Correlated Boolean operators for uncertainty logic

Miralles-Dolz, Enrique and Gray, Ander and Patelli, Edoardo and Ferson, Scott; Ciucci, Davide and Couso, Inés and Medina, Jesús and Ślęzak, Dominik and Petturiti, Davide and Bouchon-Meunier, Bernadette and Yager, Ronald R., eds. (2022) Correlated Boolean operators for uncertainty logic. In: Information Processing and Management of Uncertainty in Knowledge-Based Systems. Communications in Computer and Information Science . Springer, ITA, pp. 798-811. ISBN 9783031089718 (

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We present a correlated and gate which may be used to propagate uncertainty and dependence through Boolean functions, since any Boolean function may be expressed as a combination of and and not operations. We argue that the and gate is a bivariate copula family, which has the interpretation of constructing bivariate Bernoulli random variables following a given Pearson correlation coefficient and marginal probabilities. We show how this copula family may be used to propagate uncertainty in the form of probabilities of events, probability intervals, and probability boxes, with only partial or no knowledge of the dependency between events, expressed as an interval for the correlation coefficient. These results generalise previous results by Fréchet on the conjunction of two events with unknown dependencies. We show an application propagating uncertainty through a fault tree for a pressure tank. This paper comes with an open-source Julia library for performing uncertainty logic.