Definition: Acute τ-transitions
Acute1 τ-transitions are τ-transitions that cannot be removed from the a-role transition chart by gathering and minimisation.
The states linked by acute τ-transitions are states that provide distinct input or save behav-iours, that introduce save ambiguity, or do not enforce input ordering consistently.
Section 6.3.2 about "State gathering" has described those cases. Acute τ-transitions require special attention. They may lead to ambiguous behaviours, either as triggers of equivoque transitions as explained in Section 6.5, or when combined with other transi-tions. This section focuses on the combination of τ-transitions with other transitransi-tions.
Note that acute τ-transitions are a symptom of ambiguity, but do not necessarily mean that a machine presents ambiguity. Gathering requires the successor states triggered by iden-tical input events to be ideniden-tical. Successor states may be distinct without introducing ambiguity, but only output divergence. This is shown in Figure 6.62. States “1” and “2”
cannot be gathered as the triggering by “A” lead to distinct states. However, an external observer is able to determine the further machine behaviour after receiving “B” or “C”.
6.7.1 Mixed ambiguity
In Figure 6.63, the τ-events link states that provide distinct input behaviours. In both cases the τ-event is combined with an output event “A”. As the τ-transitions cannot be per-ceived by an external observer, the combination of these triggering events introduces ambiguity. Case (a) describes a mixed ambiguity: an external observer is not able to
deter-1. According to Merriam-Webster, the term “acute” can be associated with the ideas of sudden onset, urgent attention and uncertain outcome.
Figure 6.62 : Acute τ-transition with no ambiguity.
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mine whether output “A” or input “B” will occur. Case (b) describes a weak mixed ambiguity: output “A” can always occur, but input “B” may not always be consumed. In case (b), the behaviour “A” may lead to new ambiguity depending on the definition of the states “2a” and “2b”. Note that in both cases it is possible to transform the graph by insert-ing a state between state “1” and the sendinsert-ing of signal “A” without modifyinsert-ing the observable association behaviour. This is the reverse operation of gathering. After this transformation, state “1” describes equivoque τ-transitions. Thus, the analysis of this case can be done in a similar way as the analysis of equivoque transitions.
Similarly, a mixed ambiguity can occur when acute τ-events are combined with input events. This is illustrated in Figure 6.64. Here the insertion of a state between state “1”
and “B” modifies the observable association behaviour. Thus the graph cannot be trans-formed so that state “1” describes equivoque τ-transitions.
6.7.2 Input ambiguity
The combination of τ-events with input events may also lead to input ambiguity. This is illustrated in Figure 6.65. Here the states linked by the acute τ-transitions also provide dis-tinct input behaviours. Case (a) describes an input ambiguity: an external observer is not able to determine whether input “A” or input “B” is expected. Case (b) describes a weak input ambiguity: input “A” is always expected, but input “B” is not always expected. In case (b), the behaviour “A” may lead to new ambiguity depending on the definition of the states “2a” and “2b”.
Figure 6.63 : Acute τ-transition and mixed ambiguity (1).
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Figure 6.64 : Acute τ-transition and mixed ambiguity (2).
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A slight difference exists between input ambiguity occurring after equivoque transitions and input ambiguity occurring in relation with an acute τ-transition. While equivoque transitions lead to a particular behaviour condition (i.e. one state is entered that sets a con-dition for signal consumption), the acute τ-transition describes a change of behaviour condition. This change may occur at any time. In the case of equivoque transitions, a com-plementary a-role wonders which signals are expected. In the case of acute τ-transition, it wonders which signals and when. However, this difference does not influence the valida-tion analysis.
6.7.3 Termination ambiguity
As a special form of input or mixed ambiguity, termination ambiguity can also occur when a τ-event is combined with an input or output event. This is illustrated in Figure 6.66. An external observer is not able to determine whether the a-role state machine has terminated, or is waiting for a triggering event to occur.
6.7.4 Termination occurrence ambiguity
Termination occurrence ambiguity is a weak form or ambiguity. As gathering is not applied to exit states, τ-transitions may remain before exit states. In that case, an external
Figure 6.65 : Acute τ-transition and input ambiguity.
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Figure 6.66 : Acute τ-transition and termination ambiguity.
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observer is able to determine that the a-role will terminate, but not when. This is illustrated in Figure 6.67.
Definition: Termination occurrence ambiguity
A termination occurrence ambiguity occurs when an external observer knows that the role state machine will terminate, but is not to determine when. The behaviour of the a-role is said to present a termination occurrence ambiguity.
6.7.5 Save ambiguity
Save ambiguity is a weak form of ambiguity. As an external observer cannot determine whether or not a signal can be saved, it may reserve itself from sending the signal.
Figure 6.68 illustrates this form of ambiguity. An external observer should not send “B”.
6.7.6 Ordering ambiguity
Ordering ambiguity is also a weak form of ambiguity. As an external observer can only determine one of the input ordering, it may restrict to that order. Figure 6.69 illustrates this form of ambiguity. An external observer should restrict to sending “A”.