How to Manage Supports in Incomplete Argumentation

The growing interest in generalizations of Dung's abstract argumentation frameworks has recently led to the simultaneous and independent discovery of a combination of two of these generalizations: Bipolar Argumentation Frameworks (BAFs), where a relation representing supports between arguments is added, and Incomplete Argumentation Frameworks (IAFs), where the existence of arguments and attacks may be uncertain, resulting in the so-called Incomplete Bipolar Abstract Argumentation Frameworks (IBAFs). This paper digs deeper into such a combination by: (i) providing a thoughtful analysis of the existing notions of completion (the hypothetical removal of uncertainty used in IBAFs to reason about argument acceptability); (ii) proposing, motivating and studying new notions of completion; (iii) throwing new complexity results on argument acceptability problems associated with IBAFs; (iv) encoding these reasoning problems into a lightweight version of dynamic logic.