Model Explanation via Support Graphs

In this note, we introduce the notion of support graph to define explanations for any model of a logic program. An explanation is an acyclic support graph that, for each true atom in the model, induces a proof in terms of program rules represented by labels. A classical model may have zero, one or several explanations: when it has at least one, it is called a justified model. We prove that all stable models are justified, whereas, for disjunctive programs, some justified models may not be stable. We also provide a meta-programming encoding in Answer Set Programming that generates the explanations for a given stable model of some program. We prove that the encoding is sound and complete, that is, there is a one-to-one correspondence between each answer set of the encoding and each explanation for the original stable model.