## Abstract

Action structures have previously been proposed as an algebra for both the syntax and the semantics of interactive computation. Here, a class of concrete action structures called*action calculi* is identified, which can serve as a non-linear syntax for a wide variety of models of interactive behaviour. Each action in an action calculus is represented as an assembly of*molecules*; the syntactic binding of*names* is the means by which molecules are bound together. A graphical form,*action graphs*, is used to aid presentation. One action calculus differs from another only in its generators, called*controls*.

Action calculi generalise a previously defined action structure PIC for the π*-* calculus. Several extensions to PIC are given as action calculi, giving essentially the same power as the π-calculus. An action calculus is also given for the typed λ-calculus, and for Petri nets parametrized on their places and transitions.

An equational characterization of action calculi is given: each action calculus*A* is the quotient of a term algebra by certain equations. The terms are generated by a set of operators, including those basic to all action structures as well as the controls specific to*A*; the equations are the basic axioms of action structures together with four additional axiom schemata.

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Milner, R. Calculi for interaction.
*Acta Informatica* **33, **707–737 (1996). https://doi.org/10.1007/BF03036472

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### Keywords

- Action Structure
- Molecular Form
- Monoidal Category
- Control Rule
- Action Graph