Lecture Notes in Computer Science 3472

178 Machiel van der Bijl and Fabien Peureux
The main difference between an IOSM and an IOA is that there is no equivalence
relation in the IOSM definition. Furthermore internal actions are abstracted
into one action τ. Another difference is that the sets of states, input
labels and output labels are restricted to be finite. Phalippou also uses the notion
of locally (output and internal) and exteriorly (input) controlled actions
and strong input-enabledness.
Example. In Figure 7.1, the transition system in the middle is an IOSM. It is
very similar to the IOA on the left. The only difference is that the internal action
‘init’ is replaced by τ.
Input output transition system. This is the model as introduced by Tretmans
[Tre96c].
Definition 7.5. An input-output transition system is a 5-tuple 〈Q, I , U , T , q0〉
where
• Q is a countable, non-empty set of states.
• I is a countable set of input labels.
• U is a countable set of output labels, such that I ∩ U = ∅.
• T ⊆ Q × (I ∪ U ∪{τ}) × Q is the transition relation, whereτ /∈ I ,τ /∈ U .
Furthermore, every state is weakly input-enabled: ∀ q ∈ Q, a ∈ I : q a
=⇒ .
• q0 ∈ Q is the start state.
The input-output transition system (IOTS) is a more general version of the
IOSM. Like the IOSM it does not have the equivalence relation of the IOA and
it also models internal actions with the τ label. However, it does not restrict the
set of states and labels to be finite. Furthermore, there is a clean partitioning of
the label set in inputs and outputs, where the IOSM hides this in the transition
relation. A subtle but important difference with IOA is that an IOTS is weakly
input enabled: ∀a ∈ I , q ∈ Q : q a
=⇒ . We denote the class of input-output
transition systems over I and U by IOTS(I , U ).
Example. In Figure 7.1, the transition system on the right is an IOTS. We see
that the internal action init is replaced by τ. Notice furthermore that the states
q1 and q4 do not have the self-loops with button1 and button2. This is allowed
because an IOTS is weakly enabled. With an internal action we can go from q1
to the input enabled state q2 (note that the same holds for q4 and q5).
7.4 Implementation Relations with Inputs and Outputs
In this section, we will introduce a number of implementation relations. As was
introduced in the conformance testing framework in Section II, an implementation
relation (or conformance relation) is a relation that defines a notion of
correctness between an implementation and a specification. When the implementation
relation holds we say that the specification is implemented by the
implementation or, in other words, that the implementation conforms to the