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In this paper we explore coloring theorems for the reals, its quotients, cardinals, and their combinations. This work is done under the scope of the axiom of determinacy. We also explore generalizations of Mycielski's theorem and show how these can be used to establish coloring theorems. To finish, we discuss the strange realm of long unions.

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Yes / Critical technological systems exhibit complex dynamic characteristics such as time-dependent behavior, functional dependencies among events, sequencing and priority of causes that may alter the effects of failure. Dynamic fault trees (DFTs) have been used in the past to model the failure logic of such systems, but the quantitative analysis of DFTs has assumed the existence of precise failure data and statistical independence among events, which are unrealistic assumptions. In this paper, we propose an improved approach to reliability analysis of dynamic systems, allowing for uncertain failure data and statistical and stochastic dependencies among events. In the proposed framework, DFTs are used for dynamic failure modeling. Quantitative evaluation of DFTs is performed by converting them into generalized stochastic Petri nets. When failure data are unavailable, expert judgment and fuzzy set theory are used to obtain reasonable estimates. The approach is demonstrated on a simplified model of a cardiac assist system. / DEIS H2020 Project under Grant 732242.

Yes / Fault tree analysis (FTA) is a powerful technique that is widely used for evaluating system safety and reliability. It can be used to assess the effects of combinations of failures on system behaviour but is unable to capture sequence dependent dynamic behaviour. A number of extensions to fault trees have been proposed to overcome this limitation. Pandora, one such extension, introduces temporal gates and temporal laws to allow dynamic analysis of temporal fault trees (TFTs). It can be easily integrated in model-based design and analysis techniques. The quantitative evaluation of failure probability in Pandora TFTs is performed using exact probabilistic data about component failures. However, exact data can often be difficult to obtain. In this paper, we propose a method that combines expert elicitation and fuzzy set theory with Pandora TFTs to enable dynamic analysis of complex systems with limited or absent exact quantitative data. This gives Pandora the ability to perform quantitative analysis under uncertainty, which increases further its potential utility in the emerging field of model-based design and dependability analysis. The method has been demonstrated by applying it to a fault tolerant fuel distribution system of a ship, and the results are compared with the results obtained by other existing techniques.

Yes / Fault tree analysis is a widely used method of risk assessment in process industries. However, the classical fault tree approach has its own limitations such as the inability to deal with uncertain failure data and to consider statistical dependence among the failure events. In this paper, we propose a comprehensive framework for the risk assessment in process industries under the conditions of uncertainty and statistical dependency of events. The proposed approach makes the use of expert knowledge and fuzzy set theory for handling the uncertainty in the failure data and employs the Bayesian network modeling for capturing dependency among the events and for a robust probabilistic reasoning in the conditions of uncertainty. The effectiveness of the approach was demonstrated by performing risk assessment in an ethylene transportation line unit in an ethylene oxide (EO) production plant.

Yes / Temporal fault trees (TFTs), an extension of classical Boolean fault trees, can model time-dependent failure behaviour of dynamic systems. The methodologies used for quantitative analysis of TFTs include algebraic solutions, Petri nets (PN), and Bayesian networks (BN). In these approaches, precise failure data of components are usually used to calculate the probability of the top event of a TFT. However, it can be problematic to obtain these precise data due to the imprecise and incomplete information about the components of a system. In this paper, we propose a framework that combines intuitionistic fuzzy set theory and expert elicitation to enable quantitative analysis of TFTs of dynamic systems with uncertain data. Experts’ opinions are taken into account to compute the failure probability of the basic events of the TFT as intuitionistic fuzzy numbers. Subsequently, for the algebraic approach, the intuitionistic fuzzy operators for the logic gates of TFT are defined to quantify the TFT. On the other hand, for the quantification of TFTs via PN and BN-based approaches, the intuitionistic fuzzy numbers are defuzzified to be used in these approaches. As a result, the framework can be used with all the currently available TFT analysis approaches. The effectiveness of the proposed framework is illustrated via application to a practical system and through a comparison of the results of each approach. / This work was supported in part by the Mobile IOT: Location Aware project (grant no. MMUE/180025) and Indoor Internet of Things (IOT) Tracking Algorithm Development based on Radio Signal Characterisation project (grant no. FRGS/1/2018/TK08/MMU/02/1). This research also received partial support from DEIS H2020 project (grant no. 732242).