again, then the marking of that place is defined by the second point of the marking function. On the other hand, if the place of ϕ(N) that we examine is a control place of a step place that belongs to a different building block from that of the step place that is ‘controlled’ by it, then the marking of the control place is defined according to the formula that is given in the third part of the function. It should be noted that the initial marking of the root control place and all the other control places of the ‘starting’ place of the net is defined using the special colours given by the formula of the third condition of the initial marking relation. The following paragraphs describe different examples of both the second and third case of the construction of the initial marking. If the place that is examined is the control place of the root net of the ϕ(N) net, then the marking of this place takes the values of all the complementary colours of the tokens that reside in the step place ‘controlled’ by it, when the respective control place of the root net of N holds no tokens at all. Otherwise, if the control place of the root net of N hosts at least one token, then the control place of the root net of the ϕ(N) net takes the values of all the complementary colours of the tokens that reside in the respective step place, excluding that or those complementary value(s) that correspond(s) to the colour(s) of the token(s) that reside(s) in the control place of the root net of N. If the places that are examined are the control places of the ‘starting’ place of the ϕ(N) net, then the marking of these places takes the values of all the special colours of the tokens that reside in the step place ‘controlled’ by each of them, when the respective control place of the root net ofN holds no tokens at all. On the other hand, when the control place of the root net of N holds at least one token, then the control places of the ‘starting’ place of the ϕ(N) net take the values of all the complementary colours of the tokens that reside in the ‘starting’ place except for that or those complementary value(s) that correspond(s) to the colour(s) of the token(s) that reside(s) in the control place of the root net of N.
(xii) All the gluing places ofN are considered as gluing places of the constructed ϕ(N) net. More specifically, the places of the ϕ(N) net that correspond to the gluing places of N, are the gluing places of the constructed ϕ(N).
6.3
A Construction Example
After the definition of the construction, we demonstrate the application of the above definition through an example of the construction of a T-APN net ϕ(N) out of a given APN net N just by following the relations between the elements of
6.3 A Construction Example
the two nets that are given by that definition. The given APN netN is presented in Figure 6.1.
Figure 6.1: The APN net N.
Knowing the structure of the given APN net N in Figure 6.1 and following the construction we build the respective T-APN net ϕ(N), which is associated to N. It should be noted that in the following example, the labelling of the elements of both nets is exactly the same, which means that the label of the places, the transitions, etc. are the same for N and ϕ(N). This does not imply that two labels with the same value refer to the same element but to respective elements of the two nets. For example, place p1 of ϕ(N) is the respective place of p1 of N.
To proceed with the construction of ϕ(N), first we have to identify every place and every transition and to sort them into the proper subset of places and transitions respectively. As is known from the theory of the APNs (see section
6.3 A Construction Example
4.2.1), every composite step net has a specific structure that consists of database, response, control and step places and of emptying, retrieve and step transitions. Starting with the places ofN, we could easily define the set P of all the places of the net, where:
P = {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18}
But, it is also known that for every net of composite step nets, P consists of subsets of special places:
P = PR∪ PD∪ PC∪ PSF
In this case the response, database, control and step places of N are as follows: PR= {p5, p10, p11, p17, p18},
PD = {p4, p8, p9, p15, p16},
PC = {p0, p3, p7, p14},
and PSF = {p1, p2, p6}.
At this point we should mention that the control places of the ‘starting’ place are needed for the construction of the initial marking of the ϕ(N) net and are as follows:
p0
c = {p0}
which means that the ‘starting’ place p1 has only one control place in this par-
ticular example, the place p0. Finally, the set of inhibitor places PI in this case
is the same as the set of control places PC. So, PI = {p0, p3, p7, p14}.
Having defined all the sets of places, we continue with the construction of the places of the ϕ(N) net according to definition 21(i), where it is described that for every place of the APN net N there exists a respective place in the T-APN net ϕ(N). This means that for every database, response, control and step place of N there exists the respective database, response, control and step place in ϕ(N), which implies that firstly the number of places in both nets is the same and secondly all the following:
PR0 = PR = {p5, p10, p11, p17, p18},
PD0 = PD = {p4, p8, p9, p15, p16},
PC0 = PC = {p0, p3, p7, p14} and
PSF0 = PSF = {p1, p2, p6}.
After the description of the construction for the places of ϕ(N), we continue with the construction of the transitions of that net out of the transitions of N. As is known, N comprises retrieve, emptying and step transitions:
T = TR∪ TEM ∪ TST EP
6.3 A Construction Example T = {t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12, t13} with TR= {t3, t6, t8, t11, t13}, TEM = {t2, t5, t7, t10, t12} and TST EP = {t1, t4, t9}.
