Constraints and lifting-based (conditional) preferences in abstract argumentation

Dealing with controversial information is an important issue in several application contexts. Formal argumentation enables reasoning on arguments for and against a claim to decide on an outcome. Abstract Argumentation Framework (AF) has emerged as a central formalism in argument-based reasoning. In recent years there has been an increasing interest in extending AF to facilitate the knowledge representation and reasoning process. In this paper, we present an extension of AF that allows for the representation of labelled constraints and labelled preferences. A labelled argument is of the form $\mathrm{in}\left(a\right)$, $\mathrm{out}\left(a\right)$, or $\mathrm{und}\left(a\right)$, where a is an argument, whereas in, out, and und denote the acceptance status (i.e., accepted, rejected, undecided, respectively) of the specified argument. We start by considering an extension of AF with labelled constraints, namely Labelled Constrained AF (LCAF), then we focus on AF with labelled preferences (Labelled Preference-based AF, LPAF for short) and, finally, we introduce a general framework called Labelled Preference-based Constrained AF (LPCAF) that combines AF, labelled constraints, and labelled preferences. We also investigate an extension of AF with labelled conditional (or extended) preferences, namely Labelled extended Preference-based AF (LePAF), and its further combination with labelled constraints (Labelled extended Preference-based Constrained AF, LePCAF for short). Herein, conditional preferences are of the form $a>b\leftarrow$ body, where a and b are labelled arguments, whereas body is a propositional formula over labelled arguments. For each framework, we define its syntax and semantics, and investigate the computational complexity of four canonical argumentation problems: existence, verification, and credulous and skeptical acceptance, under the well-known complete, stable, semi-stable, and preferred semantics.