3 3.2. Modules and symmetries 63 Contributions. Our main contributions are: (i) We define modules and symmetries based on the MCSs. (ii) We present algorithms to automatically identify modules and symmetries from the MCSs. (iii) We introduce SymLearn, an approach to automatically infer FTs from failure datasets by exploiting modules and symmetries. (iv) We implemented SymLearn in Python and numerically evaluated it in several case studies. The implementation and data are available at zenodo.org/record/5571811. Outline. Section 3.2 defines modules and symmetries. Section 3.3 details the SymLearn approach. In Section 3.4, we evaluate SymLearn on truss system models and discuss the results. We conclude in Section 3.5 and present future work. 3.2 Modules and symmetries 3.2.1 Modules Instead of directly inferring an FT FCD from the MCSs CD, we aim to first partition CDinto multiple parts, infer individual FTs for each of them, and then combine the FTs into the overall FTFCD. Definition 3 (MCS partitioning). Let M1, . . . , Mn ⇔C be a partitioning of the set C of MCSs, i.e., Mi ↖Mj =↙ for all i ⇑ =j and M1 ∝· · · ∝Mn =C. For a partition Mi, we let BEsMi :=,C→Mi C denote the set of BE occurring in Mi. BE occurring in multiple partitions are called the shared BE. In the case of a large number of shared BEs, the inferred FTs—which each might be optimal individually—can yield an overall FT which is sub-optimal. For example, gates with (some of the) shared BEs as input might occur in multiple FTs. Thus, the goal is to find a partitioning such that the number of shared BEs is as small as possible. If no BE are shared, the resulting partitioning of BEs forms independent modules. In FTs, (independent) modules are independent sub-trees, where only the root node is connected to other parts of the FT (Dutuit and Rauzy, 1996). Modules can therefore be thought of as coherent entities in the context of the overall system, e.g., components. Modularisation is used to simplify the FT analysis. Definition 4 (Modules). A partitioning M1, . . .Mn of the set C of MCSs is called a module partitioning if the corresponding BEsM1, . . . , BEsMn form a partitioning of BEs. A subset Mof BEs is called an independent module if it is part of a module partitioning, i.e., all BE of Mare included in MCSs of a single Mi. An independent module Mdoes not share BE. Thus, the BEinMare not connected to other parts of the FT and they belong to an independent sub-tree.
RkJQdWJsaXNoZXIy MjY0ODMw