Lecture Notes in Computer Science 3472

7 I/O-automata Based Testing 179
specification. Several implementation relations have been defined for the automata
that were introduced in the previous section. In this section, we start
with implementation relations defined on IOA and continue with implementation
relations on labeled transition systems.
7.4.1 Preorders on IOA
In this section, we treat implementation relations on Input Output Automata.
Some of these implementation relations canalsobeexpressedonlabeledtransition
systems as we will explain in the next section. All of the implementation
relations that we treat in this section are preorders.
We first recapitulate some concepts that are used with IOA. An execution
fragment of an IOA p is an alternating, possibly infinite sequence of states and
actions α = q0a1q1a2q2 ··· such that (qi, ai+1, qi+1) ∈ steps(p). When q0 is a
start state of p we call the execution fragment an execution of p. Anexternal
trace of an IOA p is an execution (fragment) that is restricted to the set of
external actions. We use the notation etraces(p) to denote the external traces
of IOA p, whereetraces ∗ (p) denotes the set of finite external traces of p. We
use the notation a ∈ enabled(q) to denote that state q enables a transition with
action a. This means that there is a state q ′ for which (q, a, q ′ ) ∈ steps(p).
To denote the set of enabled external actions in a state q, we use the notation
wenabled(q). An IOA p is finitely branching iff each state of p enables finitely
many transitions.
We start with the trace inclusion preorder. This is a very weak relation. It
expresses that one system is an implementation of the other if its set of external
traces is a subset of the set of external traces of the specification.
Definition 7.6 (External trace inclusion). For IOA i and s:
i ≤tr s =def etraces(i) ⊆ etraces(s)
The above definition is defined in a so-called intentional way. Many implementation
relations can also be defined in an extensional way in the style of
De Nicola and Hennessy [NH84]. The term extensional refers to an external observer.
The intuition behind this idea is that an implementation conforms to a
specification if no external observer can see the difference. We will not use the
extensional definition in this section, but we refer to Chapter 5 for more information
about extensional definitions of implementation relations and to [Tre96b]
for more information on extensional definitions of implementation relations with
inputs and outputs.
Example. We give an example of the external trace inclusion preorder in Figure
7.2. On the left hand side we see a specification of a coffee machine. It
prescribes that after pressing the button at least twice we expect to observe
either coffee or tea as output. We will reuse this coffee machine specification in
other examples. On the right hand side we see two implementations. The first