Reasoning about Expectation
This work addresses foundational reasoning challenges in AI and probability theory, but it is incremental as it extends existing logical frameworks to expectation.
The authors tackled the problem of reasoning about expectation across different uncertainty representations by introducing a propositional logic with sound and complete axiomatizations for probability, sets of probability measures, belief functions, and possibility measures. They found that the logic is more expressive than likelihood logic for sets of probability measures, equally expressive for the others, and that satisfiability is NP-complete.
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.