Architecture Modeling - SPES 2020

Architecture Modeling
1. For σ ∈ Σ, the set Σ is called receptive on P ′ w.r.t. σ, if it is nonempty and for all initial
segments T ⊆ Time,
∀τ ∈ T (P). τ ↓T = σ ↓T ⇒ ∃σ ′ ∈ Σ. σ ′ ↓T = σ ↓T ∧σ ′ ↓ P ′= τ ↓ P ′
2. The receptive kernel of Σ on P ′ is the largest receptive subset of Σ, if there is a receptive
subset, or else the empty set.
Lemma 6.2.2 (Receptiveness Properties) Let Σ be a set of traces over a set of ports P, and let
P ′ ⊆ P .
1. An arbitrary nonempty union of subsets of Σ which are receptive on P ′ is again receptive.
Thus, the receptive kernel is always well defined.
2. The intersection of two subsets of Σ which are receptive on P ′ need not be receptive.
Proof:
Ad 1: This follows from the existential nature of the defining clause (1. in Definition 6.2.1).
Ad 2: Let us consider traces over one input and one output port, where each may take values
in {0,1}. We take the following two trace sets which are receptive on the input port. In
the first set, the output is always 0, in the second the output always equals the input. Both
trace sets are obviously receptive because the input is not restricted. Their intersection is
the set of traces where both output and input are always 0, which is not receptive on the
input port, because there is no trace where the input is 1. 1
Receptiveness over a port set P ′ essentially says that the values of the ports in P ′ are not restricted
– at any point in time there is a future behaviour for every possible future valuation of
P ′ . This captures the intuition about direction of ports. The reader may note that also pathological
cases like a component which guesses future outputs are excluded by requiring receptiveness
on the inports. The potential non-receptiveness (Item 2 in the lemma) should be taken seriously.
It is difficult to avoid inconsistencies in specifications, there are far more intricate cases than the
simple counterexample used in the proof.
As well as receptiveness, determinism can be expressed on the semantic level.
Definition 6.2.3 (Deterministic Trace Set) Let Σ be a set of traces over a set of ports P, and
let P ′ ⊆ P . Then Σ depends deterministically on P ′ if
6.2.3 Semantical Definitions for HRC
∀σ,σ ′ ∈ Σ. σ ↓ P ′= σ ′ ↓ P ′ ⇒ σ = σ ′ .
For a component M, we denote its semantics by [[M]]. It is a set of traces over the interface of the
component. We have already restricted ourselves to ports with just one flow. Service ports are
not considered here either, they are assumed to be implemented by more primitive constructs
(e. g. a protocol over directed ports). Furthermore, ports may only have direction in or out
and not bidirectional. With these simplifications, we sketch the semantical domain for
interpreting HRC and HRC state machines.
1 An even simpler, yet maybe less instructive, example takes just two receptive sets with constant, different output,
so that the intersection is empty.
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