Conflict-free and conflict-tolerant semantics for constrained argumentation frameworks

In this paper we incorporate integrity constraints in abstract argumentation frameworks. Two types of semantics are considered for these constrained frameworks: conflict-free and conflict-tolerant. The first one is a conservative extension of standard approaches for giving coherent-based semantics to argumentation frameworks, where in addition certain constraints must be satisfied. A primary consideration behind this approach is a dismissal of any contradiction between accepted arguments of the constrained frameworks. The second type of semantics preserves contradictions, which are regarded as meaningful and sometimes even critical for the conclusions. We show that this approach is particularly useful for assuring the existence of non-empty extensions and for handling contradictions among the constraints, in which cases conflict-free extensions are not available.Both types of semantics are represented by propositional sets of formulas and are evaluated in the context of three-valued and four-valued logics. Among others, we show a one-to-one correspondence between the models of these theories, the extensions, and the labelings of the underlying constrained argumentation frameworks.