# Fluid Net Models: From Behavioral Properties to Structural Objects
## 6 Reduction Rules  
According to what has been previously established, when looking for undesired behaviors in an MTS TCPN system, it is important to see if there exist problematic configurations that can be activated. A first step may be to verify if there are problematic configurations in the net. If there are no problematic configurations, then the considered undesired behaviors cannot appear in the system. Otherwise, the initial marking has to be considered to analyze if the problematic configurations can be activated. As already mentioned, the number of configurations of a net may be exponential with the net size. Thus, searching for problematic configurations can be computationally expensive. Even if the Algorithm 3 runs in polynomial time, the number of configurations in a net can make the evaluation of every configuration impractical. In this sense, the CPN system may be polynomially pre-processed before the computation of problematic configurations, so that the total number of configurations to be analyzed is reduced. The purpose of this pre-processing is to reduce the computational problem, but the theoretical problem still remains. Nonetheless, it results very useful for many practical systems that can be found in manufacturing systems, chemical reactions, and healthcare systems, among others. Now, a set of reduction rules (see [16] for a more general and technical presentation) are intuitively explained for very particular cases in order to present an analysis on how to reduce systems maintaining its configuration properties. As mentioned before, our running example of Figure 1b has 64 configurations. It is possible to evaluate all these configurations using Algorithm 3 in order to determine if there exist problematic configurations and their type (B- or nB-Problematic). After 64 steps, the algorithm finds (see Figure 4) that:  
1. Two of these configurations are B-Problematic, with associated T-covertures: \[\mathcal{T}_{C_{1,8;10;11,8,6;7}}=\{p_{1},p_{6},p_{7},p_{8},p_{10},p_{11}\},\] \[\mathcal{T}_{C_{9,8;10;11,8,6;7}}=\{p_{6},p_{7},p_{8},p_{9},p_{10},p_{11}\}\]
2. One of them is nB-Problematic, with associated T-coverture: \[\mathcal{T}_{C_{1,2;10;11,8,6;7}}=\{p_{1},p_{2},p_{6},p_{7},p_{8},p_{10},p_{11}\}\]
3. The other 61 configurations are suitable. Figure 13 shows a reduction process of our running example, which is explained below. The intention of this reductions is to maintain the problematic configurations or theability to recover them from the reduced net. More importantly, the reductions allow to gain insights about the structures that produce problematic configurations, which cause the undesired behaviors of the equilibrium throughput. Let us focus first on transition \(t_{7}\). One can notice that it has one input place (it is said to be _essential_ in the coverture). Hence, every configuration of the net must contain the arc \((p_{7},t_{7})\); thus, every T-coverture of the net contains place \(p_{7}\). From this fact, it is evident that a problematic configuration cannot contain the arc \((p_{12},t_{6})\), because place \(p_{12}\) would be contained in its T-coverture, and \(p_{7}\) and \(p_{12}\) are the support of a P-semiflow. This can be captured using two reduction rules:  
1. A _post-fusion_ of transitions can be performed for transitions \(t_{6}\) and \(t_{7}\). Their structure is a particular case of the third reduction rule in Figure 14. It keeps the same number of configurations in the reduced net, and it is possible to establish that every problematic configuration of the reduced net has its corresponding problematic configuration in the original one. A detailed explanation of this fact is exposed in Proposition 4 in [16]. 2. The reduction of transitions \(t_{6}\) and \(t_{7}\) makes (in the neighboring of this part of the reduced net) that \(p_{12}\) becomes a marked self-loop place. This place can be removed following the idea of the _implicit_ place for continuous Petri nets discussed in [19].