Probabilistic Abduction in a Fuzzy Logic Framework
For researchers in AI and logic, this work offers a formal treatment of abduction under probabilistic uncertainty, though it is primarily a theoretical complexity study.
The paper formalizes probabilistic abduction in a fuzzy logic framework, studying the complexity of recognizing and finding explanations for probabilistic observations. It provides a comprehensive complexity analysis for the full language and its fragments, and translates classical probabilistic abduction into this framework.
We study the problem of explaining observations about the probabilities of events, such as "it rains $20\%$ 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 $\mathsf{FP}$. 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 $\mathsf{FP}$ 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 $\mathsf{FP}$.