Only Knowing with Confidence Levels: Reductions and Complexity

Abstract

A new logic of belief (in the “Only knowing” family) with confidence levels is presented. The logic allows a natural distinction between explicit and implicit belief representations, where the explicit form directly expresses its models. The explicit form can be found by applying a set of equivalence preserving rewriting rules to the implicit form. The rewriting process is performed entirely within the logic, on the object level. We prove that the problem of deciding whether there exists a consistent explicit form is Σ$^{p}_{\rm 2}$-complete, a complexity class which many problems of nonmonotonic reasoning belong to.