Algebra of communicating processes

The algebra of communicating processes (ACP) is an algebraic approach to reasoning about concurrent systems. It is a member of the family of mathematical theories of concurrency known as process algebras or process calculi. ACP was initially developed by Jan Bergstra and Jan Willem Klop in 1982, as part of an effort to investigate the solutions of unguarded recursive equations. More so than the other seminal process calculi (CCS and CSP), the development of ACP focused on the algebra of processes, and sought to create an abstract, generalized axiomatic system for processes, and in fact the term "process algebra" was coined during the research that led to ACP.
Informal description.
ACP is fundamentally an algebra, in the sense of universal algebra. This algebra is a way to describe systems in terms of algebraic process expressions that define compositions of other processes, or of certain primitive elements.
Primitives.
ACP uses instantaneous, "atomic actions" (formula_1) as its primitives. Some actions have special meaning, such as the action formula_2, which represents deadlock or stagnation, and the action formula_3, which represents a "silent action" (abstracted actions that have no specific identity).
Algebraic operators.
Actions can be combined to form "processes" using a variety of operators. These operators can be roughly categorized as providing a "basic process algebra", "concurrency", and "communication".
Formal definition.
ACP fundamentally adopts an axiomatic, algebraic approach to the formal definition of its various operators. The axioms presented below comprise the full axiomatic system for ACPformula_3 (ACP with abstraction).
Basic process algebra.
Using the alternative and sequential composition operators, ACP defines a "basic process algebra" which satisfies the axioms
Deadlock.
Beyond the basic algebra, two additional axioms define the relationships between the alternative and sequencing operators, and the "deadlock" action, formula_2
Concurrency and interaction.
The axioms associated with the merge, left-merge, and communication operators are
When the communications operator is applied to actions alone, rather than processes, it is interpreted as a binary function from actions to actions, formula_42. The definition of this function defines the possible interactions between processes — those pairs of actions that do not constitute interactions are mapped to the deadlock action, formula_2, while permitted interaction pairs are mapped to corresponding single actions representing the occurrence of an interaction. For example, the communications function might specify that
which indicates that a successful interaction formula_45 will be reduced to the action formula_46. ACP also includes an encapsulation operator, formula_47 for some formula_48, which is used to convert unsuccessful communication attempts (i.e. elements of formula_49 that have not been reduced via the communication function) to the deadlock action. The axioms associated with the communications function and encapsulation operator are
Abstraction.
The axioms associated with the abstraction operator are
Note that the action "a" in the above list may take the value δ (but of course, δ cannot belong to the abstraction set "I").
Related formalisms.
ACP has served as the basis or inspiration for several other formalisms that can be used to describe and analyze concurrent systems, including: