Actor model and process calculi

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).

How channels work
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
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.

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

Synchronous channels in process calculi
However, simple synchronous channels do not suffice for process calculi such as Communicating Sequential Processes (CSP) [Hoare 1978 and 1985] because use of the guarded choice (after Dijkstra) command (called the alternative command in CSP). In a guarded choice command multiple offers (called guards) can be made concurrently on multiple channels to  and   messages; however at most one of the guards can be chosen for each execution of the guarded choice command. Because only one guard can be chosen, a guarded choice command in general effectively requires a kind of two-phase commit protocol or perhaps even a three-phase commit protocol if time-outs are allowed in guards (as in Occam 3 [1992]).

Consider the following program written in CSP [Hoare 1978]: [X :: Z!stop || Y :: guard: boolean; guard := true; *[guard →  Z!go; Z?guard] || Z :: n: integer; n:= 0; *[X?stop →  Y!false; print!n; [] Y?go →  n := n+1; Y!true] ] According to Clinger [1981], this program illustrates global nondeterminism, since the nondeterminism arises from incomplete specification of the timing of signals between the three processes,  , and. The repetitive guarded command in the definition of  has two alternatives: If  ever accepts the   message from , then   terminates. Accepting the  causes   to be sent false which when input as the value of its guard will cause   to terminate. When both  and   have terminated,   terminates because it no longer has live processes providing input.
 * 1) the  message is accepted from , in which case   is sent the value false and   is sent the value
 * 2) a  message is accepted from , in which case   is incremented and   is sent the value true.

In the above program, there are synchronous channels from  to ,   to  , and   to.

Analogy with the committee coordination problem
According to Knabe [1992], Chandy and Misra [1988] characterized this as analogous to the committee coordination problem:


 * Professors in a university are assigned to various committees. Occasionally a professor will decide to attend a meeting of any of her committees, and will wait until that is possible.  Meetings may begin only if there is full attendance.  The task is to ensure that if all the members of a committee are waiting, then at least one of them will attend some meeting.
 * The crux of this problem is that two or more committees might share a professor. When that professor becomes available, she can only choose one of the meetings, while the others continue to wait.

A simple distributed protocol
This section presents a simple distributed protocol for channels in synchronous process calculi. The protocol has some problems that are addressed in the sections below.

The behavior of a guarded choice command is as follows:
 * The command sends a message to each of its guards to.
 * When it receives the first response from one of its guards that it is prepared, then it sends a message to that guard to  and sends messages to all of the other guards to.
 * When it receives a message from the guard that it is, then it sends the guard a   message. However, if the guard throws an exception that it cannot  , then guarded choice command starts the whole process all over again.
 * If all of its guards respond that they cannot, then the guarded command does nothing.

The behavior of a guard is as follows:
 * When a message to  is received, then the guard sends a   message to each of the channels with which it is offering to communicate.  If the guard has booleans such that it cannot   or if any of the channels respond that they cannot , then it sends   messages to the other channels and then responds that it cannot.
 * When a message to  is received, then the guard sends a   message to each of the channels. If any of the channels respond that they cannot , then it sends   messages to the other channels and then throws an exception that it cannot.
 * When a message to  is received, then the guard sends a   message to each of the channels.
 * When a message to  is received, then the guard sends an   message to each of the channels.

The behavior of a channel is as follows:
 * When a  communication is received, then respond that it is prepared if there is a   communication pending unless a   communication has been received, in which case throw an exception that it cannot.
 * When a  communication is received, then respond that it is prepared if there is a   communication pending unless a   communication has been received, in which case throw an exception that it cannot.
 * When a  communication is received, then respond that it is prepared if there is a   communication pending unless a   communication has been received, in which case throw an exception that it cannot.
 * When a  communication is received, then respond that it is prepared if there is a   communication pending unless a   communication has been received, in which case throw an exception that it cannot.
 * When a  communication is received, then depending on which of the following is received:
 * When a  communication is received, then if not already done perform the   and   and clean up the preparations.
 * When an  communication is received, then cancel the preparations
 * When a  communication is received, then depending on which of the following is received:
 * When a  communication is received, then if not already done perform the   and   and clean up the preparations.
 * When an  communication is received, then cancel the preparations.
 * When an  communication is received, then cancel the preparations.
 * When an  communication is received, then cancel the preparations.

Starvation on getting from multiple channels
Again consider the program written in CSP (discussed in Synchronous channels in process calculi above): [X :: Z!stop || Y :: guard: boolean; guard := true; *[guard →  Z!go; Z?guard] || Z :: n: integer; n:= 0; *[X?stop →  Y!false; print!n; [] Y?go →  n := n+1; Y!true] ]

As pointed out in Knabe [1992], a problem with the above protocol (A simple distributed protocol) is that the process  might never accept the   message from   (a phenomenon called starvation) and consequently the above program might never print anything.

In contrast consider, a simple Actor system that consists of Actors X, Y, Z, and print where


 * the Actor X is created with the following behavior:
 * If the message  is received, then send Z the message
 * the Actor Y is created with the following behavior:
 * If the message  is received, then send Z the message
 * If the message true is received, then send Z the message
 * If the message false is received, then do nothing
 * the Actor Z is created with the following behavior that has a count  that is initially 0:
 * If the message  is received, then do nothing.
 * If the message  is received, then send Y the message false and send print the message the count.
 * If the message  is received, then send Y the message true and process the next message received with count   being.

By the laws of Actor semantics, the above Actor system will always halt when the Actors X, Y, are Z are each sent a  message resulting in sending print a number that can be unbounded large.

The difference between the CSP program and the Actor system is that the Actor Z does not get messages using a guarded choice command from multiple channels. Instead it processes messages in arrival ordering, and by the laws for Actor systems, the  message is guaranteed to arrive.

Livelock on getting from multiple channels
Consider the following program written in CSP [Hoare 1978]: [Bidder1 :: b: bid; *[Bids1?b →  process1!b; [] Bids2?b →  process1!b;] || Bidder2 :: b: bid; *[Bids1?b →  process2!b; [] Bids2?b →  process2!b;] ]

As pointed out in Knabe [1992], an issue with the above protocol (A simple distributed protocol) is that the process  might never accept a bid from   or   (a phenomenon called livelock) and consequently   might never be sent anything. In each attempt to accept a message,  is thwarted because the bid that was offered by   or   is snatched away by   because it turns out that   has much faster access than   to   and. Consequently,  can accept a bid, process it and accept another bid before   can commit to accepting a bid.

Efficiency
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.

Summary of Issues
The subsections above have articulated the following three issues concerned with the use of synchronous channels for process calculi:
 * 1) Starvation. The use of synchronous channels can cause starvation when a process attempts to get messages from multiple channels in a guarded choice command.
 * 2) Livelock. The use of synchronous channels can cause a process to be caught in livelock when it attempts to get messages from multiple channels in a guarded choice command.
 * 3) Efficiency. The use of synchronous channels can require a large number of communications in order to get messages from multiple channels in a guarded choice command.

It is notable that in all of the above, issues arise from the use of a guarded choice command to get messages from multiple channels.

Asynchronous channels
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.

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

Asynchronous channels in process calculi
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.

Algebras
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, ,.

Denotational semantics
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.