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In computer science, the Actor model and process calculi are two closely related approaches to the modelling of concurrent digital computation. See Actor model and process calculi history. There are many similarities between the two approaches, but also several differences (some philosophical, some technical):
There is only one Actor model (although it has numerous formal systems for design, analysis, verification, modeling, etc. ); there are numerous process calculi, developed for reasoning about a variety of different kinds of concurrent systems at various levels of detail (including calculi that incorporate time, stochastic transitions, or constructs specific to application areas such as security analysis). The Actor model was inspired by the laws of physics and depends on them for its fundamental axioms, i. e. physical laws (see Actor model theory); the process calculi were originally inspired by algebra Processes in the process calculi are anonymous, and communicate by sending messages either through named channels (synchronous or asynchronous), or via ambients (which can also be used to model channel like communications ). In contrast, actors in the Actor model possess an identity, and communicate by sending messages to the mailing addresses of other actors (this style of communication can also be used to model channel like communications—see below). The publications on the Actor model and on process calculi have a fair number of cross references, acknowledgments, and reciprocal citations (see Actor model and process calculi history).

Indirect communication using channels (e. g. Gilles Kahn and David MacQueen [1977]) has been an important issue for communication in parallel and concurrent computation affecting both semantics and performance. Some process calculi differ from the Actor model in their use of channels as opposed to direct communication.

Synchronous channels have the property that a sender putting a message in the channel must wait for a receiver to get the message out of the channel before the sender can proceed.

A synchronous channel can be modeled by an Actor that receives put and get communications. The following is the behavior of an Actor for a simple synchronous channel:
Each put communication has a message and an address to which an acknowledgment is sent when the message is received by a get communication from the channel in FIFO order. Each get communication has an address to which the received message is sent.

As pointed out in Knabe [1992], an issue with the above protocol (A simple distributed protocol) is the large number of communications that must be sent in order to perform the handshaking in order to send a message through a synchronous channel. Indeed, as shown in the previous section (Livelock), the number of communications can be unbounded.

Asynchronous channels have the property that a sender putting a message in the channel need not wait for a receiver to get the message out of the channel.

An asynchronous channel can be modeled by an Actor that receives put and get communications. The following is the behavior of an Actor for a simple asynchronous channel:
Each put communication has a message and an address to which an acknowledgment is sent immediately (without waiting for the message to be gotten by a get communication). Each get communication has an address to which the gotten message is sent.

The Join calculus programming language (published in 1996) implemented local and distributed concurrent computations. It incorporated asynchronous channels as well as a kind of synchronous channel that is used for procedure calls. Agha's Aπ Actor calculus is based on a typed version of the asynchronous π calculus.

The use of algebraic techniques was pioneered in the process calculi. Subsequently, several different process calculi intended to provide algebraic reasoning about Actor systems have been developed in , , .

Will Clinger (building on the work of Irene Greif [1975], Gordon Plotkin [1976], Henry Baker [1978], Michael Smyth [1978], and Francez, Hoare, Lehmann, and de Roever [1979]) published the first satisfactory mathematical denotational theory of the Actor model using domain theory in his dissertation in 1981. His semantics contrasted the unbounded nondeterminism of the Actor model with the bounded nondeterminism of CSP [Hoare 1978] and Concurrent Processes [Milne and Milner 1979] (see denotational semantics). Roscoe [2005] has developed a denotational semantics with unbounded nondeterminism for a subsequent version of Communicating Sequential Processes Hoare [1985]. More recently Carl Hewitt [2006b] developed a denotational semantics for Actors based on timed diagrams. Ugo Montanari and Carolyn Talcott [1998] have contributed to attempting to reconcile Actors with process calculi.