Complexity of Possible and Necessary Existence Problems in Abstract Argumentation

This work focuses on generalizing the existence problems for extensions in abstract argumentation to incomplete argumentation frameworks. In this extended model, incomplete or conflicting knowledge about the state of the arguments and attacks are allowed. We propose possible and necessary variations of the existence and nonemptiness problems, originally defined for (complete) argumentation frameworks, to extend these problems to incomplete argumentation frameworks. While the computational complexity of existence problems is already known for the standard model, we provide a full analysis of the complexity for incomplete argumentation frameworks using the most prominent semantics, namely, the conflict-free, admissible, complete, grounded, preferred, and stable semantics. We show that the complexity rises from NP-completeness to Pi(p)(2)-completeness for most "necessary" problem variants when uncertainty is allowed.