathematical Structures in Computer Science
- 1992

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…

### Concurrency and Automata on Infinite Sequences

- Computer ScienceTheoretical Computer Science
- 1981

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.

### SMoLCS-Driven Concurrent Calculi

- Computer ScienceTAPSOFT, Vol.1
- 1987

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.

### Pi-Calculus Semantics of Object-Oriented Programming Languages

- Computer ScienceTACS
- 1991

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.

### The Typed lambda-Calculus with First-Class Processes

- Computer SciencePARLE
- 1989

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.

### Towards a Lambda-Calculus for Concurrent and Communicating Systems

- Computer Science, MathematicsTAPSOFT, Vol.1
- 1989

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.

### Algebraic laws for nondeterminism and concurrency

- Computer ScienceJACM
- 1985

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.

### Algebraic theory of processes

- Computer ScienceMIT Press series in the foundations of computing
- 1988

Algebraic Theory of Processes provides the first general and systematic introduction to the semantics of concurrent systems, a relatively new research area in computer science. It develops the…