Architecture Modeling - SPES 2020

6.2 Semantics of HRC
6.2.1 Semantical Domain
Architecture Modeling
We assume a notion of time, given by a domain Time with a total order < on its instances.
Whether time is discrete (i.e., N) or continuous (R + , the non-negative real numbers) does not
matter for the general definitions in the following, whereas our more specific semantics of HRC
introduced lateron assumes a continuous domain. An initial segment of the time domain is a
downward-closed subset, i. e. a set T ⊆ Time satisfying t ∈ T ⇒ {t ′ ∈ Time|t ′ < t} ⊆ T .
There is a set of ports P, where each port has a type with an associated domain of values.
The universe of all values is V . We will assume all things being type correct and will not be
concerned with details of types and values. Also, we simplify the interface concept of HRC in
our presentation. We assume that each port contains just one flow, so that we can equate ports
and flows, and just speak of ports.
Given a set of ports P ⊆ P, a trace assigns, for any point in time, to each port a value of the
port’s type. Thus, the set T (P) of all traces T over P is a subset of
[Time → [P → V ]] .
For discrete time domains, usually every function in the set above will be a trace, whereas
with continous time and value domains, traces may be restricted e. g. to functions which are
continuous almost everywhere (continuous but for a null set).
In the following, whenever convenient we will freely apply decurrying, changing the argument
order and currying when referring to traces. Thus, a trace may also be viewed as an
element of [P → [Time → V ]].
We denote by σ ↓P (resp., Σ ↓P) the restriction of a trace (resp., trace set) to a subset of its
ports. Similarly, by σ ↓T we will denote its restriction to a subset T ⊆ Time.
A Note on the Choice of the Semantical Domain Basing the semantical definitions on sets
of (timed) traces is already some restriction. There are approaches to modeling which do not
fit into this framework. It would be, however, very difficult to incorporate all these approaches
in a general description, and it would result in a rather complex development. We will develop
our theory for timed traces and leave it to the reader (or future versions of this document, if necessary)
to perform the generalizations and modifications as appropriate for her/his semantical
domain.
One of the limitations of our approach concerns the synchronous nature of composition.
This is not always adequate, for instance it excludes the elegant abstraction of [34] and related
settings. It is, however, often possible to model asynchrony by synchrony, see e. g. [30].
6.2.2 Properties of Trace Sets
Traces do not distinguish between input and output, both are recorded in an equal manner.
Operationally, the diference is important: A component should not be able to restrict the values
on an inport, nor should its environment be able to influence the component’s outports. This
is reflected in the trace sets we get as semantic interpretations of components in HRC, and the
following definition formalizes the observation in a property of trace sets.
Definition 6.2.1 (Receptiveness) Let Σ be a set of traces over a set of ports P, and let P ′ ⊆ P .
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