3 3.3. Exploiting modules and symmetries in Fault Tree inference 65 Failure dataset MD* D MCS Compute Minimal Cut Sets 1 Identify indep. modules 2 Infer FT Yes Yes No No Split MCSs using symmetries FT* Identify symmetries 3 4 5 6 Copy symmetricFT FTD Figure 3.2: SymLearn tool chain overview. Blue boxes indicate novel steps. modules). For example, ϑ1({A, C}) = {F, H} ↓ CF. Symmetries within the modules are given by ϑ2 =(AE)(CD) ↓ZCF and ϑ3 =(FJ)(HI) ↓ZCF . 3.3 Exploiting modules and symmetries in Fault Tree inference Our SymLearn approach is outlined in Figure 3.2 and consists of 6 steps: 1. Computes the set of all MCSs CDassociated with input dataset D. 2. Finds a partitioning M1, . . . , Mn of CD s.t. the corresponding BEs form independent modules M1, . . . , Mn. In the worst case, no proper partitioning is possible and the independent module consists of all BEs. 3. Identifies the symmetries ZCD on CD. If symmetries exist between independent modules, then only one of these modules needs to be considered in the following. Otherwise, SymLearn directly goes to Step 5. 4. Tries to further split the MCSs Mi of each module Mi via a symmetry ϑ ↓ ZCD. The split into M1 i and M2 i should satisfy ϑ(M1 i ) = M2 i and preferably have a small number of shared BE. If a split is found, SymLearn recursively starts again with Step 2 for M1 i ; otherwise it proceeds with Step 5. 5. Infers an FT FM for each partition M of the MCSs. Several approaches can be used, e.g., FT-MOEA (Jimenez-Roa, Heskes, Tinga, et al., 2023) or simplification of Boolean formulas (Lazarova-Molnar, Niloofar, and Barta, 2020). 6. Creates for each set of symmetric MCSs M2 i a corresponding symmetric FT FM2 i by copying the “original” FTFM1 i and renaming the BEs according to the symmetry ϑ. Last, all inferred FTs are joined under an OR-gate. We provide details on all steps of SymLearn in the following. Step 1: Compute Minimal Cut Sets. SymLearn starts by extracting all the MCSs CDfrom the data D. We use the algorithm from Lazarova-Molnar, Niloofar, and Barta, 2020, but employ an improved computation of the MCSs from the cut sets. Here, we iteratively select a cut set C with minimal cardinality and remove all cut sets that include C. The runtime complexity of the algorithm is quadratic in D, i.e., O(D2)=O(22·|BEs|).
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