Actor model theory

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.

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].

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).

The arrival ordering of an Actor x ( -x→ ) models the (total) ordering of events in which a message arrives at x. 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).

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:

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.

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