From “Formal Methods” to System Modeling - Formal Methods and Hybrid Real-Time Systems - page 35

Taking into account the idea of a component identifier, we can also establish the
is_part relation between component identifiers that formalizes the component
hierarchy. Note that a hierarchy of composed system defines a tree (an acyclic
directed graph where each path is unique). It is called the component tree or the
component hierarchy .

4.2.9 Abstractions of Composed Systems
Easily we define abstractions for composed systems:

α CS / IF : CS IF → IF

α CS / PRC : CS PRC → PRC

SM
The relationship between composition and composed system and their abstractions
is defined as follows. Let

α CS / SM : CS SM →

be a composed system with components modeled by
interfaces and with component identifiers

ν

IK = { k 1 , ..., k n }
We assume

(k n ) architectural interface abstraction
This definition essentially assumes that composition is associative and
commutative. Composition is the key to hierarchical models. The inverse to
composition is decomposition. A decomposition of a system described by an interface
model into a family of components yields a composed system the abstraction of which
yields the original interface model.
Thus finally we get an interface abstraction for all kinds of systems represented by
the mapping

α CS / IF ( ν ) = ν (k 1 ) ⊗ IF ... ⊗ IF ν

IF
The idea of an interface abstraction is the basis for the composition and integration
of all kinds of systems represented by state machines, processes, or composed
systems.

α IF : HCS →

4.2.10 Time
Time is a very essential aspect of a system and its behavior. Many systems interact in
a time frame. This means that the time and the duration of the execution of actions are
essential for the behavior of systems. We denote the set of all timed composed
hierarchical systems by TCS and the set of nontimed composed systems by NCS. Again
we assume an abstraction function

NCS
Time is a concept that is orthogonal to all the concepts introduced so far. We may
introduce a notion of time in all the model aspects introduced above. It is in particular
interesting to relate system models with explicit time to system models without
explicit time.
Forgetting about timing aspects is again an abstraction. We may have timed system
models as well as nontimed system models (see [4]).

α T : TCS →