Probabilistic Abduction in a Fuzzy Logic Framework

TL;DR

This paper introduces a fuzzy probabilistic logic framework for explaining probabilistic observations, analyzing the complexity of abduction problems, and translating classical probabilistic abduction.

Contribution

It formalizes probabilistic abduction within a fuzzy logic framework and studies the computational complexity of solution recognition and existence.

Findings

Comprehensive complexity analysis of abduction problems in $ extsf{FP}$.

Translation of classical probabilistic abduction into $ extsf{FP}$.

Formalization of explanations for probabilistic observations.

Abstract

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 $\mathsf{FP}$.