## 1 Introduction
Moreover, such exponential growth increases the memory requirements, which, together with the extremely long computation time, renders the approach impractical for complex large-scale systems. It should be also noted that due to the size, reachability graphs may become overly complex and difficult to interpret, especially when dealing with systems with high concurrency and interaction. Finally, this method does not inherently support the analysis of certain qualitative properties, such as fairness or performance metrics, without extensive modifications or additional tools. These limitations necessitate the development of more scalable and efficient techniques, such as initial verification, to enhance the practical applicability of Petri net analysis for concurrent systems. Moving on to the linear algebra techniques, invariants are usually applied to examine properties such as boundedness and conservativeness [29]. Moreover, it is possible to perform the examination of deadlock-freeness and even liveness [30]. In general, there are two invariant types in the system specified by a Petri net. The first one, place invariants (p-invariants), formalizes invariant properties of places. The second type, transition invariants (t-invariants), corresponds to cyclic behaviour and properties regarding possible firing sequences of a Petri net [31]. Analysis of a system using both p-invariants and t-invariants has important restrictions [21; 31; 32]. The first limitation is similar to the drawbacks of reachability graph construction, since the number of invariants may grow exponentially [5; 21; 33]. Furthermore, invariants permit analysis of the structure of the system, and thus the obtained result may not always be accurate, and additional investigations may be required [30]. Let us describe the most popular techniques of invariant computation by exposing their benefits and weak points. There are several invariant computation techniques for a Petri net. Common methods usually compute a superset of reachable states while using forward propagation with widening until a fixed point is reached. One technique is based on the Farkas algorithm to solve a set of linear equations. The main logic of this method is based on creating and rewriting graphs with consideration of neighbours. It creates and deletes places while preserving a set of invariants [34]. While the original method works sequentially, there are other algorithms that implement parallel computation by sharing columns and rows of computed matrices between different processes [34]. Note, in the algorithms that as an input use the incidence matrix of a given Petri net, we can transpose the matrix according to our needs, to computer transition invariants instead of place invariants. Let us present the well-known algorithm for computing t-invariants in Petri nets, initially shown in [35]. The results are generated by a solution of a homogeneous linear equation system over nonnegative integers. This task involves significant mathematical challenges due to the lack of a closed formula for solutions. A solution requires techniques--specifically, a class of algorithms--that systematically generate the solution space by considering all possible candidates. Such a process involves the consequent transformation of the incidence matrix of the Petri net into a zero matrix through appropriate matrix operations. Let us present the algorithm in more detail. The method starts by augmentation of the incidence matrix with an auxiliary matrix to log the performed operations. This auxiliary matrix is initialized as an identity matrix and is appended to the right of the incidence matrix for computing place invariants or below it for t-invariants, ensuring that the matrix operations apply to the composite structure. The algorithm concludes once all columns of the incidence matrix are zeroed, requiring at the most as many iterations as there are transitions since each iteration zeroes one column. Let us underline that implementation of the algorithm presents two major challenges.