the binary communications operator, . For example, the actions and might be interpreted as the reading and writing of a data item , respectively. Then the process will communicate the value from the right component process to the left component process (i.e. the identifier is bound to the value , and free instances of in the process take on that value), and then behave as the merge of and . Abstraction — the abstraction operator, , is a way to "hide" certain actions, and treat them as events that are internal to the systems being modelled. Abstracted actions are converted to the silent step action . In some cases, these silent steps can also be removed from the process expression as part of the abstraction process. For example, which, in this case, can be reduced to since the event is no longer observable and has no observable effects. 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 ACP (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, 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, . 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, , 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 will be reduced to the action . ACP also includes an encapsulation operator, for some , which is used to convert unsuccessful communication attempts (i.e. elements of 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