## Action Structures

**Robin Milner**
*Abstract:* Action structures are proposed as a variety of
algebra to underlie concrete models of concurrency and interaction.
An action structure is equipped with *composition* and
*product* of actions, together with two other ingredients: an
indexed family of *abstractors* to allow parametrisation of
actions, and a *reaction* relation to represent activity. The
eight axioms of an action structure make it an enriched strict
monoidal category; however, the work is presented algebraically
rather than in category theory.

The notion of action structure is developed mathematically, and
examples are studied ranging from the evaluation of expressions to
the statics and dynamics of Petri nets. For algebraic process
calculi in particular, it is shown how they may be defined by a
uniform superposition of process structure upon an action structure
specific to each calculus. This allows a common treatment of
bisimulation congruence.

The theory of action structures emphasizes the notion of
*effect*; that is, the effect which any interaction among
processes exerts upon its participants. Effects are degenerate
actions, and constitute a sub-actionstructure with special
properties which support the general treatment of bisimulation.

Other current work on action structures is outlined, in
particular their use for the pi-calculus. Challenges are posed for
the algebraic theory, including the study of combinations of action
structures. Action structures are briefly compared with some other
general models.

*LFCS report ECS-LFCS-92-249,
December 1992.*

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