On the Relation Between Approximation Fixpoint Theory and Justification Theory

On the Relation Between Approximation Fixpoint Theory and Justification Theory

Simon Marynissen, Bart Bogaerts, Marc Denecker

Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence
Main Track. Pages 1973-1980. https://doi.org/10.24963/ijcai.2021/272

Approximation Fixpoint Theory (AFT) and Justification Theory (JT) are two frameworks to unify logical formalisms. AFT studies semantics in terms of fixpoints of lattice operators, and JT in terms of so-called justifications, which are explanations of why certain facts do or do not hold in a model. While the approaches differ, the frameworks were designed with similar goals in mind, namely to study the different semantics that arise in (mainly) non-monotonic logics. The First contribution of our current paper is to provide a formal link between the two frameworks. To be precise, we show that every justification frame induces an approximator and that this mapping from JT to AFT preserves all major semantics. The second contribution exploits this correspondence to extend JT with a novel class of semantics, namely ultimate semantics: we formally show that ultimate semantics can be obtained in JT by a syntactic transformation on the justification frame, essentially performing some sort of resolution on the rules.
Keywords:
Knowledge Representation and Reasoning: Knowledge Representation Languages
Knowledge Representation and Reasoning: Logics for Knowledge Representation
Knowledge Representation and Reasoning: Non-monotonic Reasoning