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Fuzzy Fault Trees: the Fast and the Formal

Thi Kim Nhung Dang, Benedikt Peterseim, Milan Lopuhaä-Zwakenberg, Mariëlle Stoelinga

TL;DR

The paper addresses uncertainty in fault-tree analysis by introducing fuzzy fault trees, where basic-event probabilities are fuzzy numbers and the system unreliability is defined via the Zadeh extension. The core method uses alpha-cut representations and leverages fault-tree coherence to produce a fast bottom-up algorithm for tree-structured FTs and a general result extending any unreliability algorithm to fuzzy DAGs under regular fuzzy numbers. The authors prove and rely on a key DAG extension theorem and validate their approach with empirical studies using the Storm model checker. The work provides a principled, scalable framework for simultaneous reliability assessment and uncertainty quantification in safety-critical systems.

Abstract

We provide a rigorous framework for handling uncertainty in quantitative fault tree analysis based on fuzzy theory. We show that any algorithm for fault tree unreliability analysis can be adapted to this framework in a fully general and computationally efficient manner. This result crucially leverages both the alpha-cut representation of fuzzy numbers and the coherence property of fault trees. We evaluate our algorithms on an established benchmark of synthetic fault trees, demonstrating their practical effectiveness.

Fuzzy Fault Trees: the Fast and the Formal

TL;DR

The paper addresses uncertainty in fault-tree analysis by introducing fuzzy fault trees, where basic-event probabilities are fuzzy numbers and the system unreliability is defined via the Zadeh extension. The core method uses alpha-cut representations and leverages fault-tree coherence to produce a fast bottom-up algorithm for tree-structured FTs and a general result extending any unreliability algorithm to fuzzy DAGs under regular fuzzy numbers. The authors prove and rely on a key DAG extension theorem and validate their approach with empirical studies using the Storm model checker. The work provides a principled, scalable framework for simultaneous reliability assessment and uncertainty quantification in safety-critical systems.

Abstract

We provide a rigorous framework for handling uncertainty in quantitative fault tree analysis based on fuzzy theory. We show that any algorithm for fault tree unreliability analysis can be adapted to this framework in a fully general and computationally efficient manner. This result crucially leverages both the alpha-cut representation of fuzzy numbers and the coherence property of fault trees. We evaluate our algorithms on an established benchmark of synthetic fault trees, demonstrating their practical effectiveness.

Paper Structure

This paper contains 11 sections, 7 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Fault trees.
  3. Fault trees under uncertainty.
  4. Fuzzy theory.
  5. Challenges.
  6. Contributions.
  7. Fault trees
  8. Fault tree reliability analysis
  9. Fuzzy numbers
  10. Zadeh's extension principle
  11. Fuzzy fault trees

Figures (3)

  • Figure 1: Fault tree for a road trip. The road trip fails if both the phone fails and the car fails; the latter happens when either the engine or the battery fails. Its structure function equals $S_T(\,\hbox{\boldmath$b$}\:) = b_u \wedge (b_v \vee b_w)$.
  • Figure 2: "Approximately $0.3$"
  • Figure 3: Membership functions, from left to right, of a trapezoidal$\mathsf{trap}_{0.1,\, 0.4, \,0.6, \,0.8}$, triangular$\triangle_{0.2, \,0.4,\,0.8}$, interval$\mathbb{1}_{[0.3, \,0.6]}$, and Gaussian$\mathsf{gauss}_{0.4,\,0.1}$ fuzzy number.

Theorems & Definitions (5)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition: Zadeh's Extension Principle