Correlated Boolean Operators for Uncertainty Logic

TL;DR

This paper introduces a correlated nd gate for uncertainty propagation in Boolean functions, modeling dependencies with a copula family, and demonstrates its application in fault tree analysis with an open-source Julia library.

Contribution

It presents a novel correlated nd gate based on copula theory for uncertainty propagation, extending previous models to handle partial dependency knowledge.

Findings

Enables uncertainty propagation with partial dependency info

Generalizes Fre9chet's results on event conjunctions

Provides an open-source Julia library for implementation

Abstract

We present a correlated \textit{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 \textit{and} and \textit{not} operations. We argue that the \textit{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 Fr\'echet 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.