Characterizing and Reasoning about Probabilistic and Non-Probabilistic Expectation

TL;DR

This paper introduces a propositional logic framework for reasoning about expectation across various uncertainty representations, providing axiomatizations, expressiveness comparisons, and complexity results.

Contribution

It presents a unified logic for expectation applicable to probability, belief, and possibility measures, with sound, complete axiomatizations and complexity analysis.

Findings

Logic is more expressive than likelihood logic for sets of probabilities.

Satisfiability is NP-complete for all cases.

Axiomatizations are provided for different uncertainty models.

Abstract

Expectation is a central notion in probability theory. The notion of expectation also makes sense for other notions of uncertainty. We introduce a propositional logic for reasoning about expectation, where the semantics depends on the underlying representation of uncertainty. We give sound and complete axiomatizations for the logic in the case that the underlying representation is (a) probability, (b) sets of probability measures, (c) belief functions, and (d) possibility measures. We show that this logic is more expressive than the corresponding logic for reasoning about likelihood in the case of sets of probability measures, but equi-expressive in the case of probability, belief, and possibility. Finally, we show that satisfiability for these logics is NP-complete, no harder than satisfiability for propositional logic.