## 3 Experimental Verification  
The experimental verification of the proposed algorithm was performed in terms of efficiency (run-time of the algorithm) and effectiveness (correctness of the obtained results). The results obtained by the proposed method were compared with the results gained by the reference method and by the widely used Petri net analysis tool (both are described in detail later). The evaluation was conducted with a set of benchmarks comprising 386 different classes and complexities of Petri net models, including real-life concurrent systems, cyber-physical systems, and manufacturing systems. To ensure transparency and reproducibility, all benchmarks are publicly available online at [[https://hippo.uz.zgora.pl/](https://hippo.uz.zgora.pl/)]([https://hippo.uz.zgora.pl/](https://hippo.uz.zgora.pl/)) (accessed on 8 January 2025) [51]. The experiments were performed on a machine with the following equipment: processor with Intel(R) Core(TM) i7-9750H CPU with clock, 2.60 GHz per core, 16 GB of system memory, and Windows 64-bit architecture. Moving on to the reference method to which the proposed algorithm was compared, a classical and commonly used algorithm of invariant computation was used. The method denoted as the \"Martinez-Silva\" algorithm [35] is a classical approach widely recognized [46] in Petri net analysis for computing all minimal invariants (that is, not including other invariants). This method is still one of the fundamental solutions for transition computation of invariants in a Petri net-based system [46]. It involves algebraic techniques to derive the t-invariants of a Petri net. Additionally, the proposed algorithm was compared to one of the most popular modelling and analysis tools, Platform Independent Petri Net Editor 2 (PIPE) [54; 55]. This tool is a common platform independent Petri net editor designed for modelling, simulation, and analysis of Petri nets. PIPE provides several built-in functionalities, including invariant computation, state space analysis, as well as simulation, making it a comprehensive tool for Petri net researchers in practice. In order to perform experiments, the proposed technique was compared to the built-in technique of invariant computation implemented within PIPE. Table 1 presents result of experiments achieved for the selected concurrent systems modelled by Petri nets. The particular columns are as follows. The first column, _benchmark_, denotes the name of a Petri net-based system. The subsequent two columns include information about the number of places and transitions of the particular benchmark. The fourth column includes results obtained by the reference method (Martinez-Silva algorithm), where the subcolumn _result_ denotes whether the system is covered by t-invariants, and the subcolumn _time_ presents the run-time (in milliseconds) of the algorithm. Similarly, the fifth column contains the results achieved by the PIPE tool, while the last column includes the results obtained by the method proposed in this paper. To clarify the comparison, \"true\" denotes that no uncovered transitions are found--the Petri net is covered, while \"false\" means that uncovered transitions are found. Moreover, _timeout_ indicates that the particular method or tool is not able to compute the results within one hour owing to its computational complexity (denoted as \"n/a\" in the result). Finally, let us note that although the results of selected (representative) benchmarks are presented, the experiments were conducted for all 386 benchmarks (for all three compared methods, that is, the Martinez-Silva algorithm, PIPE tool, and the proposed algorithm). Let us analyse the results of the experiments. Looking from the point of view of efficiency (computation time), the proposed method successfully computes the results for all 386 tested cases. However, neither the reference Martinez-Silva algorithm nor the PIPE tool could compute the results for all networks within the target time of one hour.