colours of tokens, i.e., a black token in places p1 and p2 respectively and a grey
token in place p5 . Firing the enabled step {(t, b)} removing the black tokens from
the places p1 and p2 and the grey token from the place p5 (see (†)) and adding
one black token to the places p3 and p4 (see (††)) respectively, it can be noticed
that the capacity of the post-places p3 and p4 is not violated as these places were
previously empty. Additionally, the guard of transition t evaluates to true as the colours of tokens that reside into the pre-places of that transition belong to the set of binding pairs. Contrary to the enabled step {(t, b)}, the steps {(t, w)} and {(t, b), (t, w)} are not enabled at M0 and cannot be fired as the complementary
coloured token of w (i.e., l) is missing from p5. Finally, the structured transition
(t, b) cannot fire twice, since the step {(t, b), (t, b)} is not enabled due to the lack of sufficient number of tokens in the places p1, p2 and p5.
5.3
Step Transition System of T-APNs
The full execution semantics of a T-APN net will be captured using a transition system where arcs are labelled by executed steps, the step transition system. The step transition system of the T-APNs is defined in exactly the same way as the step transition system of the APNs is described in definition 5. Furthermore, from that definition, it follows that the concurrent reachability graph of a T-APN net is a kind of a step transition system.
Hence, to describe the behaviour of the T-APNs through the step transi- tion systems, we take as an example the net of Figure 5.1(a) and its concurrent reachability graph (shown in Figure 5.1(b)). We start capturing the behaviour of the net by examining the enabledness of transition t, as it is the only tran- sition in this net. According to the enabling rules of the T-APNs, transition t is enabled for the black token at the initial marking M0 (see section 5.2). This
means that the transition can fire at least once under the black mode. After the firing of enabled step {(t, b)}, a new state results and is described by the marking M1 = {(p1, w), (p2, w), (p3, b), (p4, b)}. Now, checking the enabledness of t at M1,
it turns that the transition cannot be executed further since it is not enabled. So, the non-existence of enabled transition leads to a dead state for the net, which is the terminal state defined by M1. Thus, to draw the concurrent reachability
graph of N, we start from the initial marking M0 and we generate an arc that
links to a new marking (state) every time that a step is executed. This arc is labelled with the step that is executed each time. In this case, the concurrent reachability graph of N is shown in Figure 5.1(b) and consists of two states, M0
and M1, which are linked with the arc that is named after the step {(t, b)}, as
was described earlier.
5.4 Concluding Remarks
net and executing every time its enabled transitions, concurrently or not, we can construct the concurrent reachability graph of that net starting from the initial state that is given by the marking M0.
5.4
Concluding Remarks
In this chapter, we defined the class of the T-APNs by describing both the static and the dynamic aspects of it. The static aspect of the T-APNs is related to those element that determine the structure of these nets, such as the places, the transition, the arcs, the capacity, etc. On the contrary, the dynamic aspect of the T-APNs concentrates on how the marked T-APN nets behave with respect to the semantics mentioned above.
The behaviour of T-APNs can be captured by the step transition systems and in particular by the concurrent reachability graphs of the nets, as with the APNs. It should be mentioned that all the above notions are fundamental principles of the class of the T-APNs and are needed for the correct and proper functioning of the nets.
Finally, an important inference that derives from this chapter is that the theory behind the T-APN class is based on a strong mathematical background and is well written.
Chapter 6
Relating APNs with T-APNs
Chapters 3 and 5 discuss the classes of Ambient Petri Nets and Transformed Ambient Petri Nets respectively, which provide different kind of coloured Petri nets. As has already been mentioned in those chapters, the nets of these classes are used for different purposes.
In this chapter, we describe a construction of a T-APN net associated with a given APN net showing how the nets of these two classes are related. Con- sequently, demonstrating that these nets are related and more specifically that they behave in exactly the same way, it can be concluded that the verification results of the T-APN nets could be applied to the structural and the behavioural analysis of the APN nets in a reasonable and meaningful way. Hence, an APN net that cannot be verified directly by the verification tools, could be ‘verified’ through an equivalent T-APN net.
The discussion about the construction is conducted into two stages. Firstly, the two classes of nets are compared with regard to their structure, their behaviour and generally their formalisms. Finally, we provide the formal definition of the construction which is based on the comparison that is carried out in the first stage.
Thereafter, we illustrate the working of the construction by giving an example of a constructed T-APN net out of a given associated APN net. In addition, we provide another example that examines the behaviour of two nets intuitively, i.e., a constructed T-APN nets and its associated APN net. Finally, after that example, we demonstrate the translation of the APN models of the case studies into the respective T-APN models through the application of the construction.
In the last section of this chapter, we prove the behavioural equivalence of the nets of these two classes by showing that their concurrent reachability graphs are isomorphic.