In theoretical computer science, Actor model theory concerns theoretical issues for the Actor model. Actors are the primitives that form the basis of the Actor model of concurrent digital computation. In response to a message that it receives, an Actor can make local decisions, create more Actors, send more messages, and designate how to respond to the next message received. Actor model theory incorporates theories of the events and structures of Actor computations, their proof theory, and denotational models.

Events and their orderings

From the definition of an Actor, it can be seen that numerous events take place: local decisions, creating Actors, sending messages, receiving messages, and designating how to respond to the next message received. However, this article focuses on just those events that are the arrival of a message sent to an Actor. This article reports on the results published in Hewitt [2006].

Activation ordering

The activation ordering is a fundamental ordering that models one event activating another (there must be energy flow in the message passing from an event to an event which it activates).

Arrival orderings

The arrival ordering of an Actor models the (total) ordering of events in which a message arrives at. Arrival ordering is determined by arbitration in processing messages (often making use of a digital circuit called an arbiter). The arrival events of an Actor are on its world line. The arrival ordering means that the Actor model inherently has indeterminacy (see Indeterminacy in concurrent computation).

Combined ordering

The combined ordering (denoted by ) is defined to be the transitive closure of the activation ordering and the arrival orderings of all Actors. The combined ordering is obviously transitive by definition. In [Baker and Hewitt 197?], it was conjectured that the above laws might entail the following law:

Independence of the Law of Finite Chains Between Events in the Combined Ordering

However, [Clinger 1981] surprisingly proved that the Law of Finite Chains Between Events in the Combined Ordering is independent of the previous laws, i.e., Theorem. The Law of Finite Chains Between Events in the Combined Ordering does not follow from the previously stated laws. Proof. It is sufficient to show that there is an Actor computation that satisfies the previously stated laws but violates the Law of Finite Chains Between Events in the Combined Ordering. However, we know from physics that infinite energy cannot be expended along a finite trajectory. Therefore, since the Actor model is based on physics, the Law of Finite Chains Between Events in the Combined Ordering was taken as an axiom of the Actor model.

Law of Discreteness

The Law of Finite Chains Between Events in the Combined Ordering is closely related to the following law: In fact the previous two laws have been shown to be equivalent: The law of discreteness rules out Zeno machines and is related to results on Petri nets [Best et al. 1984, 1987]. The Law of Discreteness implies the property of unbounded nondeterminism. The combined ordering is used by [Clinger 1981] in the construction of a denotational model of Actors (see denotational semantics).

Denotational semantics

Clinger [1981] used the Actor event model described above to construct a denotational model for Actors using power domains. Subsequently Hewitt [2006] augmented the diagrams with arrival times to construct a technically simpler denotational model that is easier to understand.