# Fluid Net Models: From Behavioral Properties to Structural Objects
## 6 Reduction Rules
This particular structure is a marked P-semiflow of the net, even when it restricts the behavior of the system, its elimination allows to remove several configurations. In this particular case, after the transformation, the reduced net has now 32 configurations in contrast with the 64 configurations of the original one. Now, paying attention to transition \(t_{1}\), one can notice that its input places are \(p_{1}\) and \(p_{9}\). This transition is the only input of place \(p_{2}\), which in turn, only have the output transition \(t_{2}\). Roughly speaking, the enabling of \(t_{2}\) depends indirectly on the marking of places \(p_{1}\) and \(p_{9}\). A _pre-fusion_ of transitions \(t_{1}\) and \(t_{2}\) can be performed in the net, obtaining a reduced net in which the T-covertures of the problematic configurations are related to the ones of the original net, except that the \"absorved\" place \(p_{2}\) is not present in the T-covertures of the reduced net. This reduction is a particular case of the forth rule depicted in Figure 14, which is explained in detail in Proposition 5 of [16]. As in the previous reduction, a _self-loop_ place \(p_{9}\) is obtained after the pre-fusion of those transitions. Once again, this place is structurally implicit (particularly, an isolated and marked P-semiflow) and can be removed. Therefore, a reduced net in which there are 16 configurations is obtained. Again, a post-fusion between transitions \(t_{5}\) and \(t_{67}\) can be performed obtaining a reduced net in which the number of configurations remains (16 configurations). Figure 13: Reduction process of a net model. (**a**) Original net model, post-fusion of transitions \(t_{6}\) and \(t_{7}\), and pre-fusion of \(t_{1}\) and \(t_{2}\) are shaded. (**b**) First reduced net with self-loop places that can be removed. (**c**) Second reduced net, post-fusion of transitions \(t_{5}\) and \(t_{67}\) is shaded. (**d**) Third reduced net, macrotransition is shaded. (**e**) Final reduced net. Transitions \(t_{4}\) and \(t_{567}\) together with places \(p_{5}\) and \(p_{11}\) form a strongly connected choice-free net, which is also consistent (in particular, it is a simple circuit). Obviously, the P-semiflows inside this subnet should be initially marked, otherwise the system would be blocked. The reduction of this local structure is identified as a _macrotransition_ according to Definition 14 in [16]. It is shown as the second rule in Figure 14. This reduction can be performed in a net maintaining its problematic configurations, because no choice places are removed in it. The dual concept of the macrotransition is the _macroplace_ (see the first rule in Figure 14). In this case, the places and transitions in the identified subnet form a strongly connected join-free net, which is also conservative. The configurations of the reduced net are mostly the same as the original, because no join transitions are removed during the procedure. A detailed discussion about this rule can be found in [16]. Finally, for the reduced net in Figure 13e with eight configurations, Algorithm 3 can be applied to decide if these configurations are problematic. This reduced net has one B-Problematic configuration, whose associated T-coverture is \(\mathcal{T}_{\mathcal{C}_{8,108}}=\left\{p_{8},p_{10}\right\}\), and one nB-Problematic configuration, whose T-coverture is \(\mathcal{T}_{\mathcal{C}_{1,108}}=\left\{p_{1},p_{8},p_{10}\right\}\). Now, if we want to obtain the problematic configurations of the original net from the reduced one, this can be done going backwards in the transformation path carried out. * From the net in Figure 13e, configuration \(\mathcal{C}_{8,10,8}\) is B-Problematic. Its T-coverure is \(\mathcal{T}_{\mathcal{C}_{8,108}}=\left\{p_{8},p_{10}\right\}\), where we can see that places \(p_{1}\), \(p_{3}\), and \(p_{4}\) are not part of the problematic configuration in the reduced net, and thus, they are not in the original one.