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F α IF / IF (F)) syntactic compatibility of interface abstraction
Clearly not every mapping between interfaces is to be called an abstraction.
Particular behavioral properties are required.
State machine abstraction is generally not an injective function: there are several
state machines with the same interface abstraction. This shows that there does not
exist an inverse function for
α SM / IF , but we can require a canonical function
SM
such that for all we interface behaviors F
ρ IF / SM : IF
IF g e t
ρ IF / SM (F)) = F reversibility of representation
This implies that there is a state machine for each (consistent) interface specification.
α SM / IF (
4.2.4 The Process Model
A run of a system can always be understood as a family of events that are causally
connected. Each such run is called a process . Every event in a process is the instance
of an action. Which actions are considered in a process is again a question of the
chosen view and level of abstraction. We can define interface processes or processes
that reflect internal actions and internal events. A system behavior then is a set of
processes.
By PRC(Action) we denote the set of system descriptions by sets of processes over
the set Action, by PRC the set of all system descriptions by sets of processes.
Like for state machines and interfaces, we define abstractions between processes
by mappings of the form:
PRC
Again not every mapping between processes is called an abstraction. Particular
behavioral properties are required.
In fact the process view is closely related to the state view. Every state machine
defines a set of processes and thus a system behavior in terms of processes. In
essence, the process view is a mild abstraction of the state view.
We assume, in particular, the existence of two specific abstraction functions. We
assume a process abstraction for state machines:
α PRC/ PRC : PRC
PRC
and a canonical interface abstraction mapping:
α SM / PRC : SM
IF
In fact we expect that the composition of these two abstractions yields the state
machine interface abstraction. For every state machine M in SM we assume:
α PRC/ IF : PRC
α SM / IF (M)
In other words a process abstraction of a state machine followed by an interface
abstraction of the process yields the interface abstraction of a state machine.
Any functional composition of abstraction functions should yield an abstraction. In
other words, we require that the set of abstractions is closed under functional
composition.
α PRC/ IF (
α SM / PRC (M)) =
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