36 Part I: Data-driven Inference of Fault Tree models through KDD and FTs, employing causality analysis and NASA’s turbofan dataset as an example. Similarly, the methodology in Nadim, Ragab, and M. Ouali, 2023 uses interpretable machine learning and causal analysis to derive Petri Nets from event logs, with potential applications to FTs. Further advancements in the field include the method proposed in Niloofar and Lazarova-Molnar, 2023c for learning Dynamic FTs from temporal Truth Tables and time series data, and the study in Niloofar and Lazarova-Molnar, 2023a that integrates collaborative data analytics into the derivation of FTs models, showing enhanced accuracy in data-driven FT inference. I.4 Preliminaries I.4.1 Fault Trees OR Mem1 AND Computer Comp. AND M Memories Memory 1 Mem2 Memory 2 CPU1 CPU2 Central Pro. Unit 1 Central Pro. Unit 2 P Processors Pw Power Figure I.1: Example Fault Tree. A Fault Tree (FT) models how failures occur and propagate in a system, and lead to a system failure (NASA, 2002; Ruijters and Stoelinga, 2015). Formally, an FT is a directed acyclic graph where the leaves, called basic events (BE), correspond to (atomic) system components. The intermediate nodes are equipped with logical gates modelling the failure propagation. A logical AND-gate fails, if all successor nodes fail, an OR-gate fails if at least one successor fails. An FT F fails if the dedicated root node fails. Figure I.1 depicts an example FT modelling a computer. The root Computer is an OR-gate, Memory and Processor are AND-gates, circles indicate BE. Definition 1 (Fault tree). AFault Tree (FT) is a rooted directed acyclic graph (V, E) assigning the type to nodes via function Tp : V ↑{BE, AND, OR} s.t. Tp(v)=BE i" v is a leaf. The inputs of a node v, denoted I(v), are the successors of v. All nodes v ↓V must be reachable from the root Top. All nodes of type BE are denoted by BEs :={v ↓V | Tp(v)=BE}. The semantics of an FT F are given by its structure function fF. Let b = ↔b1, . . . , b|BEs|↗ ↓{ 0,1}| BEs| be a status vector where bi =1 indicates that the i-th BE has failed, and bi =0 that it functions properly, respectively.
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