Abstract Interpretation in Formal Argumentation: with a Galois Connection for Abstract Dialectical Frameworks and May-Must Argumentation (First Report)

Labelling-based formal argumentation relies on labelling functions that typically assign one of 3 labels to indicate either acceptance, rejection, or else undecided-to-be-either, to each argument. While a classical labelling-based approach applies globally uniform conditions as to how an argument is to be labelled, they can be determined more locally per argument. Abstract dialectical frameworks (ADF) is a well-known argumentation formalism that belongs to this category, offering a greater labelling flexibility. As the size of an argumentation increases in the numbers of arguments and argument-to-argument relations, however, it becomes increasingly more costly to check whether a labelling function satisfies those local conditions or even whether the conditions are as per the intention of those who had specified them. Some compromise is thus required for reasoning about a larger argumentation. In this context, there is a more recently proposed formalism of may-must argumentation (MMA) that enforces still local but more abstract labelling conditions. We identify how they link to each other in this work. We prove that there is a Galois connection between them, in which ADF is a concretisation of MMA and MMA is an abstraction of ADF. We explore the consequence of abstract interpretation at play in formal argumentation, demonstrating a sound reasoning about the judgement of acceptability/rejectability in ADF from within MMA. As far as we are aware, there is seldom any work that incorporates abstract interpretation into formal argumentation in the literature, and, in the stated context, this work is the first to demonstrate its use and relevance.