Having defined the transitions of the APN net N and following once again the construction and particularly the second point that refers to the construction of the transitions of ϕ(N), it follows that for every transition of N there exists a respective transition in ϕ(N). This means that for every retrieve, emptying and step transition of N there exists the respective retrieve, emptying and step transition in ϕ(N), which implies that the number of transitions in both nets is the same and that:
TR0 = TR = {t3, t6, t8, t11, t13},
TEM0 = TEM = {t2, t5, t7, t10, t12} and
TST EP0 = TST EP = {t1, t4, t9}.
Thus, from definition21(ii), (x), it follows that TST EP0 = TGand TR0∪T 0
EM = TN G.
The sets of TG and TN G transitions will be used later for the guards.
Next, we identify all the pre and post arcs ofN starting with the pre mapping first. The set of pre arcs, denoted by PreALL is divided into direct pre arcs and
inhibitor arcs, which are denoted by PreDIR and I respectively and are defined
as follows:
PreALL= PreDIR∪ I
where:
PreDIR = {(p1, t1), (p3, t2), (p5, t3), (p4, t2), (p2, t4), (p7, t5), (p7, t7), (p8, t5),
(p9, t7), (p10, t6), (p11, t8), (p14, t10), (p6, t9), (p14, t12), (p15, t10), (p16, t12), (p17, t11),
(p18, t13)}
and I = {(p0, t1), (p3, t4), (p7, t9), (p14, t4)}.
According to point three that defines the construction of the pre arcs of the ϕ(N) net with respect to the pre arcs of N, every arc that belongs to the inhibitor arcs of N is replaced by the respective direct pre arc and every arc that belongs to the direct pre arcs of N corresponds to a direct pre arc in ϕ(N). This implies that the total number of pre arcs of ϕ(N) is the same as the number of pre arcs of N since that replacement retains the number of the pre arcs the exactly the same.
Hence,
6.3 A Construction Example
with
Pre0DIR = PreDIR = {(p1, t1), (p3, t2), (p5, t3), (p4, t2), (p2, t4), (p7, t5), (p7, t7),
(p8, t5), (p9, t7), (p10, t6), (p11, t8), (p14, t10), (p6, t9), (p14, t12), (p15, t10), (p16, t12),
(p17, t11), (p18, t13)} and PreREP = {(p0, t1), (p3, t4), (p7, t9), (p14, t4)}, with PreREP
being the set of direct arcs that replaces the respective set of inhibitor arcs. Continuing with the post mapping of N we have the following set of all the post arcs of N, denoted by PostALL:
PostALL = {(t1, p2), (t1, p3), (t2, p5), (t3, p4), (t4, p6), (t4, p7), (t5, p10), (t6, p8),
(t7, p11), (t8, p9), (t4, p14), (t9, p2), (t10, p17), (t11, p15), (t12, p18), (t13, p16)}.
Now we provide the respective post mapping of ϕ(N) using point three again that also defines the construction of the post arcs of ϕ(N) with respect to the post arcs of the APN net N. According to the construction of the post mapping for every arc that belongs to the set of post arcs of N there exists a respective post arc in ϕ(N). Furthermore, new direct post arcs are added to the ϕ(N) net between the emptying transitions and every inhibitor place and between step transitions and the control places of step places that are revisited. In this exam- ple, the new post arcs between the emptying transition and the inhibitor places are {(t2, p3), (t5, p7), (t7, p7), (t10, p14), (t12, p14)} as these arcs comply with the re-
quirements of adding a new arc since all these transitions are emptying transition, the ‘destination’ places of these arcs are all inhibitor places and finally there exists a pre arc for every pair of the above post arcs. Finally, the remaining post arcs that are added to ϕ(N) are {(t1, p14), (t9, p3)} which are arcs that connect the
step transitions t1 and t9 with the control places p14 and p3 of the step place p2
respectively, which is a step place that is revisited as it belongs to a bidirectional step.
All the newly added post arcs will be denoted by P ostN EW and are as follows:
PostNEW = {(t2, p3), (t5, p7), (t7, p7), (t10, p14), (t12, p14), (t1, p14), (t9, p3)}.
From all the above, it can be concluded that: Post0ALL = PostALL∪PostNEW =
{(t1, p2), (t1, p3), (t2, p5), (t3, p4), (t4, p6), (t4, p7), (t5, p10), (t6, p8), (t7, p11), (t8, p9),
(t4, p14), (t9, p2), (t10, p17), (t11, p15), (t12, p18), (t13, p16)} ∪ {(t2, p3), (t5, p7), (t7, p7),
(t10, p14), (t12, p14), (t1, p14), (t9, p3)} = {(t1, p2), (t1, p3), (t2, p5), (t3, p4), (t4, p6),
(t4, p7), (t5, p10), (t6, p8), (t7, p11), (t8, p9), (t4, p14), (t9, p2), (t10, p17), (t11, p15),
(t12, p18), (t13, p16), (t2, p3), (t5, p7), (t7, p7), (t10, p14), (t12, p14), (t1, p14), (t9, p3)}.
Having completed the construction of places, transitions and arcs, we could say that we have developed the structure of ϕ(N), which will be presented later as a graph. Moving from the structural parts of the nets to the functional ones, we will deal with the colour functions and the structuring sets that are used in both nets and we will show how they are related through our example.
