64 Chapter 3. Data-Driven Inference of Fault Tree Models Exploiting Symmetry and Modularisation OR OR OR TE C A B E OR Module 1 Partition 1.1 Partition 1.2 Partition 2.1 Partition 2.2 Module 2 OR H F G J ANDAND OR AND OR D I AND AND Figure 3.1: FT with independent modules and further partitioning. Example 1 (Modules). The partitioning for the FT in Figure 3.1 is given by coloured boxes. The BEs {A, B, C, D, E} and {F, G, H, I, K} form independent modules. The corresponding MCSs can be further subdivided. For instance, Partition 1.1 with {{A, C}{B, C}} and Partition 1.2 with {{B, D}, {D, E}} share BE B. 3.2.2 Symmetries Symmetries in an FT describe components, e.g., BE or complete sub-trees, that can be swapped without changing the failure behaviour of the FT. In our setting, symmetries reduce the computational e!ort for inferring FTs as only one of the sub-trees must be constructed; other sub-tree(s) can be copied from the (original) sub-tree because of the symmetry. We define symmetries on the MCSs. Applying a symmetry on the MCSs yields the same MCSs, i.e., swapping symmetric BE does not change the structure function of the FT. Definition 5 (Symmetry on MCSs). A symmetry on the set C of all MCSs is a permutation ϑ : BEs ↑BEs which preserves C, i.e., ϑ(C) =C where ϑ(C) := {ϑ(C) | C↓C} and ϑ(C) :={ϑ(b) | b ↓C}. We denote all possible symmetries on C by ZC. Asymmetry between sets A, B⇔ BEs is a symmetry ϑ ↓ZC with ϑ(A) ⇔B and ϑ(B) ⇔A. Note that we define symmetries only on BEs and not on gates. The definition is thus more general and allows symmetries even in cases where sub-trees are not isomorphic. Lemma 1 (Necessary condition for symmetry). If ϑ ↓ ZC is a symmetry on the MCSs C, then count(b) = count(ϑ(b)) for all b ↓ BEs, where count(b) := |{C↓C| b ↓C}| denotes the number of occurrences of b in C. Example 2 (Symmetry). Consider again the FTF in Figure 3.1. The permutation ϑ1 = (AF)(BG)(CH)(DI)(EJ) is a symmetry in F (between the independent
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