Generalized possibilistic logic: Foundations and applications to qualitative reasoning about uncertainty

Abstract

This paper introduces generalized possibilistic logic (GPL), a logic for epistemic reasoning based on possibility theory. Formulas in GPL correspond to propositional combinations of assertions such as “it is certain to degree λ that the propositional formula α is true”. As its name suggests, the logic generalizes possibilistic logic (PL), which at the syntactic level only allows conjunctions of the aforementioned type of assertions. At the semantic level, PL can only encode sets of epistemic states encompassed by a single least informed one, whereas GPL can encode any set of epistemic states. This feature makes GPL particularly suitable for reasoning about what an agent knows about the beliefs of another agent, e.g., allowing the former to draw conclusions about what the other agent does not know. We introduce an axiomatization for GPL and show its soundness and completeness w.r.t. possibilistic semantics. Subsequently, we highlight the usefulness of GPL as a powerful unifying framework for various knowledge representation formalisms. Among others, we show how comparative uncertainty and ignorance can be modelled in GPL. We also exhibit a close connection between GPL and various existing formalisms, including possibilistic logic with partially ordered formulas, a logic of conditional assertions in the style of Kraus, Lehmann and Magidor, answer set programming and a fragment of the logic of minimal belief and negation as failure. Finally, we analyse the computational complexity of reasoning in GPL, identifying decision problems at the first, second, third and fourth level of the polynomial hierarchy.