Probabilistic Abduction in a Fuzzy Logic Framework

We study the problem of explaining observations about the probabilities of events, such as “it rains $20\mathrm{\%}$ of the time”, “rain and snow are equally likely”, etc. We explain these statements with a probability distribution or a statement about probabilities of (other) events that are consistent with our knowledge and entail the observation. We formalise this problem in a fuzzy probabilistic logic $\mathsf{F}\mathsf{P}$. We define and motivate the notions of abduction problems and their solutions. Our main technical contribution is a comprehensive study of the complexity of solution recognition and existence for a given abduction problem in $\mathsf{F}\mathsf{P}$ for the case of full language and its disjunctive-clause fragments. We also obtain a translation of classical probabilistic abduction (finding the most likely explanation of a given event) to $\mathsf{F}\mathsf{P}$.