I.4.2 Failure dataset 37 Definition 2 (Semantics of FT). Let b be a status vector and F an FT. The structure function fF : {0,1}| BEs| ↘V ↑{0,1} returns the status of node v and is defined as: fF(b, v) := bi if Tp(v)=BE and v is the i-th BE, %v→→I(v) f(b, v↑) if Tp(v)=AND, &v→→I(v) f(b, v↑) if Tp(v)=OR. We say FTF fails for b if fF(b, Top)=1. A status vector b can also be given as the set C={bi ↓b | bi =1} of failed BE, and we write fF(C) instead of fF(b). C is as Minimal Cut Set (MCS) if fF(C)=1 and ≃C↑ ⇐C: fF(C↑)=0. For the FT in Figure I.1, the set C={Mem1, Mem2, CPU1} of failed BE leads to a failure of the overall FT: fF(C)=1. The FT has three MCS: {Mem1, Mem2}, {Power} and {CPU1, CPU2}. I.4.2 Failure dataset We assume the failure data is given in a format such that a series of data points (vector) represents the possible state of each component as well as the overall system state. Table I.1: Example failure dataset D. Mem1 Mem2 Pw CPU1 CPU2 Comp. b0 0 0 0 0 1 0 b1 0 0 0 1 1 1 b2 0 0 1 0 0 1 .. . .. . .. . .. . .. . .. . .. . Table I.1 gives an example dataset corresponding to the FT in Figure I.1. A row in the failure dataset D corresponds to a status vector bk—giving the status of each (atomic) component— together with the overall system status fD(bk). For instance, the first row (status vector b0) represents that only component CPU2 has failed, and the system “Computer” is still operational. The second row (status vector b1) represents the failure of components CPU1 and CPU2, leading to a system failure. We can also identify MCSs in the failure data D. A (minimal) cut set C of Dis a (minimal) set of BEs s.t. the corresponding status vector b yields a system failure in D. The set of all MCSs in Dis denoted by MD. We assume the dataset is coherent, i.e., a failed system stays failed for further component failures, and noise-free, i.e., the same status of components always yields the same system state. I.4.3 Inference of Fault Tree models We define the inference of FT models as the process of finding a compact FTFD that matches a given failure dataset D. In a perfectly accurate FT, assigning the
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