Abstract: Operations like belief change or merging have been adapted to the context of abstract argumentation. However, these operations may require to express some uncertainty or some disjunction in the result, which is not representable in classical AFs. For this reason, some of these earlier works require a set of AFs or a set of extensions as the outcome of the operation, somehow to represent a “disjunction” of AFs or extensions. In parallel, the notion of Incomplete AFs (IAFs) has been developed recently. It corresponds to AFs where the existence of some arguments or attacks may be uncertain. Each IAF can be associated with a set of classical AFs called completions, that correspond to different ways of resolving the uncertainty. While these IAFs could be good candidates for a compact representation of a disjunction of AFs, we prove that this model is not expressive enough. Then we introduce Constrained IAFs, that include a propositional formula allowing to select the set of completions used for reasoning. We prove that this model is expressive enough for representing any set of AFs, or any set of extensions. Moreover, we study the complexity of various decision problems related to the verification of extensions and the acceptability of arguments. While some of them are one level higher in the polynomial hierarchy (compared to their counterpart with standard IAFs), most of them have the same complexity than in the case of IAFs. Finally, we show that CIAFs can be used to model a new form of extension enforcement, where the possible evolutions of an AF are taken into account and modeled by the completions of the CIAF.
