Incomplete Bipolar Argumentation Frameworks

We introduce Incomplete Bipolar Argumentation Frameworks (iBAFs), the extension of Dung's Abstract Argumentation Frameworks (AAFs) allowing the simultaneous presence of supports (borrowed from BAFs - Bipolar AAFs) and of uncertain elements of the argumentation graph (borrowed from iAAFs - incomplete AAFs). We investigate the computational complexity of the acceptance problem and the verification problem (under the possible perspective), by performing an analysis of sensitivity to the semantics of supports and the semantics of extensions. On the one hand, we show that adding supports on top of incompleteness does not affect the complexity of the acceptance. On the other hand, surprisingly, we show that the joint use of bipolarity and incompleteness has a deep impact on the verification's complexity: for the Dungean semantics under which the verification over AAFs is polynomial-time solvable, although moving from AAFs to BAFs or to iAAFs does not change the complexity, the verification's complexity over iBAFs may increase up to NP-complete.