## 1 Introduction and Problem Formulation  
Petri nets, a versatile and powerful formal modeling tool, have been instrumental in advancing the design process of control and complex systems [1]. Originally introduced by [PERSON] in 1962, Petri nets provide an intuitive graphical representation for describing and analyzing the dynamics of concurrent systems [2]. Over the years, Petri nets have found widespread applications in various domains, including computer science [3], manufacturing [4; 5; 6], biology [7], and cyber-physical systems [8]. Recently, the Petri net-based modeling approach has become especially popular in the modeling of real-life industrial systems (e.g., they are applied by manufacturing companies) [9; 10; 11]. The main advantage of Petri nets lies in their simplicity and graphical formal representation, which can be constructed relatively very easily. In addition, a Petri net can also be considered as a mathematical modeling framework that reflects the behavior of systems through the directed bipartite graph. Such a graph consists of places and transitions that are connected by directed arcs [12]. A place is active (marked) if it contains a token. Placesthat are marked at the same time represent states (markings). Furthermore, transitions reflect actions in a system. In particular, a transition may fire, by removing tokens from all its input places and relocating them to all its output places. Consequently, such an action changes the state in the system [13]. Furthermore, tokens may be used to model resources of the system elements [14] or indicate a logical behavior of the system [13; 15]. Finally, Petri nets are heavily supported by formal analysis techniques [1; 16; 17; 18]. There exist methods that allow for verification (sometimes even validation) of the system at the early specification stage. Such a combination of graphical and mathematical approaches makes Petri nets an intuitive and comfortable tool for the modeling, analysis, and verification of systems that strongly utilize concurrency [19; 20]. Moving on to the concurrency aspects, it should be emphasized that Petri nets reflect such a relation by their nature [21]. In particular, one or more places can be marked at the same time. It means that operations associated with those places are executed concurrently. This makes Petri nets applicable in many areas of human life, especially including industry applications [22]. For example, they can be found in modern industries, such as resilient manufacturing, where 3D printing techniques are applied [23]. It should also be noted that Petri nets provide a robust means of modeling the intricate interactions and dependencies in such systems, allowing for a detailed examination of their behavior [24]. On the other hand, the modeled concurrent system ought to be carefully examined against possible errors and designer mistakes. Systems that involve concurrency are exposed to serious risks, including deadlocks or an infinite number of states (if wrongly designed). There are two fundamental properties of Petri nets that are related to the above mentioned problems: liveness and boundedness [25]. The first one, liveness, guarantees that a system can reach any desirable state under certain conditions (formally after a final firing sequence), ensuring that the system remains responsive and operational at all times [26]. Furthermore, boundedness implies that the number of tokens in a Petri net does not exceed predefined limits in each place, preventing uncontrolled behavior or resource depletion [27]. An unbounded Petri net with those specification assumptions could lead to uncontrolled behavior [28]. Such a condition can be defined more strictly. There exist Petri net systems that are required to be 1-bounded (safe) [29]. This means that in each place, in any state, there can be at most only one token. This restriction is especially useful in the case of binary control systems, where safeness assures the accurate logical behavior of the system [30; 31]. Let us move on to the interpreted Petri nets.