Abstract
The paper presents a novel approach for discovering Petri nets (PN) that include silent transitions from logs of event sequences. We propose a repairing method that extends existing discovery techniques that do not deal with silent transitions; such techniques may yield substructures that involve deadlocks. Such substructures, called inconsistent (IS), are detected through a structural pattern. IS are rewritten by adding new transitions labelled with event symbols already assigned to transitions in IS; the rewritten model has no deadlocks. Afterwards, the PN with duplicated event labels is transformed into an equivalent model with silent transitions. The algorithms derived from the technique, which have polynomial-time complexity, have been implemented and tested on examples of diverse structures.
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02 March 2022
The original version of this paper was updated to fix the numbering of the section headings.
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Román Pomares-Angelino is supported by CONACYT, México. PhD Grant 901440
Appendix
Appendix
This appendix contains additional tests of the proposed method. The tests were performed using the scheme described in the paper using artificial logs. Besides, we include the discovered models by the algorithm Alpha# for comparison.
1.1 Example A.1
The purpose of this example is to show the outcomes of all the steps of the repairing technique. Consider the net in Fig. 20, which has silent transitions (in black); the artificial log generated by PIPE from this net is the following:
Proposed PN with silent transitions in PIPE
From such a log, the following data has been computed from CoMiner algorithm where the discovered dependencies are: [t3,t23 + t24 + t27][t23,t31 + t25][t1,t15][t1,t12][t13,t14][t14,t37][t15,t16 + t17] [t37,t13 + t18][t17,t18][t24,t29 + t35][t29,t30][t30,t36 + t33][t33,t34][t0,t4 + t8][t4,t6 + t10][t6,t7 + t11][t7,t5][t3 + t25,t27][t23 + t27,t31][t31 + t24,t35][t35 + t30 + t34,t36][t2 + t8,t9][t9 + t4,t10][t10 + t6,t11][t11 + t5,t39][t12 + t37,t13][t15 + t16,t17]
Then, the PN in Fig. 21 built with these dependencies; this PN has deadlocks.
PN with IS
After the creation of the net, the dependencies that form IS are given below.
Based on these IS the PN is repaired using duplicated labels. It is shown in Fig. 22, where tr’ indicates that tr is duplicated.
PN with duplicated transitions
Now, the dependencies of the Petri net transformed with duplicated labels are:
The last step, where the PN with duplicated labels is transformed into a PN with silent transitions, yields a set of repaired dependencies given below. Then the PN is shown in Fig. 23, where epk designates silent transitions.
PN with silent transitions discovered from λA1
Now, in Fig. 24 we include the WFN obtained using the algorithm Alpha#. This model is the same that obtained by our method. In both cases the PN used for generation of the artificial log is rediscovered.
WFN discovered using the algorithm Alpha# from λA1
1.2 Example A.2
The processed log is λA2 = {xabcdefgy, xabcefgy, xabcdefbcdefgy, xABCDy, xabCDy, xABCfgy, xAy}. The model discovered including silent transitions is shown in Fig. 25. It has nine silent transitions; five Skip transitions, two Redo transitions, and one Switch transition.
The PN used for generating the log is rediscovered. The processing time of the repairing algorithm in this test is 2.7866 ms.
The WFN found by the algorithm Alpha# is shown in Fig. 26. Notice that the model is not the same; it has twelve silent transitions and transition p has no input place whilst transition n has no output place.
PN discovered from λA2
PN discovered from λA2 using the algorithm Alpha#
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Pomares-Angelino, R., López-Mellado, E. Discovering petri nets including silent transitions. A repairing approach based on structural patterns. Discrete Event Dyn Syst 32, 291–315 (2022). https://doi.org/10.1007/s10626-021-00358-w
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DOI: https://doi.org/10.1007/s10626-021-00358-w