Firstly, we start with the structuring set Cl. In our example, Cl for N is the structuring set of both places and transition and is as follows:
6.3 A Construction Example
Cl = {b, w}
As is known, this means that the colours used in N are black and white. In definition 21(iv), Cl0 is equal to Cl signifying that the same structuring set is also used in ϕ(N).
For the construction of ϕ(N) we need the complementary set of Cl, the Cl which is a set that consists of the complementary values of the Cl given by the bijection f . We assume that in our example, Cl = {g, l}, which means that the complementary colours that will be used in ϕ(N) will be the grey (g) and the light grey (l), with f (b) = g and f (w) = l.
In N, the colour function C structures all the places of the net using the structuring set Cl allowing the places to host either black or white tokens or both. For the structuring of the places of ϕ(N) we use the colour function CP
which is related to the colour function C of N as stated in definition 21(vi). Checking each place of the T-APN net ϕ(N), we can define how it is structured with respect to function C depending on where this place belongs to. For example all the response, database and step places of ϕ(N) are structured using CP which
is set equal to C. On the other hand, all the control places of ϕ(N) are structured by taking values from both C and the complementary set of colours.
Thus in ϕ(N), the places PC0 = {p0, p3, p7, p14} can host tokens of black,
white, grey and light grey colour and the places PR0 = {p5, p10, p11, p17, p18},
PD0 = {p4, p8, p9, p15, p16} and PSF0 = {p1, p2, p6} host tokens of black and white
colours only.
Applying the construction to the colour function CT for the structuring of
the transitions of ϕ(N), we notice that all the transitions of ϕ(N) are structured in the same way as the transitions of N, which means that all the transition of the constructed net work under black and white modes, as they take their values from function C, since CT = C.
The last colour function that is needed for the functioning of the T-APN net ϕ(N) is the Ins function that structures the arcs of ϕ(N). To find the inscription, we first check where each arc of ϕ(N) belongs to and according to that we decide the inscription of the arc. For example, all the arcs of ϕ(N) that correspond to the arcs of N that belong to the pre and post arcs apart from those arcs that belong to the inhibitor arcs take their inscription from the Cl structuring set. In our case, all these arcs carry black and white tokens and their inscription will be written as {b, w}. Contrary to the arcs above, all the newly added arcs and the arcs that replace the inhibitor arcs of N carry tokens of the complementary colours of Cl. So, these arcs carry grey and light grey tokens and their inscription will be written as {g, l}.
Regarding the capacities of the places of the constructed net, we define the capacity of each place of ϕ(N) in accordance with the capacity of the respective
6.3 A Construction Example
place in N only if the colour of token of the place is an identity colour. In that case, the capacity is the same for both places. Otherwise, if the colour of a token is a special colour then the capacity of the place is 1. In our example, we consider that every place of N has capacity 1 per colour of token, which implies that all the places of ϕ(N) have also capacity equal to one for any colour of token, either identity colour or special colour.
The guards of the transitions TGof ϕ(N) evaluate to true for the set of binding
pairs Bt and the guards of the transition TN G are always set to true for that
specific net. Considering that the transitions of N have no guards and that the guards of ϕ(N) are Boolean functions that either depend on the set of binding pairs or not, we can only relate the transitions of the two nets only by showing that the set of binding pairs Bt derives from function C, as is described in definition
21(x). In our example, we know that function C takes values from Cl, which means that the variable x belongs to that set, i.e., x ∈ {b, w}. Furthermore, f (x) is given by bijection f , which means that f (x) takes values from ClSP EC, which
means that f (x) ∈ {g, l}.
The set of binding pairs of ϕ(N) is as follows: Bt= {(b, g), (w, l)}
with f (b) = g and f (w) = l, as has already been mentioned.
All the step transitions are followed by guards that evaluate to true for any of the binding pairs. For all the other transitions of ϕ(N), the guards always evaluate to true.
The most important part of the construction is the translation of the initial marking ofN into the respective initial marking for the T-APN net ϕ(N). Having defined all the places of both nets, we can easily proceed to the construction of the initial marking of ϕ(N). As is described by definition 21(xi) , every place of ϕ(N) retains the colours and the number of tokens that reside in the respective places of N, except some special cases for the control places of the step places including the control places of the ‘starting’ place of the net, which will take values according to the second and third condition of the construction for the initial marking.
Specifically, checking every place of ϕ(N) we concluded that the initial mark- ing of the net is:
M00 = {(p0, g), (p0, l), (p1, b), (p1, w), (p4, b), (p4, w), (p8, b), (p9, w), (p15, b), (p16, w)}
as the places p4, p8, p9, p15 and p16 are database places that are marked in N, so
they should remain marked with the same colours in ϕ(N). Additionally, checking all the response places p5, p10, p11, p17 and p18 of N we noticed that none of them